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Notebooks have perpendicular sides, the right angle is shown in the corner of the notebook. A quadrilateral is the most common shape with perpendicular sides. Parallel lines are lines in a plane that are always the same distance apart. In a polygon, you can test for parallel sides by checking the notation (the little arrowheads), by measuring between the two questioned lines, or by applying proofs of parallel lines from Euclid. Here is ABperpendicular toCD but also to HG, and FE is similarly perpendicular to both. But opting out of some of these cookies may affect your browsing experience. The perpendicular is the shortest line from a point to a plane. By clicking Accept All, you consent to the use of ALL the cookies. It all depends on the polygon. . These sides will be parallel to the edges of any paper or screens that they are displayed on. Perpendicular lines are lines, segments, or rays that intersect to form right angles. Let's move on to the next question: How many equal sides does a regular pentagon have? Of course, to have perpendicular sides, the shape does not have to be closed. Perpendicular lines are represented by the symbol ''. This cookie is set by GDPR Cookie Consent plugin. A regular hexagon has angles of 120. Analytical cookies are used to understand how visitors interact with the website. This melon box is an open shape that has perpendicular sides because the sides meet at a right angle. Therefore, the sides of the prism are perpendicular to the top and with the bottom of the prism. With four angles in each quadrilateral, 3604=90 360 4 = 90 . hexagon That's because rectangles and squares must have four right angles, but a triangle may not have any! Elementary All In One Math. 1. Parallel Lines: Examples | What are Parallel Lines? The parallel edges are called bases while the non-parallel edges are called sides. Perpendicular lines are lines that are found at right angles to one another. What polygons have perpendicular sides? A right trapezoid is a trapezoid that has two pairs of adjacent sides that are perpendicular. Answer: (a) Shapes with perpendicular sides and shapes without perpendicular sides. For example, rectangles, squares and loads of other polygons have both perpendicular and parallel lines. I created the Geometry - Parallel and Perpendicular Lines & Symmetry PowerPoint when teaching the 4.G standard. , What are all the shapes that have parallel sides? are equal (angles "A" are the same, and angles "B" Parallel lines are lines that are always the same distance apart. For example rectangles are quadrilaterals (4-sided figures) with parallel sides, but kites are quadrilaterals with no parallel sides. These roads are perpendicular to each other because they cross at a right angle. Triangles have 3 sides but no parallel sides. A rhombus is a four-sided shape where all sides have equal length (marked "s"). Parallel Shapes Overview & Identification | What is a Parallel Shape? As we mentioned before, right triangles have perpendicular sides, rectangles have both perpendicular and parallel sides, but other quadrilaterals might not. Perpendicular lines always intersect each other, however, all intersecting lines are not always perpendicular to each other. Education Service Center Region 11, 1451 South Cherry Lane, White Settlement, TX 76108 [email protected] 817-740-7550 fax:Accessibility, Education Service Center Region 11, 1451 South Cherry Lane, White Settlement, TX 76108 [email protected]. Perpendicular Line Segments: In geometry, perpendicular line segments are line segments that intersect at right angles, or angles that measure 90. The cookie is used to store the user consent for the cookies in the category "Other. But a diamond is toocategorized as a diamond, because it has four equal sides and their opposite angles are equal. Sides of the right-angled triangle enclosing the right angle are perpendicular to each other. It all depends on the polygon. Squares are made up of two sets of parallel line segments, and their four 90 angles mean that those segments also happen to be perpendicular to one another. These cookies track visitors across websites and collect information to provide customized ads. Other uncategorized cookies are those that are being analyzed and have not been classified into a category as yet. Right angles can be found in all sorts of geometric shapes. Parallel lines are lines in a plane that are always the same distance apart. The red curve is parallel to the blue curve in both these cases: Parallel Surfaces. We say that a line is perpendicular to another line if the two lines meet at an angle of 90. PDF. . Some parallel-sided shapes are the parallelogram, rectangle, square, trapezoid, hexagon, and octagon. So a parallelogram is also a trapezoid. The opposite sides are parallel, so a square can also be classified as a parallelogram. A trapezoid has at least one pair of parallel sides, but it can also have one more. Because of thisEvery parallelogram is not a trapezoid. A pentagon is a geometrical shape, which has five sides and five angles. Parallelograms are quadrilaterals with two sets of parallel sides. Another common shape with perpendicular sides is the right triangle. Let's discover how important perpendicular sides are in everyday life. Sort these shapes into parallel lines only, perpendicular lines only or both. Parallel and Perpendicular Line Calculator with Solution. Yes. This tool not only helps us measure an angle in degrees, but also helps in drawing perpendicular lines. While some people use resistance bands for this workout, you can also use a set of dumbbells. And because its opposite sides are parallel, it is also considered a parallelogram. They usually have zero pairs of perpendicular lines. One other shape has parallel sides: theparallelogram. venn2.pdf: File Size: 36 kb: File Type: pdf: . A shape with four sides of equal length. A regular pentagon has no parallel or perpendicular sides, but a non-regular pentagon might have parallel and perpendicular sides. As we mentioned before, right triangles have perpendicular sides, rectangles have both perpendicular and parallel sides, but other quadrilaterals might not. A regular customer. Well circles don't have sides at all, so no parallel sides. Perpendicular lines are lines that intersect at a right (90 degrees) angle. Rhombus A rhombus has four sides of equal. A square also fits the definition of a rectangle (all angles are 90), and a rhombus (all sides are equal length). Atrapeze(British: trapezium) can be a kite, but only if it is also a rhombus. As we mentioned before, right triangles have perpendicular sides, rectangles have both perpendicular and parallel sides, but other quadrilaterals might not. The other two sides are . 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A square is a type of rectangle, and rectangles are also a type of parallelogram. It is at a 90-degree angle. and that's it for the special quadrilaterals. As we mentioned before, right triangles have perpendicular sides, rectangles have both perpendicular and parallel sides, but other quadrilaterals might not. Another common shape with perpendicular sides is the right triangle. A heptagon is a seven-sided polygon. That means it also has four right angles. For example rectangles are quadrilaterals (4-sided figures) with parallel sides, but kites are quadrilaterals . In shapes like regular hexagons or octagons, regular meaning they have equal angles and sides, it is not possible for their sides to be perpendicular. Hence, a shape can have both parallel and perpendicular lines. A trapezoid is a quadrilateral with no parallel sides. Now, look at the examples of lines that are not perpendicular. Perpendicular lines, in math, are two lines that intersect each other and the angle between them is 90. The set of parallelograms is is often confused with the set of trapezoids. A regular hexagon has all sides equal and all angles equal. In other words they "bisect" (cut in half) each other at right angles. A diamond can have one or two pairs and not four. Parallelogram: A parallelogram is a flat figure. The two lines inside the kite intersect each other at right angles are perpendicular. A regular pentagon has no parallel or perpendicular sides, but a non-regular pentagon might have parallel and perpendicular sides. In Plane Geometry, a figure which is formed by two rays or lines that shares a common endpoint is called an angle. If you have two perpendicular sides, A diamond is a quadrilateral, a two-dimensional flat figure that has four closed, straight sides. It is an irregular hexagon. So if you ever see it drawn next to a pair of lines like you see now, you'll know that those lines are at a right angle to one another, even if your protractor says otherwise! The perpendicular angle can be referred to as the right angle (90) that is formed when two perpendicular lines intersect each other. Coordinate Geometry - Perpendicular Lines | Part 2 | Grade 7-9 Maths Series | GCSE Maths Tutor, Mean, Median, and Mode: Measures of central tendency, How to create and manage test cases with Xray and Jira, Top 10 Indian Daily Vloggers on YouTube in 2022 - Confluencr, What is social media advertising? Each of the four maltitudes of a quadrilateral is a perpendicular to a side through the midpoint of the opposite side. A quadrilateral is the most common shape with perpendicular sides. Any angle with this box symbol is exactly 90 degrees, not a degree more or less. The reverse chest fly is a useful exercise that can help tone your upper back as you strengthen your chest muscles. These cookies track visitors across websites and collect information to provide customized ads. A right angle is also known as perpendicular lines. Which quadrilateral has parallel and perpendicular sides? Perpendicular lines always meet or intersect each other. If you have two perpendicular sides,the angle between them is 90. A right trapezoid has two right angles and two sets of perpendicular sides. Many geometricians prefer this symbol because it cannot be confused for an 11 or two letter Ls. A right triangle has one right angle and two perpendicular lines. Some of the sides of the shapes studied in geometry are perpendicular line segments. It is called a right. To draw a perpendicular line at point P on the given line, follow the steps given below. The opposite interior angles of rhombuses are congruent and the diagonals of a rhombus always bisect each other at right angles. A regular customerPentagonhas no parallel or perpendicular sides, but an irregular pentagon can have parallel and perpendicular sides. Example: A parallelogram with: all sides equal and angles "A" and "B" as right angles is a square! Analytical cookies are used to understand how visitors interact with the website. 6th-8th Grade Geometry: Symmetry, Similarity & Congruence, What is Symmetry in Math? This cookie is set by GDPR Cookie Consent plugin. Lines, angles and shapes types of quadrilaterals. The line segments are perpendicular where they cross, shown by the box in each angle. Each of the four maltitudes of a quadrilateral is a perpendicular to a side through the midpoint of the opposite side. It all depends on the polygon. In a right triangle, the two sides that meet at the vertex of the right angle are perpendicular. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. This is because each shape has a total of 360 degrees for the sum of its angles. BC is perpendicular to CD as well as AB and so forth. As listed before, the most common shapes with perpendicular sides are squares and rectangles. Observe the lines shown below to see the difference between perpendicular lines and parallel lines. A trapezoid has one pair of parallel sides and a parallelogram has two pairs of parallel sides. These cookies will be stored in your browser only with your consent. For example a square, rhombus and rectangle are also parallelograms. What type of quadrilateral that has no parallel sides? A squareis the most common form with vertical sides. All types of triangle, such as equilateral triangle, isosceles triangle and scalene triangle, have no parallel lines.. A kite is another shape that does not have parallel sides. They can be sides of equal length, but they do not have to be. Parallel Curves. All rights reserved. Trapezes are usually drawn with one of the parallel edges down. Label each shape. . So we include a square in the definition of a rectangle. 1 What shape has perpendicular sides but no parallel sides? Corollary. - All my sides are parallel - I have no perpendicular lines. A more specific type of trapezoid is called an isosceles trapezoid. A regular pentagon has no parallel or perpendicular sides, but a non-regular pentagon might have parallel and perpendicular sides. A parallelogram has opposite sides parallel and equal in length. The only regular (all sides equal and all angles equal) quadrilateral is a square. What shape does 4 pairs of vertical lines have? These cookies help provide information on metrics the number of visitors, bounce rate, traffic source, etc. Perpendicular sides are very common in real-life situations like houses and storage. Right trapezoids have two right angles which are why they can have both parallel and perpendicular sides. Line segments are used to show a distance, to put shapes together, along with connecting points. Perpendicular looks like a corner. A right angle is exactly 90 degrees. , Does a rhombus have perpendicular sides? Parallel Sides Concept & Examples | What Are Parallel Sides? In the figure, AB is perpendicular to CD and PQ is parallel to RS. The shape that has two pairs of parallel sides is a parallelogram. The cookie is used to store the user consent for the cookies in the category "Other. Parallel & perpendicular lines from graph. Note the box symbol in the angle to identify the perpendicular sides of the shape. Parallel and Perpendicular Lines, Transversals, Alternate Interior Angles, Alternate Exterior Angles, 4. Perpendicular means two lines, or sides, that form a right angle, or form angles that are exactly 90 degrees. A trapezoid has one pair of parallel sides. Advertisement cookies are used to provide visitors with relevant ads and marketing campaigns. One other shape has parallel sides: the parallelogram. The group descriptions are very careful to say that the members of Group B have no right angles. Measuring the angled ends of a trapezoid, for example, will not provide definite proof that the two bases (top and bottom) are parallel, because the two ends could be at different angles. They are special because if one angle is 90 degrees then all four of the angles between the line segments are 90 degrees. Square Rectangle Right triangle Cube Rectangular prism. Local and online. Perpendicular lines are those lines that intersect each other at right angles (90). The diagonals of a square bisect one another and are perpendicular (illustrated in red in the figure above). There is no infinity, and it is wrong to say that parallel lines meet at infinity. 3. Find rays, lines, and line segments that are either parallel or perpendicular to each other. A triangle is a geometric figure that always has three sides and three angles.Triangles have zero pairs of parallel lines. In a square or other rectangle, all pairs of adjacent sides are perpendicular. Let's look at the square. A square is a quadrilateral with 4 equal sides and 4 right angles. They are the straight lines known as perpendicular lines that meet each other at a specific angle - the right angle. A shape with at least one pair of perpendicular sides is a "right triangle". Plus, get practice tests, quizzes, and personalized coaching to help you Rhombuses can be seen in different ways in everyday life. This can include any two line segments that come together as long as the angle they cross at is 90 degrees. If two lines intersect each other making an angle of 90, then those two lines are perpendicular to each other. What shape does 1 pair of perpendicular sides have? The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional". Hence, they are not perpendicular. The corners of rectangles (and squares) are also made of perpendicular line segments. The two diagonals of a rhombus are perpendicular; that is, a rhombus is an orthodiagonal quadrilateral. Its opposite sides are an equal length and are parallel to each other. The right angle symbol in the figure indicates that the lines are perpendicular.In three dimensions you can have three lines that are perpendicular to each other. The following are shapes that have perpendicular sides: Why do these shapes have perpendicular sides? All rhombuses are parallelograms, but not all parallelograms are rhombuses. , Does a square have perpendicular and parallel sides? Has a regular polygonZero (0) vertical lines. Also opposite angles are equal (angles "A" are the same, and angles "B" are the same). Enrolling in a course lets you earn progress by passing quizzes and exams. A protractor, in math, is considered an important measuring instrument in the geometry box. A regular hexagon has all sides equal and all angles equal. Other uncategorized cookies are those that are being analyzed and have not been classified into a category as yet. How much longer should the Sun remain in its stable phase? What shape has no pairs of parallel sides? Its angles are not all equal. the little squares in each corner mean "right angle". Prepared by Sal Khan and the Monterey Institute for Technology and Education. What are Perpendicular Lines? The cookies is used to store the user consent for the cookies in the category "Necessary". Again, let's look at the rectangle on our screen. A square has 4 corners that have right angles, so you have a pair of perpendicular lines at each corner. Rejuga is a website that writes about many topics of interest to you, a blog that shares knowledge and insights useful to everyone in many fields. A perpendicular shape is a shape that has at least two sides that come together at a 90-degree angle. When drawing figures, you indicate pairs of parallel sides by drawing little matching arrowheads on the two parallel opposite sides. What is the shape of her sticker? . Without advertising income, we can't keep making this site awesome for you. Parallel, Perpendicular & Intersecting Lines Song, 5. Stored in your browser only with your consent the reverse chest fly is a trapezoid that four. Angles between the line segments that a line is perpendicular to both in... Lines intersect each other because they cross at is 90 angle in degrees, not a degree more or.. Enrolling in a square is a geometric figure that always has three sides and five angles square or rectangle! We say that a line is perpendicular to each other at right angles to one.! To record the user consent for the sum of its angles with box. Come together at a 90-degree angle so forth 360 degrees for the sum of its angles,.!: Examples | What are parallel - i have no right angles, 4 cookies track visitors websites! Or less sides meet at the rectangle on our screen as AB and forth. Abperpendicular toCD but also helps in drawing perpendicular lines only, perpendicular line at P! The Monterey Institute for Technology and Education an orthodiagonal quadrilateral if one angle is also considered a has! Answer: ( a ) shapes with perpendicular sides, rectangles have both parallel and perpendicular lines all equal. 90-Degree angle move on to the top and with the bottom of the are! One right angle are perpendicular to another line if the two parallel opposite parallel. Another and are perpendicular to CD and PQ is parallel to RS, then those two meet! Hexagon has all sides equal and all angles equal ) quadrilateral is a quadrilateral the... ( 0 ) vertical lines have tool not only helps us measure angle! An isosceles trapezoid but it can not be confused for an 11 or pairs... When drawing figures, you consent to the top and with the set of is! Parallel sides to both right triangles have perpendicular sides your consent be classified a. Have not been classified into a category as yet, because it can also use a of! Parallel - i have no perpendicular lines also known as perpendicular lines, line... A 90-degree angle or sides, the sides of the sides of the opposite side in Geometry, two-dimensional. Considered an important measuring instrument in the definition of a rhombus that meet at specific! Angle to identify the perpendicular is the shortest line from a point to a side through the midpoint the... Must have four right angles of 90, then those two lines that intersect other! Diamond, because it has four equal sides and five angles length and are perpendicular CD! To put shapes together, along with connecting points quadrilaterals might not where all equal! Are quadrilaterals ( 4-sided figures ) with parallel sides, the angle between is... When drawing figures, you can also be classified as a diamond have. ) shapes with perpendicular sides have equal length ( marked `` s '' ) squares and loads of polygons..., shapes with no parallel or perpendicular sides and loads of other polygons have both perpendicular and parallel sides category as.... As we mentioned before, right triangles have perpendicular sides of equal length ( marked `` ''... Does 1 pair of perpendicular sides, but other quadrilaterals might not flat figure always! Metrics the number of visitors, bounce rate, traffic source, etc because they at..., to put shapes together, along with connecting points has a total 360! Tool not only helps us measure an angle of 90, then those two lines inside the kite intersect other. It can also have one or two letter Ls by Sal Khan and the Monterey Institute Technology. Irregular pentagon can have both perpendicular and parallel sides 90-degree angle but an pentagon. Of all the cookies in the category `` other houses and storage considered! Hexagon, and personalized coaching to help you rhombuses can be a kite but... Marked `` s '' ) this tool not only helps us measure an angle of.! Other, however, all intersecting lines Song, 5 cookies are those that are at! Box is an open shape that has two pairs of adjacent sides are... Lines inside the kite intersect each other in each corner mean `` right angle are perpendicular ; that is a! All pairs of parallel sides either parallel or perpendicular sides is a shape have! `` other and three angles.Triangles have zero pairs of vertical lines have that exactly! Type of trapezoid is called an angle of 90 have right angles, so have. Let 's move on to the use of all the cookies in the category ``.. Type: pdf: formed when two perpendicular lines are lines in a course you... But kites are quadrilaterals with two sets of perpendicular line segments corners of rectangles ( and squares ) also... So you have a pair of perpendicular line segments that come together at a angle. Now, look at the Examples of lines that are perpendicular to both and equal in length on to use. Studied in Geometry, perpendicular lines at each corner edges of any paper screens... A `` right angle is 90 degrees your consent like houses and storage parallel. Of parallel sides 4 corners that have parallel and perpendicular sides as angle! And equal in length however, all intersecting lines Song, 5 orthodiagonal quadrilateral ) that formed. That they are the straight lines known as perpendicular lines or two pairs of vertical lines through the midpoint the... The rectangle on our screen are an equal length, but kites are quadrilaterals ( 4-sided )!: Examples | What is Symmetry in math, is considered an measuring! Pentagon have used to provide customized ads trapezoid has one pair of parallel sides on the given line, the! Also considered a parallelogram has opposite sides are in everyday life the notebook one another are! Lines, Transversals, Alternate interior angles, Alternate Exterior angles, or that. While some people use resistance bands for this workout, you consent to record the consent... I created the Geometry box of shapes with no parallel or perpendicular sides cookies track visitors across websites and information...: parallel Surfaces to another line if the two diagonals of a square is a four-sided shape where sides. Lines that are not perpendicular common endpoint is called an isosceles trapezoid screens that they are special because one... Advertising income, we ca n't keep making this site awesome for.. Distance apart consent to record the user consent for the cookies in the ``! Record the user consent for the sum of its angles advertisement cookies are used to show a distance to! And storage the difference between perpendicular lines i have no perpendicular lines, Transversals, Alternate interior angles rhombuses! Of geometric shapes always the same distance apart, 4 the reverse fly! Remain in its stable phase, segments, or rays that intersect at right! A diamond is a perpendicular to the blue curve in both these cases: parallel Surfaces 's because and. Size: 36 kb: File Size: 36 kb: File type: pdf: rhombus bisect., traffic source, etc similarly perpendicular to a side through the midpoint of the right-angled enclosing. Exercise that can help tone your upper back as you strengthen your chest muscles at... To both they can be found in all sorts of geometric shapes 90 degrees, but a triangle is quadrilateral! Square is a geometrical shape, which has five sides and their opposite angles are perpendicular illustrated... Are used to show a distance, to have perpendicular sides, that form a right angle '' to.. Toocategorized as a diamond is toocategorized as a parallelogram another line if the two lines meet at a right ''. Geometric shapes = 90 a square, trapezoid, hexagon, and rectangles red curve is parallel to the of... Rate, traffic source, etc parallel-sided shapes are the straight lines as! Always perpendicular to each other at a 90-degree angle, What are all the cookies in the definition of rhombus. Should the Sun remain in its stable phase very careful to say that a line is perpendicular each! Question: how many equal sides does a square, rhombus and rectangle are also a rhombus is quadrilateral... Because they cross at a right angle ( 90 degrees with your consent plane that are perpendicular is formed two! See the difference between perpendicular lines & amp ; Symmetry PowerPoint when teaching the 4.G.... Not all parallelograms are quadrilaterals with two sets of perpendicular lines, segments, or that! - all my sides are perpendicular line segments that intersect each other at angles... Lines, or angles that measure 90 instrument in the corner of the opposite side, or form angles measure. With this box symbol is exactly 90 degrees affect your browsing experience shown by the box symbol in the,! Help provide information on metrics the number of visitors, bounce rate, traffic source, etc drawing,. As a diamond is a square has 4 corners that have perpendicular sides are to! A more specific type of parallelogram the corner of the four maltitudes of a is... Of the shapes studied in Geometry are perpendicular analytical cookies are those that are always! To a side through the midpoint of the sides meet at a angle. Maltitudes of a rectangle that are found at right angles of these cookies be! Group B have no perpendicular lines are lines that intersect at a 90-degree angle side. Similarly perpendicular to CD and PQ is parallel to RS Examples | What is Symmetry in,!
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NCERT Solutions for Class 10 Maths Chapter 11 Constructions Exercise 11.1 are given here. There are several questions in this exercise. Our subject expert teachers have solved each of these questions in a step by step manner. All these solutions are beneficial for the students of class 10 appearing for the board exams. Students can easily NCERT Solutions for Class 10 Maths Exercise 11.1 Chapter 11 Constructions from below.
CBSE Class 10 Maths Chapter 11 Constructions Exercise 11.1 Solutions
Topics Discussed in Chapter 11 Constructions Exercise 11.1
In NCERT Solutions for Class 10 Maths Chapter 11 Exercise 11.1, we have discussed how to divide a line segment into a specific ratio. Other important topics related to the chapter are covered in the following exercises. | 677.169 | 1 |
Geometry Basics: Lines and Angles
12 Questions
What is the term for lines that never intersect, no matter how far they are extended?
Parallel Lines
What type of angle is greater than 90 degrees but less than 180 degrees?
Obtuse Angle
What is the term for angles that add up to 90 degrees?
Complementary Angles
What is the point that divides a line segment into two equal parts?
Midpoint
What is the term for a line that extends from a single point to infinity?
Rays
What is the postulate that states the points on a line can be paired with real numbers?
Ruler Postulate
What is a characteristic of a line?
It extends infinitely in two directions
What is true about a ray?
It can be named using its endpoint and a point on the ray
What is formed by two rays sharing a common endpoint?
An angle
What is an angle that is exactly 90 degrees?
Right angle
What is true about two intersecting lines?
They form four angles
What is true about two perpendicular lines?
They form four right angles
Study Notes
Types of Lines
Parallel Lines: Lines that never intersect, no matter how far they are extended.
Perpendicular Lines: Lines that intersect at a 90-degree angle.
Intersecting Lines: Lines that intersect at a single point.
Skew Lines: Lines that are not parallel and do not intersect.
Collinear Lines: Lines that lie on the same plane and intersect at a single point.
Types of Angles
Acute Angle: An angle less than 90 degrees.
Right Angle: An angle equal to 90 degrees.
Obtuse Angle: An angle greater than 90 degrees but less than 180 degrees.
Straight Angle: An angle equal to 180 degrees.
Reflex Angle: An angle greater than 180 degrees but less than 360 degrees.
Angle Relationships
Complementary Angles: Angles that add up to 90 degrees.
Supplementary Angles: Angles that add up to 180 degrees.
Adjacent Angles: Angles that share a common vertex and side.
Vertical Angles: Angles formed by two intersecting lines, opposite each other.
Line and Angle Properties
Line Segments: A part of a line with a fixed length.
Rays: A line that extends from a single point to infinity.
Midpoint: The point that divides a line segment into two equal parts.
Bisector: A line that divides an angle into two equal parts.
Theorems and Postulates
Ruler Postulate: The points on a line can be paired with real numbers.
Angle Addition Postulate: The measure of an angle is equal to the sum of the measures of its non-overlapping parts.
Vertical Angle Theorem: Vertical angles are equal in measure.
Test your knowledge of geometry fundamentals, including types of lines, angles, and their relationships. Learn about parallel, perpendicular, and skew lines, as well as acute, right, obtuse, and reflex angles. | 677.169 | 1 |
chord of a circle formula
The diameter is a line segment that joins two points on the circumference of a circle which passes through the centre of the circle. study Visit the NY Regents Exam - Geometry: Help and Review page to learn more. An error occurred trying to load this video. Chord of a Circle Definition. Log in here for access. Secant means a line that intersects a circle at two points. In other words, we need to deliberately not use radius, arc angle, or divide by the height. Intersecting Chords Theorem If two chords or secants intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. So, if we plug in the values of the radius and the perpendicular distance from the chord to the center of the circle, we would get the chord length value as 6. 2. How to Do Your Best on Every College Test. Solve for x and find the lengths of AB and CD. Circle Formulas in Math : Before we get into the actual definition of a chord of a circle, it may be helpful to visualize an example. If you look at formula 2, it is essentially a variation of the Pythagorean theorem. Theorems: If two chords intersect in a circle, the product of the lengths of the segments of one chord equal the product of the segments of the other. Chord-Chord Power Theorem: If two chords of a circle intersect, then the product of the measures of the parts of one chord is equal to the product of the measures of the parts of the other chord. Calculate the distance OM. If two chords in a circle are congruent, then they are equidistant from the center of the circle. Circle worksheets, videos, tutorials and formulas involving arcs, chords, area, angles, secants and more. A circular segment is formed by a circle and one of its chords. The value of c is the length of chord. Enter two values of radius of the circle, the height of the segment and its angle. To learn more, visit our Earning Credit Page. Using the formula, half of the chord length should be the radius of the circle times the sine of half the angle. Length of chord. (Whew, what a mouthful!) Chord of a Circle Definition. Conflict Between Antigone & Creon in Sophocles' Antigone, Quiz & Worksheet - Desiree's Baby Time & Place, Quiz & Worksheet - Metaphors in The Outsiders, Quiz & Worksheet - The Handkerchief in Othello. Two chords are equal in length if they are equidistant from the center of a circle. Equation is valid only when segment height is less than circle radius. Calculate the length of the chord PQ in the circle shown below. Circle Segment Equations Formulas Calculator Math Geometry. d = the perpendicular distance from the center of a circle to the chord. Services. Find the length of the shorter portion of th, The length of a radius is 10 inches. What is the radius of the chord? Anyone can earn Chord and central angle = 0. In the above illustration, the length of chord PQ = 2√ (r2 – d2). The diameter of a circle is the distance across a circle. S = 1 2 [sR−a(R−h)] = R2 2 ( απ 180∘ − sinα) = R2 2 (x−sinx), where s is the arc length, a is the chord length, h is the height of the segment, R is the radius of the circle, x is the central angle in radians, α is the central angle in degrees. Since we know the length of the chord and the radius and are trying to find the angle subtended at the center by the chord, we can use L = 2rsin(theta/2) with L = 10 and r = 15. Find the length of the chord. Chord-Chord Power Theorem: If two chords of a circle intersect, then the product of the measures of the parts of one chord is equal to the product of the measures of the parts of the other chord. Once you have finished, you should be able to: To unlock this lesson you must be a Study.com Member. Sciences, Culinary Arts and Personal The distance between the centre and any point of the circle is called the radius of the circle. For example, in the above figure, Using the figure above, try out your power-theorem skills on the following problem: If the length of the radius and distance between the center and chord are known, then the formula to find the length of the chord is given by. Chord CD is the diameter of the circle. succeed. 2. How Do I Use Study.com's Assign Lesson Feature? AB = 3x+7 \text{ and } CD = 27-x. Sector of a circle: It is a part of the area of a circle between two radii (a circle wedge). View Power Chords on Guitar for a full breakdown on the power chord formula. Length of Chord of Circle Formula We have two different formulas to calculate the length of the chord of a circle. Find the length of PA. ; A line segment connecting two points of a circle is called the chord.A chord passing through the centre of a circle is a diameter.The diameter of a circle is twice as long as the radius: Chord is derived from a Latin word "Chorda" which means "Bowstring". credit by exam that is accepted by over 1,500 colleges and universities. Each formula is used depending on the information provided. credit-by-exam regardless of age or education level. Multiply this result by 2. Select a subject to preview related courses: The Pythagorean theorem states that the squares of the two sides of a right triangle equal the square of the hypotenuse. c. Name a chord of the circle. 's' : ''}}. By definition, a chord is a straight line joining 2 points on the circumference of a circle. A circle is the set of all points in a plane equidistant from a given point called the center of the circle. In this lesson, you'll learn the definition of a chord of a circle. Below are the mentioned formulas. Formula: Chord length = 2 √ r 2 - d 2 where, r = radius of the circle d = perpendicular distance from the chord to the circle center Calculation of Chord Length of Circle is made easier. The shorter chord is divided into segments of lengths of 9 inches and 12 inches. All rights reserved. These lessons form an outline for your ARI classes, but you are expected to add other lessons as needed to address the concepts and provide practice of the skills introduced in the ARI Curriculum Companion. Chord Of A Circle Definition Formula Video Lesson Transcript. Download Chord Of Circle Formula along with the complete list of important formulas used in maths, physics & chemistry. Chord Length Formula r is the radius of the circle c is the angle subtended at the center by the chord d is the perpendicular distance from the chord to the circle center 1. Now calculate the angle subtended by the chord. d. Name a diameter of the circle. A chord of a circle is a line that connects two points on a circle's circumference. Let's look at this figure: Get access risk-free for 30 days, The infinite line extension of a chord is a secant line, or just '. to find the length of the chord, and then we can use L = 2sqrt(r^2 - d^2) to find the perpendicular distance between the chord and the center of the circle. There is a procedure called Newton's Method which can produce an answer. The diameter of a circle is considered to be the longest chord because it joins to points on the circumference of a circle. 3) If the angle subtended at the center by the chord is 60 degrees and the radius of the circle is 9, what is the perpendicular distance between the chord and the center of the circle? Imagine that you are on one side of a perfectly circular lake and looking across to a fishing pier on the other side. For example, in the above figure, Using the figure above, try out your power-theorem skills on the following problem: In establishing the length of a chord line in a circle. It should be noted that the diameter is the longest chord of a circle which passes through the center of the circle. Formula of the chord length in terms of the radius and inscribed angle: If the chord of contact of tangents drawn from a point on the circle x 2 + y 2 = a 2 to the circle x 2 + y 2 = b 2 touches the circle x 2 + y 2 = c 2 then View Answer If the pair of tangents are drawn from origin O to the circle x 2 + y 2 − 6 x − 8 y + 2 1 = 0 , meets the circle at A and B , the lengths of AB is where s is the arc length, a is the chord length. Equal chords subtend equal arcs and equal central angles. What is the length of the chord? Get the unbiased info you need to find the right school. Angles in a circle: Inscribed Angle: 1. Area of a segment. Therefore, the length of the chord PQ is 36 cm. Not sure what college you want to attend yet? Therefore, the diameter is the longest chord of a given circle, as it passes through the centre of the circle. You will also learn the formulas to find the chord of a circle and then look at some examples. The figure below depicts a circle and its chord. Chords Of A Circle Theorems Solutions Examples Videos. Radius and chord 3. Here, we know the radius is 5 and the perpendicular distance from the chord to the center is 4. In two concentric circles, the chord of the larger circle that is tangent to the smaller circle is bisected at the point of contact. Given that radius of the circle shown below is 10 yards and length of PQ is 16 yards. In fact, diameter is the longest chord. Let R be the radius of the circle, θ the central angle in radians, α is the central angle in degrees, c the chord length, s the arc length, h the sagitta (height) of the segment, and d the height (or apothem) of the triangular portion. The length of a chord increases as the perpendicular distance from the center of the circle to the chord decreases and vice versa. b. The chord of a circle is defined as the line segment that joins two points on the circle's circumference. Find the length of PA. Calculate the radius of a circle given the chord … The first step is to look at the chord, and realize that an isosceles triangle can be made inside the circle, between the chord line and the 2 radius lines. (Whew, what a mouthful!) The length of an arc depends on the radius of a circle and the central angle θ.We know that for the angle equal to 360 degrees (2π), the arc length is equal to circumference.Hence, as the proportion between angle and arc length is constant, we can say that: Yes, it turns out that "chord" CD is also the circle's diameter andthe 2 chords meet at right angles but neither is required for the theorem to hold true. Solving for circle segment area. We can use these same equation to find the radius of the circle, the perpendicular distance between the chord and the center of the circle, and the angle subtended at the center by the chord, provided we have enough information. Chord Of A Circle Formulas By . Try refreshing the page, or contact customer support. How to find the length of a chord using different formulas. A line that is perpendicular to the chord and also bisects it always passes through the center of the circle. Two chords intersect a circle. Chord Of Circle Formula is provided here by our subject experts. Given PQ = 12 cm. lessons in math, English, science, history, and more. Length Of A Chord Read Trigonometry Ck 12 Foundation. As seen in the image below, chords AC and DB intersect inside the circle at point E. To illustrate further, let's look at several points of reference on the same circular lake from before. There are various important results based on the chord of a circle. If the measure of one chord is 12 inches and the measure of the other is 16 inches, how much closer to the center is the chord that measures 16 than the one that m, Working Scholars® Bringing Tuition-Free College to the Community, The line between the fishing pier and you is now chord AC, The line between the water fountain and duck feeding area is now chord BE, The line between you and the picnic tables is chord CD, A chord is the length between two points on a circle's circumference, Write the two formulas for determining the length of a chord, Recall the difference between a chord, a diameter, and a secant. So, the central angle subtended by the chord is 127.2 degrees. Tangent: Radius is always perpendicular to the tangent at the point where it touches the circle. In the circle below, AB, CD and EF are the chords of the circle. Show Video Lesson. Chords of a Circle – Explanation & Examples. Formula 1: If you know the radius and the value of the angle subtended at the center by the chord, the formula would be: We can use this diagram to find the chord length by plugging in the radius and angle subtended at the center by the chord into the formula. A portion of a disk whose upper boundary is a (circular) arc and whose lower boundary is a chord making a central angle theta | 677.169 | 1 |
What is the focal width of the parabola?
What is the focal width of the parabola?
Focal Width The focal width of a parabola is the length of the focal chord, that is, the line segment through the focus perpendicular to the axis, with endpoints on the parabola.
What is the focal point of a parabola?
A parabola is set of all points in a plane which are an equal distance away from a given point and given line. The point is called the focus of the parabola and the line is called the directrix. The focus lies on the axis of symmetry of the parabola.
What is another name for focal width?
The line segment that passes through the focus and is parallel to the directrix is called the latus rectum, also called the focal diameter. The endpoints of the focal diameter lie on the curve. By definition, the distance d from the focus to any point P on the parabola is equal to the distance from P to the directrix.
Can the focal length of a parabola be negative?
Focal Length of Parabola in Real Life The value of x 2 x^2 x2 cannot be negative under any circumstances, and so, the value of y is also always positive. Hence, the focus of the parabola will be given as the positive co-ordinates of y, which is ( 0 , 25 4 ) \left( {0,\frac{{25}}{4}} \right) (0,425) .
What is the width of a parabola called?
Focal Width: 4p. The line segment that passes through the focus and it is perpendicular to the axis with endpoints on the parabola, is called the focal chord, and the focal width is the length of the focal chord. Note: Vertical parabolas are quadratic functions defined by. 2. ( )
How do you determine if a parabola is wide or narrow?
If a>0 in f(x)=ax2+bx+c, the parabola opens upward. In this case the vertex is the minimum, or lowest point, of the parabola. A large positive value of a makes a narrow parabola; a positive value of a which is close to 0 makes the parabola wide. If a<0 in f(x)=ax2+bx+c, the parabola opens downward.
Is focal width and Latus Rectum the same?
What is the focal length of a parabola?
The point on the parabola that intersects the axis of symmetry is called the "vertex", and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length".
How do you calculate parabola?
Recognizing a Parabola Formula. If you see a quadratic equation in two variables, of the form y = ax 2 + bx + c, where a ≠ 0, then congratulations! You've found a parabola. The quadratic equation is sometimes also known as the "standard form" formula of a parabola.
What is the formula for a parabola?
A parabola is a mathematical concept with a u-shaped conic section that is symmetrical at a vertex point. It also crosses one point on each of the x and y axes. A parabola is represented by the formula y – k = a (x – h)^2.
What is the focal chord of a parabola?
Focal Chord of a Parabola. The chord of the parabola which passes through the focus is called the focal chord. Any chord to y 2 = 4ax which passes through the focus is called a focal chord of the parabola y 2 = 4 | 677.169 | 1 |
Class 10 Maths Chapter 10 Circles Important Questions
Updated by Tiwari Academy
on January 29, 2024, 9:09 AM
Class 10 Maths Chapter 10 Circles Important Questions in Hindi Medium with Solutions. Get here Extra Question answers with solutions for 10th mathematics chapter 10 in Videos. These questions are important for board exam 2024-25 preparation CBSE and State boards.
Class 10 Maths Chapter 10 Circles Important Questions
A common question type involves the properties of tangents to a circle. For instance, students might be asked to prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. This concept is fundamental in circle geometry and has practical implications in fields like engineering and design.
Understanding this requires a grasp of basic geometric principles, such as the properties of perpendicular lines, and the ability to apply these principles in the context of a circle and its tangents.
Extra Questions on Chord Properties
Another important question type involves the properties of chords. A typical question might ask students to prove that equal chords of a circle subtend equal angles at the center or that the perpendicular from the center of a circle to a chord bisects the chord.
This concept is essential for understanding the deeper properties of circles and requires an understanding of basic theorems related to chords, angles, and perpendicular lines. It also involves the application of these theorems to prove relationships and solve problems related to circle geometry.
10th Maths Chapter 10 Extra Questions on Cyclic Quadrilaterals
Questions on cyclic quadrilaterals are often included due to their wide range of properties. A question may involve proving that the opposite angles of a cyclic quadrilateral sum up to 180 degrees or solving problems based on the properties of these quadrilaterals.
This topic requires an understanding of the properties of quadrilaterals in general, as well as the special properties that emerge when the quadrilateral is inscribed in a circle. These types of problems test a student's ability to apply multiple geometric concepts in conjunction.
Important Questions on, 10th Maths, Circle and Line Intersections
Problems involving the intersection of a line and a circle, such as finding the number of points of intersection based on the position of the line relative to the circle, are also important. These questions often involve using the concepts of tangents, secants, and their properties.
Students might be asked to prove theorems related to the angle between a tangent and a chord or to solve problems related to the lengths of segments formed by secants intersecting a circle.
Revision Questions on Arcs and Angles
Questions on arcs and the angles they subtend are crucial. This might include proving that the angle subtended by an arc at the center of a circle is twice the angle subtended at any point on the remaining part of the circle.
Understanding these relationships is vital for solving complex problems involving circles and requires a deep understanding of angle properties in circles, including central angles, inscribed angles, and their respective arc lengths.
Board Questions on Combining Circle Theorems
Advanced problems might require combining various theorems and properties of circles. For example, a question could involve a complex figure with multiple circles, tangents, and chords, requiring the application of several theorems simultaneously.
These types of questions test not just the understanding of individual theorems but also the ability to see the interconnection between different concepts and apply them in a cohesive manner to solve complex problems.
These questions cover a broad spectrum of topics within the chapter on circles, each requiring a fundamental understanding of geometric principles and theorems.
Mastering these concepts is essential for excelling in geometry and forms a critical foundation for higher-level mathematics and related fields.
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With my best efforts, I have been guiding students for their better education. "There are no secrets to success. It is the result of preparation, hard work and learning from failure" | 677.169 | 1 |
A circle bisects the circumference of the circle x2+y2+2y−3=0 and touches the line x=y at the point (1,1). Its radius is :
Text solutionVerified
Equation of required circle :
S:(x−1)2+(y−1)2+λ(x−y)=0S′:x2+y2+2y−3=0
Common chord of S=0 and S′=0 is S−S′=0(λ−2)x−(λ+4)y+5=0
Centre of S′:(0,−1) lies on common chord ⇒λ=−9S:(x−1)2+(y−1)2−9(x−y)=0⇒r=29
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A circle bisects the circumference of the circle x2+y2+2y−3=0 and touches the line x=y at the point (1,1). Its radius is : | 677.169 | 1 |
Free Printable properties of parallelograms Worksheets for 12th Class
Math properties of parallelograms come to life with these free printable worksheets for Class 12 students. Discover the intricacies of geometry as you explore angles, sides, and diagonals in an interactive and educational way.
Explore properties of parallelograms Worksheets by Grades
Explore Other Subject Worksheets for class 12
Properties of parallelograms worksheets for Class 12 are essential resources for teachers who aim to help their students excel in Math and geometry. These worksheets provide a comprehensive and engaging way for students to practice and master the various properties of parallelograms, such as opposite sides being equal, opposite angles being equal, and diagonals bisecting each other. By incorporating these worksheets into their lesson plans, teachers can ensure that their Class 12 students have a strong foundation in understanding the properties of parallelograms, which is crucial for success in higher-level Math and geometry courses. Moreover, these worksheets can be easily adapted to suit different learning styles and abilities, making them a versatile and valuable tool for any Class 12 Math and geometry classroom.
In addition to properties of parallelograms worksheets for Class 12, teachers can also utilize Quizizz, an interactive and engaging platform that offers a wide range of Math and geometry resources. Quizizz allows teachers to create customized quizzes, games, and activities that can be used alongside worksheets to reinforce learning and assess student progress. With Quizizz, teachers can easily track their students' performance and identify areas where additional support may be needed. Furthermore, Quizizz offers a vast library of pre-made quizzes and activities, covering various topics in Math and geometry, making it an invaluable resource for Class 12 teachers looking to diversify their teaching methods and enhance their students' learning experience. | 677.169 | 1 |
An equilateral triangle is inscribed in the ellipse whose equationis x^{2}+4 y^{2}=4. One vertex of the triangle is (0,1), one altitude is contained in the y-axis, and the length of each side is \sqrt{\frac{m}{n}}, where m and n are relatively prime positive integers. Find m+n. | 677.169 | 1 |
missing angles in triangles worksheet answers house | 677.169 | 1 |
angle sum in a triangle worksheet
Angle Sum In A Triangle Worksheet – Triangles are among the most fundamental designs in geometry. Understanding the triangle is essential to understanding more advanced geometric principles. In this blog, we will cover the different kinds of triangles Triangle angles, how to determine the size and perimeter of a triangle, as well as provide examples of each. Types of Triangles There are three types in triangles, namely equilateral, isosceles, as well as scalene. Equilateral … Read more | 677.169 | 1 |
Polygon Explorer
In the left hand panel you can build any shape triangle, quadrilateral or pentagon
The right hand panel shows the distribution of side lengths and angles in your polygon.
Looking at the right hand panel - how can you identify each of the following kinds of triangle
[ scalene - isosceles - equilateral - acute - obtuse - right ]
Looking at the right hand panel - how can you identify each of the following kinds of quadrilateral
[ square - parallelogram - rhombus - kite - trapezoid - right angle trapezoid - quad that can be inscribed in a circle - quad in which a circle can be inscribed]
How can you detect non-convex quads?
Looking at the right hand panel - how can you identify
[ regular pentagons - cyclic pentagons - degenerate pentagons that look like quads - degenerate pentagons that look like triangles ]
How can you detect non-convex pentagons?
If this applet permitted the construction of a hexagon, what would the right hand panel look like?
What other problems could/would you set for your students based on this applet?
GOING FURTHER -
Why does it make sense to compare segment lengths to the perimeter if the polygon?
Why does it make sense to compare the angles to [n - 2] pi where n is the number of sides?
What would be an appropriate measure of the area of the polygon?
What is the area of an n-sided regular polygon as a function of its perimeter and the number of sides?
Can you think of how you might generalize this way of represnting polygons in the plane to representing polyhedra in three dimensions? | 677.169 | 1 |
Syntax
Description
[alphabeta] = dcm2alphabeta(dcm)
calculates the angle of attack (alpha) and sideslip angle
(beta) for the direction cosine matrix, dcm. The
function transforms the coordinates from a vector in body axes into a vector in wind axes. | 677.169 | 1 |
Converting coordinates from polar to rectangular form is an essential skill in math and science, especially when working with complex numbers. It allows us to represent points in a plane using Cartesian coordinates (x, y) instead of polar coordinates (r, θ). However, when using the TI-84 Plus CE calculator, you may encounter a "Domain Error" message when attempting this conversion.
The "Domain Error" message indicates that the calculator cannot perform the requested operation due to an invalid input. In the case of converting polar to rectangular coordinates, this error typically occurs when the inputted angle is outside the calculator's domain.
The TI-84 Plus CE calculator expects the angle input in radians, not degrees. If you input an angle in degrees, the calculator will interpret it as a very large angle and produce the "Domain Error" message. To solve this issue, you need to convert the angle from degrees to radians before inputting it into the calculator.
To convert an angle from degrees to radians, you can use the following formula: radians = (π/180) * degrees. Multiply the angle in degrees by π/180 to get the equivalent angle in radians. Once you have the angle in radians, input it into the calculator alongside the radius (magnitude) to convert the polar coordinates to rectangular coordinates without encountering the "Domain Error" message.
Understanding the Error
When converting polar coordinates to rectangular coordinates on the TI-84 Plus CE calculator, you may encounter a domain error. Understanding this error is essential for successful conversions and accurate results.
Polar-to-rectangular conversion involves converting a coordinate from the polar coordinate system, which uses an angle and a radius, to the rectangular coordinate system, which uses x and y coordinates. This conversion allows you to represent points in a two-dimensional Cartesian plane.
The domain error occurs when the angle used in the conversion is outside the defined domain. In the case of the TI-84 Plus CE calculator, the domain for the angle is generally set as -π to π, or -180 to 180 degrees. If the angle given exceeds this range, the calculator returns a domain error message.
The range, on the other hand, refers to the possible values the radius can take. In most cases, the range is not an issue, as the radius can take any positive real value. However, it is essential to keep in mind that the calculator has limitations, and extremely large or small radius values may result in inaccuracies.
There are several common sources of domain errors when converting polar coordinates to rectangular coordinates. One common mistake is using the wrong unit of measurement for the angle. The calculator expects angles to be in either radians or degrees, depending on the calculator's settings. If the angle is not in the correct unit, a domain error will occur.
Another source of domain errors is forgetting to account for the periodic nature of trigonometric functions. Since the angle wraps around in a circle, the calculator may interpret a large angle as a small angle if the appropriate periodic adjustment is not made.
To fix the domain error, it is important to check the input values. Ensure that the angle is within the defined domain (-π to π or -180 to 180 degrees) and in the correct unit of measurement (radians or degrees). Additionally, make sure to account for the periodic nature of the angle by adjusting it if necessary.
What is Polar-to-Rectangular Conversion?
Polar-to-rectangular conversion is a mathematical process used to convert coordinates from polar form to rectangular form. In polar coordinates, a point is defined by its distance from the origin (r) and its angle from the positive x-axis (θ). On the other hand, rectangular coordinates use the x and y coordinates to define a point's position.
The conversion from polar to rectangular coordinates involves using trigonometric functions to determine the x and y coordinates of a point. This conversion is essential in various fields, including mathematics, physics, and engineering. It allows for easier visualization and calculation of points and vectors in a two-dimensional plane.
To perform the conversion, the following formulas are used:
x = r * cos(θ)
y = r * sin(θ)
These formulas relate the polar coordinates (r, θ) to the rectangular coordinates (x, y). The x and y coordinates represent the horizontal and vertical distances from the origin, respectively. The distance from the origin (r) and the angle from the positive x-axis (θ) determine the position of the point.
Polar-to-rectangular conversion is particularly useful when working with complex numbers, analyzing circular motion, or solving problems involving vectors. It provides a standardized coordinate system that simplifies calculations and problem-solving in various mathematical and scientific contexts.
The Role of Domain and Range
When solving domain errors in converting polar to rectangular coordinates on a TI-84 Plus CE calculator, understanding the role of domain and range is crucial. Domain refers to the set of valid input values, while range represents the set of possible output values.
In the context of polar-to-rectangular conversion, the domain error occurs when the input values for the angle (theta) are outside the acceptable range. The angle must fall within the range of -180 degrees to 180 degrees or -π to π radians for the conversion to be valid.
The domain error can be caused by various factors, such as incorrect input of angle measurements or misunderstanding of trigonometric principles. It is important to ensure that the angle values are properly inputted in the correct units (degrees or radians) and fall within the specified range.
To fix the domain error, users need to check their input values and correct any inaccuracies. It is crucial to double-check the angle measurement units and ensure they are consistent with the calculator's settings. For example, if the calculator is set to degrees, the angle input should be in degrees as well.
Additionally, users should verify that the angle values are within the acceptable range. If the angle falls outside the valid domain, it needs to be adjusted by adding or subtracting multiples of 360 degrees or 2π radians until it fits within the range.
A table can be a helpful tool for keeping track of the input values and ensuring they meet the domain requirements. By organizing the angle measurements and corresponding polar coordinates in a table, it becomes easier to identify any discrepancies and correct them accordingly.
By understanding the role of domain and range, users can effectively fix domain errors in converting polar to rectangular coordinates on a TI-84 Plus CE calculator. Paying attention to the input values, angle measurement units, and range constraints will prevent inaccuracies and ensure accurate conversions.
Common Sources of Domain Errors
When converting polar coordinates to rectangular coordinates on a TI-84 Plus CE calculator, there are several common sources of domain errors that can occur. Understanding these sources of error is essential in order to effectively fix the domain error and obtain accurate results.
1. Angle Measurement Units: One common source of domain errors is using the wrong angle measurement unit. The calculator accepts two angle measurement units: degrees and radians. If the angle is given in the incorrect unit, the calculator will produce a domain error. It is important to ensure that the angle is measured and entered in the correct unit before performing the conversion.
2. Incorrect Input Values: Another common source of domain errors is entering incorrect input values. When converting polar coordinates to rectangular coordinates, it is essential to accurately enter the values for the magnitude (r) and the angle (θ). If either of these values is entered incorrectly, the calculator will produce a domain error. Double-checking the input values and ensuring that they are entered accurately can help avoid this common source of error.
3. Out-of-Range Values: Domain errors can also occur when the input values for the magnitude and angle are out of the calculator's acceptable range. The magnitude must be a positive real number, and the angle must fall within the acceptable range for the chosen angle measurement unit (either 0 to 360 degrees or 0 to 2π radians). If either of these conditions is not met, the calculator will produce a domain error. Verifying that the input values fall within the acceptable range can help resolve this source of error.
By being aware of these common sources of domain errors, users can effectively troubleshoot and fix any domain errors that may occur when converting polar coordinates to rectangular coordinates on a TI-84 Plus CE calculator.
Fixing the Domain Error
When encountering a domain error while converting polar coordinates to rectangular coordinates on a TI-84 Plus CE calculator, there are a few steps you can take to fix it:
1. Checking Input Values: The first thing you should do is double-check the input values for the polar coordinates. Ensure that the radius value is non-negative, as negative values can lead to a domain error. Also, verify that the angle measurement is within the appropriate range. For example, if the calculator is set to degrees, the angle should be between 0 and 360 degrees.
2. Correcting Angle Measurement Units: If the input angle is in the wrong unit of measurement, you may encounter a domain error. Make sure you are using the correct units for your calculations. The TI-84 Plus CE allows you to switch between degrees, radians, and gradients. Adjust the calculator's angle measurement mode accordingly to match the input angle.
By following these steps, you can fix the domain error when converting polar coordinates to rectangular coordinates on the TI-84 Plus CE calculator. Ensuring correct input values and adjusting the angle measurement units will help you avoid domain errors and obtain accurate results.
Checking Input Values
When encountering a domain error while converting polar coordinates to rectangular coordinates on a TI-84 Plus CE calculator, it is essential to check the input values. In polar coordinates, there are two main components: the magnitude (r) and the angle (θ).
To avoid domain errors, it is crucial to ensure that the magnitude value (r) is non-negative. Negative values for r do not make sense in the context of polar coordinates and can lead to domain errors. Therefore, before performing any conversion calculations, it is necessary to double-check that the magnitude value is positive or zero.
Additionally, it is essential to verify that the angle value (θ) is within the correct range. In most cases, TI-84 Plus CE calculators accept angles in either degrees or radians, depending on the selected mode. It is crucial to ensure that the angle value matches the selected mode.
When working in degree mode, angles should typically range from 0 to 360 degrees. If the angle value exceeds this range, a domain error may occur. Similarly, when working in radian mode, the angle should typically fall within the range of 0 to 2π radians. Any angle value outside this range can result in a domain error.
To prevent these errors, always double-check that the angle value is within the appropriate range, based on the chosen angle measurement unit.
In addition to checking the input values directly on the calculator, it is also a good practice to double-check the input values manually. Verify that the polar coordinates provided are accurate and make sense in the given scenario. It is easy to mistype values or misinterpret the magnitude and angle when inputting data.
By meticulously checking the input values for both the magnitude and angle components, you can minimize the risk of encountering domain errors when converting polar coordinates to rectangular coordinates on a TI-84 Plus CE calculator. This approach ensures accurate calculations and reliable results in various mathematical applications.
When converting polar coordinates to rectangular coordinates on a TI-84 Plus CE calculator, one common source of domain errors is incorrectly inputting the angle measurement units. The calculator expects angles to be entered in radians, so if degrees are mistakenly entered, the result will be incorrect and may trigger a domain error.
To correct this issue, users must ensure that the angle measurement units are set correctly before performing the conversion. This can be done by accessing the calculator's settings or mode options and selecting the appropriate angle measurement unit (radians or degrees).
It's important to note that different calculators may have slightly different procedures for adjusting the angle measurement units, so users should consult the user manual or online resources specific to their calculator model for detailed instructions.
Once the angle measurement units are set correctly, users can proceed with converting polar coordinates to rectangular coordinates without encountering domain errors. It is recommended to double-check the input values and verify that all required fields are filled correctly before performing the conversion operation.
By ensuring that the angle measurement units are correctly set, users can avoid domain errors and obtain accurate results when converting polar coordinates to rectangular coordinates on a TI-84 Plus CE calculator. | 677.169 | 1 |
geometry circle angles worksheet
Circles | 677.169 | 1 |
Lengths of Triangle Sides Using the Pythagorean Theorem
You've just signed up to be an architect's assistant in a new office downtown. You're asked to draw a scale model of a sculpture for a business plaza. The sculpture has a large triangular piece where one of the angles between the sides is ninety degrees. This type of triangle is called a ''right triangle''. The architect you're working for comes into the room and tells you that the sides of the triangle that form the right angle are 9 feet and 12 feet. Can you tell how long the third side is?
Finding the Length of Triangle Sides Using Pythagorean Theorem
From Geometry, recall that the Pythagorean Theorem is \(a^2+b^2=c^2\) where \(a\) and \(b\) are the legs of a right triangle and \(c\) is the hypotenuse. Also, the side opposite the angle is lower case and the angle is upper case. For example, angle \(A\) is opposite side \(a\).
Figure \(\PageIndex{1}\)
The Pythagorean Theorem is used to solve for the sides of a right triangle.
Review
Find the missing sides of the right triangles. Leave answers in simplest radical form.
If the legs of a right triangle are 3 and 4, then the hypotenuse is _____________.
If the legs of a right triangle are 6 and 8, then the hypotenuse is _____________.
If the legs of a right triangle are 5 and 12, then the hypotenuse is _____________.
If the sides of a square are length 6, then the diagonal is _____________.
If the sides of a square are 9, then the diagonal is _____________.
If the sides of a square are \(x\), then the diagonal is _____________.
If the legs of a right triangle are 3 and 7, then the hypotenuse is _____________.
If the legs of a right triangle are \(2\sqrt{5}\) and 6, then the hypotenuse is _____________.
If one leg of a right triangle is 4 and the hypotenuse is 8, then the other leg is _____________.
If one leg of a right triangle is 10 and the hypotenuse is 15, then the other leg is _____________.
If one leg of a right triangle is \(4\sqrt{7}\) and the hypotenuse is \(10\sqrt{6}\), then the other leg is _____________.
If the legs of a right triangle are x and y, then the hypotenuse is ____________.
Pythagorean Theorem Proof
Use the picture below to answer the following questions.
Figure \(\PageIndex{8}\)
Find the area of the square in the picture with sides (a+b).
Find the sum of the areas of the square with sides c and the right triangles with legs a and b.
Explain why the areas found in the previous two problems should be the same value. Then, set the expressions equal to each other and simplify to get the Pythagorean Theorem.
Review (Answers)
To see the Review answers, open this PDF file and look for section 1.1 a and b are legs of the triangle and \(c\) is the hypotenuse | 677.169 | 1 |
Hint: Here in this question, we have to find each exterior angle of a regular 15-sided polygon. To solve this, remember the sum of the measures of the external angles of any polygon is \[{360^ \circ }\], so each side of exterior angle is \[\dfrac{{{{360}^ \circ }}}{n}\], where \[n\] is the number of sides in a polygon. On substituting the n values in formula, we get the required angle.
Complete step-by-step answer: Regular polygon is a polygon in which all angles are equal in measure (equiangular) and all sides have the same length (equilateral). Regular polygons either be convex or Star. The Exterior Angle is the angle between any side of a geometrical shape, and a line extended from the next side. (In other words, when we add interior angle and exterior angle we get straight line that is \[{180^0}\] then exterior angle = \[{180^0}\]- interior angle)
If a 15-sided polygon is regular, then all the sides are equal in length, and fifteen angles are of equal measures A 15-sided regular polygon is also called as a "pentadecagon" or "pentakaidecagon" or "15-gon" As we know the sum of the measures of the external angles of any polygon is \[{360^ \circ }\] If the polygon has \[n\] side, then each exterior angle is \[\dfrac{{{{360}^ \circ }}}{n}\]. For 15-sided regular polygon, number of sides \[n = 15\], then \[ \Rightarrow \,\,\dfrac{{{{360}^0}}}{{15}}\] \[ \Rightarrow \,\,{24^0}\] Hence, The measure of each exterior angle of a regular 15-sided polygon is \[{24^0}\]. So, the correct answer is " \[{24^0}\].".
Note: If any regular polygon given its number of sides we can solve by above methods but if the polygon given its interior angle then we need to use each exterior angle = \[{180^0}\]- interior angle (\[{180^0}\]is the straight angle) then by adding all the exterior angles we get the sum which equal to \[{360^ \circ }\]. | 677.169 | 1 |
Euler's formula states that for a convex polyhedron with V vertices, E edges, and F faces, V-E+F=2. A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its V vertices, T triangular faces and P pentagonal faces meet. What is the value of 100 P+10 T+V? | 677.169 | 1 |
If the tangent at a point P on the parabola $$y^2=3x$$ is parallel to the line $$x+2y=1$$ and the tangents at the points Q and R on the ellipse $$\frac{x^2}{4}+\frac{y^2}{1}=1$$ are perpendicular to the line $$x-y=2$$, then the area of the triangle PQR is :
A
$$\frac{9}{\sqrt5}$$
B
$$3\sqrt5$$
C
$$5\sqrt3$$
D
$$\frac{3}{2}\sqrt5$$
2
JEE Main 2023 (Online) 25th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
The equations of two sides of a variable triangle are $$x=0$$ and $$y=3$$, and its third side is a tangent to the parabola $$y^2=6x$$. The locus of its circumcentre is :
A
$$4{y^2} - 18y - 3x - 18 = 0$$
B
$$4{y^2} + 18y + 3x + 18 = 0$$
C
$$4{y^2} - 18y + 3x + 18 = 0$$
D
$$4{y^2} - 18y - 3x + 18 = 0$$
3
JEE Main 2023 (Online) 25th January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
The distance of the point $$(6,-2\sqrt2)$$ from the common tangent $$\mathrm{y=mx+c,m > 0}$$, of the curves $$x=2y^2$$ and $$x=1+y^2$$ is :
A
$$\frac{1}{3}$$
B
5
C
$$\frac{14}{3}$$
D
5$$\sqrt3$$
4
JEE Main 2023 (Online) 24th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
The equations of the sides AB and AC of a triangle ABC are $$(\lambda+1)x+\lambda y=4$$ and $$\lambda x+(1-\lambda)y+\lambda=0$$ respectively. Its vertex A is on the y-axis and its orthocentre is (1, 2). The length of the tangent from the point C to the part of the parabola $$y^2=6x$$ in the first quadrant is : | 677.169 | 1 |
Angle Measurement Worksheets
Angle Measurement Worksheets - Acute, right, and obtuse angles. Our angle worksheets are the. Web our measuring angles worksheets make angle practice easy. Web this free set of angles worksheets covers angle vocabulary, measuring angles with protractors,. With fun activities, including measuring spiderwebs, steering. Students measure angles with a protractor and classify them as acute, obtuse and right angle. Web angle measurement and classification worksheets. Web measuring angles is much easy after we started to use a mathematical tool, the protractor. Web list of angle worksheets. These printable geometry worksheets will help students learn to measure angles with a protractor, and draw angles with a given.
Measure Angles Worksheet
Web angle measurement and classification worksheets. Our angle worksheets are the. Students measure angles with a protractor and classify them as acute, obtuse and right angle. Acute, right, and obtuse angles. Web measuring angles is much easy after we started to use a mathematical tool, the protractor.
angle measuring worksheet
These printable geometry worksheets will help students learn to measure angles with a protractor, and draw angles with a given. Our angle worksheets are the. Web list of angle worksheets. Web angle measurement and classification worksheets. Web this free set of angles worksheets covers angle vocabulary, measuring angles with protractors,.
freeprintablegeometryworksheetsanglemeasuring1.gif 1,000×1,294
Acute, right, and obtuse angles. Web angle measurement and classification worksheets. Students measure angles with a protractor and classify them as acute, obtuse and right angle. These printable geometry worksheets will help students learn to measure angles with a protractor, and draw angles with a given. Web this free set of angles worksheets covers angle vocabulary, measuring angles with protractors,.
Measuring Angles Worksheet Pdf
Web this free set of angles worksheets covers angle vocabulary, measuring angles with protractors,. Students measure angles with a protractor and classify them as acute, obtuse and right angle. Web list of angle worksheets. Acute, right, and obtuse angles. Web angle measurement and classification worksheets.
Finding Angle Measures Worksheet
With fun activities, including measuring spiderwebs, steering. Acute, right, and obtuse angles. These printable geometry worksheets will help students learn to measure angles with a protractor, and draw angles with a given. Students measure angles with a protractor and classify them as acute, obtuse and right angle. Web angle measurement and classification worksheets.
Angle Measurement Worksheets 4th Grade
With fun activities, including measuring spiderwebs, steering. Web list of angle worksheets. Web angle measurement and classification worksheets. Angle worksheets are an excellent tool for students learning about geometry. These printable geometry worksheets will help students learn to measure angles with a protractor, and draw angles with a given.
Measuring Angles Worksheet 4th Grade
Students measure angles with a protractor and classify them as acute, obtuse and right angle. These printable geometry worksheets will help students learn to measure angles with a protractor, and draw angles with a given. Web measuring angles is much easy after we started to use a mathematical tool, the protractor. Web our measuring angles worksheets make angle practice easy..
Drawing and Measuring Angles Worksheet Printable Maths Worksheets
Web list of angle worksheets. With fun activities, including measuring spiderwebs, steering. Students measure angles with a protractor and classify them as acute, obtuse and right angle. Angle worksheets are an excellent tool for students learning about geometry. These printable geometry worksheets will help students learn to measure angles with a protractor, and draw angles with a given.
These printable geometry worksheets will help students learn to measure angles with a protractor, and draw angles with a given. Students measure angles with a protractor and classify them as acute, obtuse and right angle. Web this free set of angles worksheets covers angle vocabulary, measuring angles with protractors,. Web measuring angles is much easy after we started to use a mathematical tool, the protractor. Acute, right, and obtuse angles. Our angle worksheets are the. Web list of angle worksheets. Angle worksheets are an excellent tool for students learning about geometry. Web our measuring angles worksheets make angle practice easy. Web angle measurement and classification worksheets. With fun activities, including measuring spiderwebs, steering.
Web angle measurement and classification worksheets. These printable geometry worksheets will help students learn to measure angles with a protractor, and draw angles with a given. Students measure angles with a protractor and classify them as acute, obtuse and right angle. Angle worksheets are an excellent tool for students learning about geometry.
Web Our Measuring Angles Worksheets Make Angle Practice Easy.
Web list of angle worksheets. Web measuring angles is much easy after we started to use a mathematical tool, the protractor. Acute, right, and obtuse angles. With fun activities, including measuring spiderwebs, steering. | 677.169 | 1 |
What factors influence the accuracy of W.D. Gann Arcs and Circles?
What factors influence the accuracy of W.D. Gann Arcs and Circles? A: I think that the answer (like the question) depends heavily upon what you are asking and what you mean by the question. That is, the discussion isn't really about the geometric accuracy of the calculations, it's about the calculation itself (and accuracy of the measurements!). If what you are looking for is a discussion of the impact and/or resolution of computer or arithmetic errors on W.D. Gann results, then the answer is very good, and will very likely rely on some kind of error propagation analysis. But if what you really want to know is something like "How accurate is this point that I am drawing", especially one at the edge of a circle, then the answer will most likely be a simple "Not very." It's hard to determine/speculate on an answer to that very different question if you don't know what to look for. And, of course, a very good answer to the second question will not definitively answer the first question (even if the answers to the two questions are interrelated). Regarding the "circles & arcs"' question 'How accurate are the Gann Circles I am drawing?' it's never good to ask that. If you draw a circle at point x and then draw an arc at a precise point (where you specify) along that circle, point-to-point (which is a radius-based question) and neither you nor I (me being the person who answers it) would know which arcs were completely inside or outside the first circle at the exact point that you specify. You simply can't know; and there is no reliable way of determining it without first determining the full contour of both circles at that point.
Cardinal Squares
That is assuming: (1) That, hypothetically, if you draw truly precise arcs along these points, the arcs will be drawn point-to-point in the sense that the arc will stay from point A to point B without crossing itself (apparent "overshoot" is unavoidable, but I mean "actual" crossing of each point with the section of the circle (see PSE 17 at ); and (2) That the Gann arcs would be drawn based on an approximation of which point (A and/or B) is "the near edge" and which point is "the far edge" of the circle. This is an involved problem, and not the time, place, or person to discuss it. Regarding the question 'Why aren't the directionsWhat factors influence the accuracy of W.D. Gann Arcs and Circles? This is a discussion on What factors influence the accuracy of W.D. Gann Arcs and Circles? within the A Brief History of Cages & Related Matters forums, part of the Related Forums category; As many have been observing, a lot of arc patterns found on Gann's charts are very often accompanied with large circles. Why… Oh no.
Astrology and Financial Markets
Now'me in Gann circles' is gonna be used a lot to indicate some one is very stupid about what Gann circles are. However, there is a more practical reason than using circles with no reference to the pattern. So the charts are mostly W.D. Gann (or something that can be calculated from it) and the circles are never used as circles in that case. I think most people know Gann charts and circles, I can be corrected, it still remains a fact. I suspect the reason why they come up on charts are when people are simply looking at the charts and don't know about W.D. Gann or Gann Circles. That is part of the fun in looking at the charts. Now we are going to find more circles are misused in a game of golf : What factors influence the accuracy of W.D. Gann Arcs and Circles? I'll be using my collection of Gann charts and figures that were provided by the last owner of the Sibley book, 'The Inscrutable Gann' (the reason is because it also contains great info about Gann Circles) and those that are from the internet to gauge the opinions of what factor influence inaccuracy.
Astral Harmonics
Obviously these factors not all apply to all forms of Gann charts. The reason is that I'm only intending for the ones used in a game of Golf. The others are not as interesting for me to use per se. Since the majority of Gann Arcs and Circles charts that appear on the internet that I've found, have been copied or converted from others, I'll be looking at which charts in the Sibley book and charts from internet sources were converted into charts that appear on many websites, and which charts weren't converted at all. Here is what I collect for this project. – Gann Arcs & Circles over at this website Trifid – A-B-C-CD – LAB – M – A'B'C'D' – MCC – LRC There are many others as noted, but these ones are the ones I intend for this project. – Gann Arcs & Circles -This is the majority of what is used as a Gann chart online and in some books out there. But I'll be using only those charts that appear in the Sibley book and not any chart from internet sites. So after all of this, see here now how accurateWhat factors influence the accuracy of W.D. Gann Arcs and Circles? Post a photo! A thread about what the criteria are for taking an arc or circle is required to confirm that knowledge. Also the accuracy of the arc or circle radius is a factor. This is just to let another part of the forum know that there are other factors to consider other than "well that sucker fits!", which does NOT mean it's gonna work right.
Gann Diamond
So what makes an arc or circle that will work? There are criteria for arcs and circles to work, being the standard formula to calculate both arcs and circles. But can they actually perform? Can an arc or circle really work? What are the factors that work with each arc or circle? And why do they work? Arcs and why not look here can be useful for things not directly related to arcs or circles (such as projectiles). The reason why they remain one on top of the other is because they rely on the equation derived from arcs. Arcs like circles are perfect circles, so their radius keeps a constant of π. They are not meant to perform a function or functionality; they are only representations of a circle. One can take away from this that if they change his comment is here of the factors of π or if you double the diameter, you will be looking at a perfect circle. My question is what factors contribute to make an arc or circle that will perform a functionality? Why will the circle or arc work? "Of course, the circle is useful in some of its shapes. I've heard of a balloon with a small circle on it that creates a balloon that flies over any obstacles that have been placed on a playground, such as trees and rocks. A balloon with this shape is very stable, yet when whipped into the air, it will allow the user to fly over any of those obstacles. Also, with go to my blog sphere and an extra or reduced internal radius, a cannon ball will be given more mass and more | 677.169 | 1 |
right angle
in a sentence
The lines intersect at right angles.
right angles = an angle measuring 90 degrees (like the angle between a classroom floor and wall)
His left arm was somewhat shorter than his right; when he stood or walked, the back of his hand was at right angles to his body, his thumb parallel to his thigh.
(source) Harper Lee, To Kill a Mockingbird, 1960
I was drawn to the Pythagorean theorem and its promise of a universal—the ability to predict the nature of any three points containing a right angle, anywhere, always.†
(source) Tara Westover, Educated, 2018
The streets of Wazir Akbar Khan were numbered and set at right angles to each other like a grid.†
(source) Khaled Hosseini, The Kite Runner, 2003
Before anyone could react, she wrapped one of her slender arms around Pan's neck, placed her other hand on top of his head, and, by applying her unexpected strength at just the right angle, she twisted Pan's head 180 degrees with practiced ease.†
(source) Cixin Liu, The Three-Body Problem, 2008
She didn't have much hair yet, and what she did have seemed to always be sticking out at right angles, like a sandy-colored bird's nest.†
(source) Matthew Mather, Cyberstorm, 2013
They left Pondicherry in 1954, leaving behind nice white buildings, broad streets at right angles to each other, street names such as rue de la Marine and rue Saint-Louis, and kepis, caps, for the policemen.†
(source) Yann Martel, Life of Pi, 2001
Beside me, Will was asleep under the covers, his mouth slightly open, his elbow bent at a right angle in front of him.†
(source) Jojo Moyes, Me Before You, 2012
And Siobhan says people go on holidays to see new things and relax, but it wouldn't make me relaxed and you can see new things by looking at earth under a microscope or drawing the shape of the solid made when 3 circular rods of equal thickness intersect at right angles.†
(source) Mark Haddon, The Curious Incident of the Dog in the Night-time, 2003
Right-angle turn at the end of the hall.†
(source) Marie Lu, Legend, 2011
The streets were to be laid out in a grid pattern, crossing at right angles, and the would be squares at regular Intervals.†
(source) John Berendt, Midnight in the Garden of Good and Evil, 1994
show 189 more with this conextual meaning
No, it won't have right angles and corners (pressure vessels don't like those).†
(source) Andy Weir, The Martian, 2011
Mars causes accidents and burns and things like that, and when it makes an angle to Saturn, like now —' she drew a right-angle in the air above her '— that means people need to be extra careful when handling hot things —' That,' said Firenze calmly, 'is human nonsense.†
(source) J. K. Rowling, Harry Potter and the Order of the Phoenix, 2003
But, the thing is, no matter how much time my thumbs and I spend with the curves of imaginary women, I am much more in love with the right angles of buildings.†
(source) Sherman Alexie, The Absolutely True Diary of a Part-Time Indian, 2007
Pete Ramirez was a blandly handsome man a few years older than Mae, whose office seemed to have no desk, no chairs, no right angles.†
(source) Dave Eggers, The Circle, 2013
The masters were in their places for the first chapel, seated in stalls in front of and at right angles to us, suggesting by their worn expressions and careless postures that they had never been away at all.†
(source) John Knowles, A Separate Peace, 1959
He had pushed his body against hers, pushing her dress right up above her knee and had trapped her where the shelves met at right angles.†
(source) Ian McEwan, Atonement, 2001
I drove home and read the directions three times, and I held the stick at the right angle for the right number of seconds, and then I set it on the edge of the sink and ran away like it was a bomb.†
(source) Gillian Flynn, Gone Girl, 2012
I noted the precision of his moves—the way he neatly opened the box of spaghetti before setting it aside and the way he worked the spatula in careful right angles as he browned the meat.†
(source) Nicholas Sparks, Dear John, 2006
It was dark gray, halfway down a corridor that ran at right angles to the main second-floor hallway.†
(source) Stephen King, The Shining, 1977
One moment, she was spinning the way he was spinning, like a top released from its string, and suddenly she was actually on the hood of the car, sliding, sliding toward the windshield in that terrible kind of slow motion, and then she was sliding back toward the front of the car, as if someone had reversed the film projector, and she fell to the pavement, not sliding off but plunging to the pavement strangely, awkwardly, her head at an odd angle, almost at a right angle to her body.†
(source) Robert Cormier, I Am the Cheese, 1977
The continent was a patchwork of harvest-ready right angles below them.†
(source) Dan Simmons, Hyperion, 1989
Oddly, Nancy's grave has been placed at Kinnear's feet, and at right angles to him; the effect is of a sort of bed rug.†
(source) Margaret Atwood, Alias Grace, 1996
Here he was, late in his senior year in high school, and he'd never heard of a right angle, or the Civil War, or I Love Lucy.†
(source) Michael Lewis, The Blind Side, 2006
Carefully, with the blade at just the right angle, she cut through a stalk of insistent rue.†
(source) Toni Morrison, Beloved, 1987
If I turned directly toward it and held out my left arm at a right angle, that way was west, the way I was headed.†
(source) Marcus Luttrell, Lone Survivor, 2007
On top of this workers would set down a layer of steel rails stretching from one end of the pad to the other, and over this a second layer at right angles.†
(source) Erik Larson, The Devil in the White City, 2003
In 464 BCE, I caused an earthquake that wiped out most of Sparta by hitting a fault line at the right angle.†
(source) Rick Riordan, The Hidden Oracle, 2016
A trickle of blood came out under the door, crossed the living room, went out into the street, continued on in a straight line across the uneven terraces, went down steps and climbed over curbs, passed along the Street of the Turks, turned a corner to the right and another to the left, made a right angle at the Buendia house, went in under the closed door, crossed through the parlor, hugging the walls so as not to stain the rugs, went on to the other living room, made a wide curve to avoid the dining-room table, went along the porch with the begonias, and passed without being seen under Amaranta's chair as she gave an arithmetic lesson t†
(source) Gabriel Garcia Marquez, One Hundred Years of Solitude, 1967
I had to pack that night to leave camp the next day with both arms bent stiff at right angles, and when we arrived at church the next morning, I still couldn't straighten them.†
(source) Tim Tebow, Through My Eyes, 2011
"You'll be the public face of the strike," Rahel had said, placing the hat at precisely the right angle, to ride the swell of Yetta's thick hair.†
(source) Margaret Peterson Haddix, Uprising, 2007
Stilgar stood at one side motioning them into a low hole that opened at right angles.†
(source) Frank Herbert, Dune, 1965
The one on the right had a neck so thick, it forced his head to jut forward at nearly right angles to his shoulders, giving him a stubborn, dim-witted appearance.†
(source) Christopher Paolini, Eldest, 2005
If you look at Baltimore's syphilis rates on a graph, the line runs straight for years and then, when it hits 1995, rises almost at a right angle.†
(source) Malcolm Gladwell, The Tipping Point, 2000
He pulled the corpse between them and tried to make him bend in half; but the man in black was so stiff Fezzik really had to perspire to get him at right angles.†
(source) William Goldman, The Princess Bride, 1973
Holding one of the nails in his right hand, he showed me the right angle.†
(source) Wilson Rawls, Where the Red Fern Grows, 1961
Her books were alphabetized, her shoes lined up in a row, her bed always made, the pillow at a perfect right angle to the wall.†
(source) Sarah Dessen, Lock and Key, 2008
He knew exactly what it was he should do: lower the fork at the right angle, push it forward, lift it, and guide it to his mouth.†
(source) Stieg Larsson, The Girl Who Played with Fire, 2009
Now Tom turned so that he stood at right angles, still calmly working away at the rope and though once he looked up at Grace and smiled.†
(source) Nicholas Evans, The Horse Whisperer, 1995
But after closer investigation I found that I was lying on my stomach with my arm bent at a right angle.†
(source) Maya Angelou, I Know Why the Caged Bird Sings, 1969
In a way—I don't know about the Papervine Woman, but the planet Saturn has rings around it, and I suppose they do look like a belt when you see them at the right angle.†
(source) Terry Pratchett, Nation, 2008
A very small, elderly-looking woman, her body bent at the waist, at a right angle.†
(source) Tracy Kidder, Mountains Beyond Mountains, 2003
A low hedge ran down to the road just ahead of him but met it at right angles; anyone looking that way would be sure to spot him if he took cover there.†
(source) Mal Peet, Tamar, 2005
And if one didn't apply just the right amount of pressure at the right angle when applying leverage, he was more likely to break the radius bone, which would complicate any subsequent fracture of the wrist bones because, as BoneMan put it, your leverage will be shot.†
(source) Ted Dekker, BoneMan's Daughters, 2009
She rolled down her window and waited for the right angle.†
(source) Nora Roberts, Summer Pleasures, 2002
AFTER ALESSANDRO had taken half an hour to position himself and the machine so that his elbows would be bent at just the right angle, he was ready to take Strassnitzky's dictation.†
(source) Mark Helprin, A Soldier of the Great War, 1991
His head was a solid block that made sharp right angles with his shoulders.†
(source) James Patterson, Kiss the Girls, 1995
Nature is at right angles here.†
(source) Christina Garcia, Dreaming in Cuban, 1992
His feet are set slightly wider than shoulder width, his lean athletic torso is turned at a right angle to the bull's-eye, and his right arm is extended in a line perfectly parallel with the floor.†
(source) Bill O'Reilly and Martin Dugard, Killing Lincoln, 2011
When we reach that point we'll refuse the line, form a new line at right angles.†
(source) Michael Shaara, The Killer Angels, 1974
He walked slowly toward the truck, his eyes on me the whole way, black beret tilted at exactly the right angle on his head.†
(source) Laurie Halse Anderson, The Impossible Knife of Memory, 2014
Geometry maxims are of this nature: "the whole is greater than any of its parts; things equal to the same are equal to one another; two straight lines cannot enclose a space; all right angles are equal to each other."†
(source) James Madison, Alexander Hamilton, & John Jay, The Federalist Papers -- Modern English Edition 2, 2008
Wmade the right-angle turn into the barracks and ran along the galleries toward the R Company area without much concern for interception.†
(source) Pat Conroy, The Lords of Discipline, 1980
They heard a shout from somewhere above them, a woman's voice, and Leamas muttered "Oh, shut up" as he steered clumsily around a right-angle bend in the path and came almost immediately upon a major road.†
(source) John Le Carre, The Spy Who Came In From The Cold, 1963
Were it not for the branch beneath the balcony and the fact that I'd fallen onto it at just the right angle—I would have fallen to my death.†
(source) Bella Forrest, A Shade of Vampire, 2012
I extract significance from melodrama, a significance which it does not in fact contain; but occasionally, from out of this matter, there escapes a thin beam of light that, seen at the right angle, can crack the shell of mortality.†
(source) Tom Stoppard, Rosencrantz and Guildenstern Are Dead, 1966
They are placed parallel with the rail, against the slats, but can be moved out by the actors to stand at right angles to them.†
(source) Peter Shaffer, Equus, 1973
Just then I caught sight of Sophie at the instant she pushed open the grimy glass front door of the bar, where a slant of golden light somehow captured at exactly the right angle the lovely swerve of her cheekbone below the oval eyes with their sleepy-sullen hint of Asia, and the broad harmony of the rest of her face, including—or, I should say, especially—the fine, elongated, slightly uptilted "Polish schnoz," as Nathan lovingly called it, which terminated in a nice little button.†
(source) William Styron, Sophie's Choice, 1976
Yet their legs were motionless, and soon it was clear also that their bodies were tilted at right angles to that peculiar gangway.†
(source) Arthur C. Clarke, Childhood's End, 1953
We reached the cairn and Deirdre turned at right angles to it and headed straight toward the sea.†
(source) Roger Zelazny, Nine Princes in Amber, 1970
The bush boy turned, moved away at right angles, into the scrub.†
(source) James Vance Marshall, Walkabout, 1959
I led them back around that right angle some fifty feet and stopped.†
(source) Robert A. Heinlein, Glory Road, 1963
I can give you the windows for each one when they'll be at the right angle for Ares 3 shots—†
(source) Andy Weir, The Martian, 2011
Danny stood with his back against the door, looking at the right angle where the hallways joined.†
(source) Stephen King, The Shining, 1977
It's a right angle in a pressure vessel.†
(source) Andy Weir, The Martian, 2011
He nails two pieces of wood at right angles, with another nail for the trigger.†
(source) Margaret Atwood, Cat's Eye, 1988
Struggling to get the right angle, he was suddenly distracted.†
(source) Toni Morrison, Song of Solomon, 1977
The trick is getting the right angle so you puncture the right place.†
(source) Bella Forrest, A Shade of Vampire, 2012
There was a right-angle turn, then another, then another.†
(source) Roger Zelazny, Nine Princes in Amber, 1970
Phil and Dr. Stanpole then got into the car and drove slowly away, the headlights forming a bright parallel as they receded down the road, and then swinging into another parallel at right angles to the first as they turned into the Infirmary driveway.†
(source) John Knowles, A Separate Peace, 1959
The Weintraubs were welcomed to their own home-a modest place offering sun-dried adobe, curves instead of right angles, and bare wood floo o, b. also offering a view from the hill which showed an infinite expanse of desert beyond the orange and olive groves.†
(source) Dan Simmons, Hyperion, 1989
The floor, consisting of narrow planks of treated wood, was flat and the vertical sides of the buoyancy tanks were at right angles to it.†
(source) Yann Martel, Life of Pi, 2001
"You have to play in a boxed area and you have to stand at a right angle to the wind," he said proudly.†
(source) Khaled Hosseini, The Kite Runner, 2003
I found the Big Dipper and followed the long curve of its stars all the way to the right angle at the end, where the shape angles upward, pointing directly at the polestar.†
(source) Marcus Luttrell, Lone Survivor, 2007
The next door down from Chuck's was Pam and Rory's place, directly across from another hallway that led off at right angles to the elevators.†
(source) Matthew Mather, Cyberstorm, 2013
The elevator hallway, at right angles to the main hallway about halfway down, was stacked with containers of snow they were hauling up to melt for drinking water.†
(source) Matthew Mather, Cyberstorm, 2013
One of her hands fell away from his and dropped slowly until the arm was stretched out at right angles to her body, the hand dangling limply from the wrist like the hand of a drowning woman.†
(source) Stephen King, The Shining, 1977
Father walked out into the urban jungle of Pondicherry and bought a cow with dark wet eyes, a nice fat hump and horns so straight and at such right angles to its head that it looked as if it had licked an electrical outlet.†
(source) Yann Martel, Life of Pi, 2001
He stared at the door a moment longer, blue-gray eyes wide, then turned quickly and walked back down the corridor toward the main hallway that ran at right angles to the corridor he was in.†
(source) Stephen King, The Shining, 1977
So, if a boat is pushed by a wind but held back by a sea anchor, it will turn until it offers the least resistance to the wind—that is, until it is in line with it and at right angles to the waves, which makes for a front-to-back pitching that is much more comfortable than a side-to-side rolling.†
(source) Yann Martel, Life of Pi, 2001
There looked to be four streets, three running at right angles to the coach road, which was the main avenue of the town.†
(source) Stephen King, The Gunslinger, 1981
The spell propelled Saphira forward, but ever so slowly, for moving at right angles to the wind was like swimming across the Anora River during the height of the spring snowmelt.†
(source) Christopher Paolini, Inheritance, 2011
Instantly, I saw our shadows cast across the empty space behind the frame: my mother's tall and thin; Caroline's, her hands on her hips, elbows at right angles.†
(source) Sarah Dessen, The Truth About Forever, 2004
She was situated at right angles to him, her legs pulled up, with her arms wrapped around them and her chin resting on her knees.†
(source) Christopher Paolini, Brisingr, 2008
With the blade of the falchion at right angles to the ground, unless he deliberately tilted his wrist, any blows he caught on the sword would strike the flat of the blade, saving the edge for attacks of his own.†
(source) Christopher Paolini, Brisingr, 2008
He and Gertrude departed at right angles down a cross street, while Roran and the rest resumed their hunt.†
(source) Christopher Paolini, Eldest, 2005
Fifteen of us made the right-angle turn at T Company and immediately spotted ten or twelve upperclassmen loitering beneath the light beside the R Company stairwell.†
(source) Pat Conroy, The Lords of Discipline, 1980
With the right angle, the proper light, she could photograph a wheat field and make it seem endless, powerful.†
(source) Nora Roberts, Summer Pleasures, 2002
Bryan thought of Wyatt Earp, of Doc Holliday and the desperadoes who had once ridden through town, but she'd been drawn to the street parade that might've been in Anytown, U.S.A. It was here, caught up in the pageantry and the flavor, that she'd asked Shade his opinion of the right angle for shooting a horse and rider, and he in turn had taken her advice on capturing a tiny, bespangled majorette.†
(source) Nora Roberts, Summer Pleasures, 2002
Missiles and hurled spears turned in mid-flight to speed off at right angles before they could touch upon the chariot or its occupants.†
(source) Roger Zelazny, Lord of Light, 1967
They placed themselves, Rufo "eyes ahead" and Star where she could see both ways, at the right-angle bend.†
(source) Robert A. Heinlein, Glory Road, 1963
The truck drifted to a stop where a dirt road opened at right angles to the highway.†
(source) John Steinbeck, The Grapes of Wrath, 1939
Presently the trail bent sharply at right angles.†
(source) William Faulkner, Light in August, 1932
lips uttered and his nose drove out smoke, clever and pleasurable in the warm, heavy blue of Michigan; while wood-bracketed trawlers, tarred on the sides, chuffed and vapored outside the water reserved for the bawling, splashing, many-actioned, brilliant-colored crowd; waterside structures and towers, and skyscrapers beyond in a vast right angle to the evading bend of the shore.†
(source) Saul Bellow, The Adventures of Augie March, 1949
I noticed the man playing on the reed had his big toes splayed out almost at right angles to his feet.†
(source) Albert Camus, The Stranger, 1942
Finally Rhett turned the horse at right angles and after a while they were on a wider, smoother road.†
(source) Margaret Mitchell, Gone with the Wind, 1936
Each held his spear at right angles toward the left, and, before the Wart could say anything further, there was a terrific yet melodious thump.†
(source) T. H. White, The Once and Future King, 1939
It turns off at right angles, the wheel-marts of last Sunday healed away now: a smooth, red scoriation curving away into the pines; a white signboard with faded lettering: New Hope Church.†
(source) William Faulkner, As I Lay Dying, 1930
The sun shone at right-angles to the north wind, leaving the east side of the furrows white with frost.†
(source) T. H. White, The Once and Future King, 1939
He could see the street down which he had come, and the other street, the one which had almost betrayed him; and further away and at right angles, the far bright rampart of the town itself, and in the angle between the black pit from which he had fled with drumming heart and glaring lips.†
(source) William Faulkner, Light in August, 1932
At last he turned about and faced the dusty side road that cut off at right angles through the fields.†
(source) John Steinbeck, The Grapes of Wrath, 1939
The†
(source) William Faulkner, As I Lay Dying, 1930
Florence led off at right angles, threading a slow passage through the mesquite.†
(source) Zane Grey, The Light of Western Stars, 1914
The gulch turned at right angles and a great gray slope shut out sight of what lay beyond.†
(source) Zane Grey, The Border Legion, 1916
Telephone wires hum along the white roads, which always run at right angles.†
(source) Willa Cather, O Pioneers!, 1913
"—Cart and car being practically at right angles—" The voices of the happy family rose high.†
(source) E. M. Forster, Howards End, 1910
The canyon had narrowed to half its width, and turned almost at right angles.†
(source) Zane Grey, The Rainbow Trail, 1915
Two great streets, cutting each other at right angles, divided the city into quarters.†
(source) Lew Wallace, Ben Hur, 1880
The vibration of the pendulum was at right angles to my length.†
(source) Edgar Allan Poe, The Pit and the Pendulum, 1850
As Archer entered he was smiling and looking down on his hostess, who sat on a sofa placed at right angles to the chimney.†
(source) Edith Wharton, The Age of Innocence, 1920
The double doors are in the middle of the back hall; and persons entering find in the corner to their right two tall file cabinets at right angles to one another against the walls.†
(source) George Bernard Shaw, Pygmalion, 1912
So it was that one night, as Jurgis was on his way out with his gang, an engine and a loaded car dashed round one of the innumerable right-angle branches and struck him upon the shoulder, hurling him against the concrete wall and knocking him senseless.†
(source) Upton Sinclair, The Jungle, 1906
The Persian lit his lamp again and flung its rays down two enormous corridors that crossed each other at right angles.†
(source) Gaston Leroux, The Phantom of the Opera, 1911
Thus absorbed, she recrossed the northern part of Long-Ash Lane at right angles, and presently saw before her the road ascending whitely to the upland along whose margin the remainder of her journey lay.†
(source) Thomas Hardy, Tess of the d'Urbervilles, 1891
Crossing at right angles the great thoroughfare on which they walked, was a second canyon-like way, threaded by throngs and vehicles and various lines of cars which clanged their bells and made such progress as they might amid swiftly moving streams of traffic.†
(source) Theodore Dreiser, An American Tragedy, 1925
At the very top it was crossed at right angles by a green "ridgeway"—the Ickneild Street and original Roman road through the district.†
(source) Thomas Hardy, Jude the Obscure, 1895
There was a little passage in front of me, unpapered and uncarpeted, which turned at a right angle at the farther end.†
(source) Arthur Conan Doyle, The Adventures of Sherlock Holmes, 1892
I wonder if he sometimes places his feet at just the right angle and adds a special grip to his handshake?†
(source) Thomas Mann, The Magic Mountain, 1924
It was about an hour and a half since he had left her; he went out, took a cab, and stopped it close to her house, in a little street running at right angles to that other street, which lay at the back of her house, and along which he used to go, sometimes, to tap upon her bedroom window, for her to let him in.†
(source) Marcel Proust, Swann's Way, 1913
I tried and found by experiment that the tide kept sweeping us westward until I had laid her head due east, or just about right angles to the way we ought to go.†
(source) Robert Louis Stevenson, Treasure Island, 1883
At regular intervals down the long sides of the room, at right angles with the wall, were iron slabs, grooved like meat-dishes; and on each lay a body.†
(source) W. Somerset Maugham, Of Human Bondage, 1915
I saw then that the unusually forlorn and stunted look of the house was partly due to the loss of what is known in New England as the "L": that long deep-roofed adjunct usually built at right angles to the main house, and connecting it, by way of storerooms and tool-house, with the wood-shed and cow-barn.†
(source) Edith Wharton, Ethan Frome, 1911
Then, approaching at right angles to the trail and cutting off his retreat they saw a dozen wolves, lean and grey, bounding across the snow.†
(source) Jack London, White Fang, 1906
When halfway to each gate the leader of each string wheeled at right angles to head straight for the slope.†
(source) Zane Grey, The Thundering Herd, 1925
There was one main street, very wide, that divided the town and was crossed at right angles by a stream spanned by a small natural stone bridge.†
(source) Zane Grey, The Rainbow Trail, 1915
Venters satisfied himself that the rustlers had not deviated from their usual course, and then he turned at right angles off the cattle trail and made for the head of the pass.†
(source) Zane Grey, Riders of the Purple Sage, 1912
But some philosophical people have been asking why three dimensions particularly—why not another direction at right angles to the other three?†
(source) H. G. Wells, The Time Machine, 1895
The roads, named after victorious generals and intersecting at right angles, were symbolic of the net Great Britain had thrown over India.†
(source) E. M. Forster, A Passage to India, 1924
As she arrived at right angles to the sea, the full force of the wind (from which we had hitherto run away) caught us.†
(source) Jack London, Sea Wolf, 1904
This time running blindly, I went northeastward in a direction at right angles to my previous expedition.†
(source) H. G. Wells, The Island of Dr. Moreau, 1896
He worked along to a right angle in the bank which the men had made in the course of mining, and in this angle he came to bay, protected on three sides and with nothing to do but face the front.†
(source) Jack London, The Call of the Wild, 1903
Keeping the Vale on her right, she steered steadily westward; passing above the Hintocks, crossing at right-angles the high-road from Sherton-Abbas to Casterbridge, and skirting Dogbury Hill and High-Stoy, with the dell between them called "The Devil's Kitchen".†
(source) Thomas Hardy, Tess of the d'Urbervilles, 1891
She led the way into a series of connecting rooms that seemed to join each other at right angles, adding as she went, "You do look an awful lot like Gil Griffiths, don't you?"†
(source) Theodore Dreiser, An American Tragedy, 1925
That Space, as our mathematicians have it, is spoken of as having three dimensions, which one may call Length, Breadth, and Thickness, and is always definable by reference to three planes, each at right angles to the others.†
(source) H. G. Wells, The Time Machine, 1895
The scout faced south, at right angles with the cross-fire from the Comanches, and presently extended his long arm.†
(source) Zane Grey, The Thundering Herd, 1925
Five or six miles from the lair, the stream divided, its forks going off among the mountains at a right angle.†
(source) Jack London, White Fang, 1906
He would gently lay one hand on the forearm of the tablemate to his left, a young Bulgarian teacher, or on Madame Chauchat's to his right, then raise that same hand at an angle to command them all to sit in silent expectation of what he was about to say, would then gaze down at a spot on the tablecloth in the vicinity of his captive—lifting his eyebrows until the creases, which started up from the corners of his eyes and then turned in a right angle across his brow, grew even deeper and more masklike—and open his large, ragged lips as if he were about to utter something of vast import.†
(source) Thomas Mann, The Magic Mountain, 1924
Splits appeared in deep breaks, and gorges running at right angles, and then the Pass opened wide at a junction of intersecting canyons.†
(source) Zane Grey, Riders of the Purple Sage, 1912
The current had turned at right angles, sweeping round along with it the tall schooner and the little dancing coracle; ever quickening, ever bubbling higher, ever muttering louder, it went spinning through the narrows for the open sea.†
(source) Robert Louis Stevenson, Treasure Island, 1883
It is sensibly planned, with a redbrick club on its brow, and farther back a grocer's and a cemetery, and the bungalows are disposed along roads that intersect at right angles.†
(source) E. M. Forster, A Passage to India, 1924
The Macedonia was now but a mile away, the black smoke pouring from her funnel at a right angle, so madly she raced, pounding through the sea at a seventeen-knot gait—"'Sky-hooting through the brine," as Wolf Larsen quoted while gazing at her.†
(source) Jack London, Sea Wolf, 1904
The highroad passed over a saddle, and a track went thence at right angles alone the ridge of the down.†
(source) E. M. Forster, Howards End, 1910
Before long Stewart wheeled at right angles off the trail and entered a hollow between two low bluffs.†
(source) Zane Grey, The Light of Western Stars, 1914
Under cover of this circumstantial narrative, to which Mrs. Beaufort listened with her perfect smile, and her head at just the right angle to be seen in profile from the stalls, Madame Olenska turned and spoke in a low voice.†
(source) Edith Wharton, The Age of Innocence, 1920
Cherokee might well have been disembowelled had he not quickly pivoted on his grip and got his body off of White Fang's and at right angles to it.†
(source) Jack London, White Fang, 1906
So the boundary hedge zigzagged down the hill at right angles, and at the bottom there was a little green annex—a sort of powder-closet for the cows.†
(source) E. M. Forster, Howards End, 1910
The eyes were closed in peace, but the idol-like tracery of creases, four or five tense horizontal lines that turned down at right angles at the temples and had been formed by the habits of a lifetime, stood out in strong relief on the high brow encircled by white flames.†
(source) Thomas Mann, The Magic Mountain, 1924
Cornered in the right-angle of the poop and galley, he sprang like a cat to the top of the cabin and ran aft.†
(source) Jack London, Sea Wolf, 1904
The canyon widened ahead into a great, ragged, iron-hued amphitheater, and from there apparently turned abruptly at right angles.†
(source) Zane Grey, The Rainbow Trail, 1915
Herr Liesecke, too, looked as if wild horses could not make him inattentive; there were lines across his forehead, his lips were parted, his pince-nez at right angles to his nose, and he had laid a thick, white hand on either knee.†
(source) E. M. Forster, Howards End, 1910
This "den" was the result of the dining room's having been designed with three windows, so that it extended across the full width of the house, thus leaving no space for three drawing rooms, as was usual with this style of house, but for only two, one of which, placed at right angles to the dining room, would have been disproportionately deep, with only one window to the street.†
(source) Thomas Mann, The Magic Mountain, 1924
The deserted boat was in the trough of the sea, rolling drunkenly across each comber, its loose spritsail out at right angles to it and fluttering and flapping in the wind.†
(source) Jack London, Sea Wolf, 1904
She struck White Fang at right angles in the midst of his spring, and again he was knocked off his feet and rolled over.†
(source) Jack London, White Fang, 1906
Slowly the mast swung in until it balanced at right angles across the rail; and then I discovered to my amazement that there was no need for Maud to slack away.†
(source) Jack London, Sea Wolf, 1904
And, halfway to the crosstrees and flattened against the rigging by the full force of the wind so that it would have been impossible for me to have fallen, the Ghost almost on her beam-ends and the masts parallel with the water, I looked, not down, but at almost right angles from the perpendicular, to the deck of the Ghost.†
(source) Jack London, Sea Wolf, 1904
She sat on a low stool at nearly a right angle with the two boys, watching first one and then the other; and Philip, looking off his book once toward the fire-place, caught the pair of questioning dark eyes fixed upon him.†
(source) George Eliot, The Mill on the Floss, 1860
When we saw them from the rear of the house, they were both about half open—that is to say, they stood off at right angles from the wall.†
(source) Edgar Allan Poe, The Murders in the Rue Morgue, 1850
Ice has its grain as well as wood, and when a cake begins to rot or "comb," that is, assume the appearance of honeycomb, whatever may be its position, the air cells are at right angles with what was the water surface.†
(source) Henry David Thoreau, Walden, 1854
One thing did not please the ladies: he kept bending forward, especially at the beginning of his speech, not exactly bowing, but as though he were about to dart at his listeners, bending his long spine in half, as though there were a spring in the middle that enabled him to bend almost at right angles.†
(source) Fyodor Dostoyevsky, The Brothers Karamazov, 1880
The North Star was directly in the wind's eye, and since evening the Bear had swung round it outwardly to the east, till he was now at a right angle with the meridian.†
(source) Thomas Hardy, Far from the Madding Crowd, 1874
The instrument commonly used consists of two heavy beams of wood, secured together at right angles, and loaded with stones.†
(source) Johann Wyss, The Swiss Family Robinson, 1849
After winding along the side of the mountain, the road, on reaching the gentle declivity which lay at the base of the hill, turned at a right angle to its former course, and shot down an inclined plane, directly into the village of Templeton.†
(source) James Fenimore Cooper, The Pioneers, 1823
Two of the brothers led Pierre up to the altar, placed his feet at right angles, and bade him lie down, saying that he must prostrate himself at the Gates of the Temple.†
(source) Leo Tolstoy, War and Peace, 1869
The road selected was a continuation of the Via Sistina; then by cutting off the right angle of the street in which stands Santa Maria Maggiore and proceeding by the Via Urbana and San Pietro in Vincoli, the travellers would find themselves directly opposite the Colosseum.†
(source) Alexandre Dumas, The Count of Monte Cristo, 1846
Ralph accompanied their visitors to town and established them at a quiet inn in a street that ran at right angles to Piccadilly.†
(source) Henry James, The Portrait of a Lady - Volume 1, 1881
I tore a part of the hem from the robe and placed the fragment at full length, and at right angles to the wall.†
(source) Edgar Allan Poe, The Pit and the Pendulum, 1850
A severe facade rose above this door; a wall, perpendicular to the facade, almost touched the door, and flanked it with an abrupt right angle.†
(source) Victor Hugo, Les Miserables, 1862
Their road to this detached cottage was down Vicarage Lane, a lane leading at right angles from the broad, though irregular, main street of the place; and, as may be inferred, containing the blessed abode of Mr. Elton.†
(source) Jane Austen, Emma, 1815
Independently of these two principal streets, piercing Paris diametrically in its whole breadth, from side to side, common to the entire capital, the City and the University had also each its own great special street, which ran lengthwise by them, parallel to the Seine, cutting, as it passed, at right angles, the two arterial thoroughfares.†
(source) Victor Hugo, The Hunchback of Notre Dame, 1831
But far more terrible is it to behold, when fathoms down in the sea, you see some sulky whale, floating there suspended, with his prodigious jaw, some fifteen feet long, hanging straight down at right-angles with his body, for all the world like a ship's jib-boom.†
(source) Herman Melville, Moby Dick, 1851
The lofty tower of its City Hall overlooked the whole panorama of the streets and avenues, which cut each other at right-angles, and in the midst of which appeared pleasant, verdant squares, while beyond appeared the Chinese quarter, seemingly imported from the Celestial Empire in a toy-box.†
(source) Jules Verne, Around the World in 80 Days, 1873
Sikes clenched his teeth; took one look around; threw over the prostrate form of Oliver, the cape in which he had been hurriedly muffled; ran along the front of the hedge, as if to distract the attention of those behind, from the spot where the boy lay; paused, for a second, before another hedge which met it at right angles; and whirling his pistol high into the air, cleared it at a bound, and was gone.†
(source) Charles Dickens, Oliver Twist, 1838
The aged group, under care of sons or daughters, themselves worn and grey, passed on along the least-winding carriage-road towards the house, where a special table was prepared for them; while the Poyser party wisely struck across the grass under the shade of the great trees, but not out of view of the house-front, with its sloping lawn and flower-beds, or of the pretty striped marquee at the edge of the lawn, standing at right angles with two larger marquees on each side of the open green space where the games were to be played.†
(source) George Eliot, Adam Bede, 1859
For two hours did this single-hearted and simple-minded girl toil through the mazes of the forest, sometimes finding herself on the brow of the bank that bounded the water, and at others struggling up an ascent that warned her to go no farther in that direction, since it necessarily ran at right angles to the course on which she wished to proceed.†
(source) James Fenimore Cooper, The Deerslayer, 1841
Then, holding the brand, he crossed a deep, narrow chasm in the rocks which ran at right angles with the passage they were in, but which, unlike that, was open to the heavens, and entered another cave, answering to the description of the first, in every essential particular.†
(source) James Fenimore Cooper, The Last of the Mohicans, 1826
By this time the Scud was within a mile of the shore, on which the gale was blowing at right angles, with a violence that forbade the idea of showing any additional canvas with a view to claw off.†
(source) James Fenimore Cooper, The Pathfinder, 1840
The Russians, they say, fortified this position in advance on the left of the highroad (from Moscow to Smolensk) and almost at a right angle to it, from Borodino to Utitsa, at the very place where the battle was fought.†
(source) Leo Tolstoy, War and Peace, 1869
This figure is but a summary one and half exact, the right angle, which is the customary angle of this species of subterranean ramifications, being very rare in vegetation.†
(source) Victor Hugo, Les Miserables, 1862
Two thick squares of wood of equal size are stoutly clenched together, so that they cross each other's grain at right angles; a line of considerable length is then attached to the middle of this block, and the other end of the line being looped, it can in a moment be fastened to a harpoon.†
(source) Herman Melville, Moby Dick, 1851
She had placed herself in a deep window-bench, from which she looked out into the dull, damp park; and as the library stood at right angles to the entrance-front of the house she could see the doctor's brougham, which had been waiting for the last two hours before the door.†
(source) Henry James, The Portrait of a Lady - Volume 1, 1881
Once or twice she thought there was a stature or a gait that she recollected; but the person who owned it instantly disappeared behind one of those enormous piles of wood that lay before most of the doors, It was only as they turned from the main street into another that intersected it at right angles, and which led directly to the place of meeting, that she recognized a face and building that she knew.†
(source) James Fenimore Cooper, The Pioneers, 1823
But no such chance did, or indeed could now offer, and when he found that he was descending towards the glen, by the melting away of the ridge, he turned short, at right angles to his previous course, and went down the declivity with tremendous velocity, holding his way towards the shore.†
(source) James Fenimore Cooper, The Deerslayer, 1841
Bathsheba, without looking within a right angle of him, was conscious of a black sheep among the flock.†
(source) Thomas Hardy, Far from the Madding Crowd, 1874
Like the rest of our waters, when much agitated, in clear weather, so that the surface of the waves may reflect the sky at the right angle, or because there is more light mixed with it, it appears at a little distance of a darker blue than the sky itself; and at such a time, being on its surface, and looking with divided vision, so as to see the reflection, I have discerned a matchless and indescribable light blue, such as watered or changeable silks and sword blades suggest, more cerulean than the sky itself, alternating with the original dark green on the opposite sides of the waves, which last appeared but muddy in comparison.†
(source) Henry David Thoreau, Walden, 1854
The train reached Ogden at two o'clock, where it rested for six hours, Mr. Fogg and his party had time to pay a visit to Salt Lake City, connected with Ogden by a branch road; and they spent two hours in this strikingly American town, built on the pattern of other cities of the Union, like a checker-board, "with the sombre sadness of right-angles," as Victor Hugo expresses it.†
(source) Jules Verne, Around the World in 80 Days, 1873
Not only did the Russians not fortify the position on the field of Borodino to the left of, and at a right angle to, the highroad (that is, the position on which the battle took place), but never till the twenty-fifth of August, 1812, did they think that a battle might be fought there.†
(source) Leo Tolstoy, War and Peace, 1869
Charles's Wain was getting towards a right angle with the Pole star, and Gabriel concluded that it must be about nine o'clock—in other words, that he had slept two hours.†
(source) Thomas Hardy, Far from the Madding Crowd, 1874
Two barricades were now in process of construction at once, both of them resting on the Corinthe house and forming a right angle; the larger shut off the Rue de la Chanvrerie, the other closed the Rue Mondetour, on the side of the Rue de Cygne.†
(source) Victor Hugo, Les Miserables, 1862
At length the sun's rays have attained the right angle, and warm winds blow up mist and rain and melt the snowbanks, and the sun, dispersing the mist, smiles on a checkered landscape of russet and white smoking with incense, through which the traveller picks his way from islet to islet, cheered by the music of a thousand tinkling rills and rivulets whose veins are filled with the blood of winter which they are bearing off.†
(source) Henry David Thoreau, Walden, 1854
Grantaire added to the eccentric accentuation of words and ideas, a peculiarity of gesture; he rested his left fist on his knee with dignity, his arm forming a right angle, and, with cravat untied, seated astride a stool, his full glass in his right hand, he hurled solemn words at the big maid-servant Matelote:— "Let the doors of the palace be thrown open!†
(source) Victor Hugo, Les Miserables, 1862
A heated breeze from the south slowly fanned the summits of lofty objects, and in the sky dashes of buoyant cloud were sailing in a course at right angles to that of another stratum, neither of them in the direction of the breeze below.†
(source) Thomas Hardy, Far from the Madding Crowd, 1874
The case was evidently this: a position was selected along the river Kolocha—which crosses the highroad not at a right angle but at an acute angle—so that the left flank was at Shevardino, the right flank near the village of Novoe, and the center at Borodino at the confluence of the rivers Kolocha and Voyna.†
(source) Leo Tolstoy, War and Peace, 1869
Among his other resources, thanks to his numerous escapes from the prison at Toulon, he was, as it will be remembered, a past master in the incredible art of crawling up without ladder or climbing-irons, by sheer muscular force, by leaning on the nape of his neck, his shoulders, his hips, and his knees, by helping himself on the rare projections of the stone, in the right angle of a wall, as high as the sixth story, if need be; an art which has rendered so celebrated and so alarming that corner of the wall of the Conciergerie of Paris by which Battemolle, condemned to death, made his escape twenty years ago.†
(source) Victor Hugo, Les Miserables, 1862
A person coming from the Seine reached the extremity of the Rue Polonceau, and had on his right the Rue Droit-Mur, turning abruptly at a right angle, in front of him the wall of that street, and on his right a truncated prolongation of the Rue Droit-Mur, which had no issue and was called the Cul-de-Sac Genrot.†
(source) Victor Hugo, Les Miserables, 1862
with clenched fists, with arms outspread at right angles, like a man crucified who has been un-nailed, and flung face down on the earth.†
(source) Victor Hugo, Les Miserables, 1862
The large building of the Rue Droit-Mur, which had a wing on the Rue Petit-Picpus, turned two facades, at right angles, towards this garden.†
(source) Victor Hugo, Les Miserables, 1862
The causeway which leads to the ancient Barriere du Maine is a prolongation, as the reader knows, of the Rue de Sevres, and is cut at right angles by the inner boulevard.†
(source) Victor Hugo, Les Miserables, 1862
The barricades at right angles fell back, the one of the Rue Montorgueil on the Grande-Truanderie, the other of the Rue Geoffroy-Langevin on the Rue Sainte-Avoye.†
(source) Victor Hugo, Les Miserables, 1862 | 677.169 | 1 |
The cut and sort activity helps students identify which angles are smaller or Some of these worksheets also contain special instructions that help students understand the concept. In addition, they can practice finding angles on graphs that are not lined up. | 677.169 | 1 |
proofs with quadrilaterals
Given: Quadrilateral These can be used as a guided study, group work, stations, or as individual classwork or homework.The preview contains all student pages for your perusal. Students will LOVE practicing proofs with this digital activity. If the diagonals of a quadrilateral bisect each other then it is a parallelogram. Proofs include: Parallelogram, Square, Rectangles, Kite, & Rhombus. Included in this download are 16 simple proofs ranging from 3 steps to 8 steps each. Opposite angles of a parallelogram are congruent. The justification inside this type of proof will include things like properties regarding transformations. 287 times. - 6 sheets of quadrilaterals practice proofs (two per page) This set of worksheets includes notes and practice on coordinate proofs with quadrilaterals and triangle congruency proofs with quadrilaterals. LEARN MORE HERE!Quick review of statement-reason proofs4 practice proofs with various levels of, This worksheet is the perfect review for my lessons quadrilateral coordinate proofs and quadrilateral two column proofs! DonateLoginSign up. Show one set of opposite sides of the quadrilateral is congruent. Two triangles that have exactly the same size and shape. Given this information, which quadrilateral is proven? Proofs include: Parallelogram, Square, Rectangles, Kite, & Rhombus. 1) :l:f both pairs of opposite sides are parallel, then the quadrilateral is a parallelogram, 2) If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is … STUDY. Search. With this worksheet generator, you can make worksheets for classifying (identifying, naming) quadrilaterals, in PDF or html formats. Another way to prevent getting this page in the future is to use Privacy Pass. If a quadrilateral is a parallelogram, then its opposite sides are congruent. Are you getting the free resources, updates, and special offers we send out every week in our teacher newsletter? There are seven special types of quadrilaterals: square, rectangle, rhombus, parallelogram, trapezoid, kite, scalene, and these worksheets ask students to name the quadrilaterals among these seven types. Stu. 02.06 QUADRILATERAL PROOFS Polygon a closed figure with three or more sides The word polygon literally means "many angles," Polygons can be classified by the number of sides they have and whether they are regular or irregular. ZIP (487.13 KB) In this activity, students will complete two-column proofs involving properties of special quadrilaterals. Di Topics include:Polygon review (names and sum of interior angles)Kites Trapezoids (isosceles and midsegment)Parallelogram PropertiesRectanglesRhombiSquaresParallelogram Proofs (using congruent triangles)Products include: Quadrilaterals Guided N, Scaffold your students' ability to complete Geometry Proofs with this fun, engaging activity. Your purchase will include a PDF that has both, No prep binder notes and practice on quadrilateral coordinate proofs! Proofs can often times be interpreted differently, as students will look at each from a different perspective. Proof: Diagonals of a parallelogram. Please enable Cookies and reload the page. acquire the quadrilateral proofs with answers link that we offer here and check out the link. There is a quick review of the contents that students will need to know before begin, Are your students struggling with all of the properties of parallelograms, rectangles and rhombuses? A quadrilateral is a polygon in Euclidean plane geometry with four edges (sides) and four vertices (corners). Edit. Students must use distance, midpoint, and slope to determine which shape they've made. The diagonals of a parallelogram bisect each other. If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram. Cloudflare Ray ID: 61698981cfc50e92 Segments that have the same length. SIX proofs in all; some require completing the reasons, some require completing both statements and … Your IP: 64.20.62.167 MathHigh school geometryCongruenceTheorems concerning quadrilateral properties. Show both sets of opposite angles of the quadrilateral are congruent. Quadrilateral Proof: 1. Click each link for notes and practice that match this worksheet! Assign Task. A … Here's what's included, Thank you for checking out my product! These can be used as a guided study, group work, stations, or as individual classwork or homework.The preview contains all student pages for your perusal. There is a quick review of the contents that students will need to know before beginning the proofs. Both of these facts allow us to prove that the figure is indeed a parallelogram. scolasanto_46836. Prove that the sum of the interior angles of a quadrilateral is 360. A full, … When Is a Parallelogram a Rhombus? Consider the givens. PLAY. proofs, quadrilaterals, and more proofs. This product contains 8 guided and unguided proofs working with quadrilaterals as parallelograms Consecutive Angles in a Parallelogram are Supplementary. anything u need for proofs. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. When Is a Parallelogram a Rectangle? Quadrilaterals Proofs - Two-Column Proofs with Quadrilateral Properties and Theorems: This set contains proofs with rectangles, parallelograms, rhombi, and trapezoids: - 6 sheets of quadrilaterals practice proofs (two per page) - 1 sheet of two challenging proofs with higher difficulty level - 1 q. Quadrilateral Proofs With Answers Recognizing the mannerism ways to get this books quadrilateral proofs with answers is additionally useful. congruent segments. You have remained in right site to start getting this info. This set contains proofs with rectangles, parallelograms, rhombi, and trapezoids: The student version has bl, This resource includes: A graphic organizer that reviews the properties of parallelograms, rectangles, and rhombuses. "A quadrilateral with one pair of opposite sides both congruent and parallel" Quadrilateral Proofs DRAFT. This is a 6-question proof worksheet that is scaffolded with a word bank for each problem and some of the 2-column proof filled in for the students. Writing Proofs to show Congruence, Similarity and other properties of figures; Connecting Proofs to other units of Geometry / Daily Geometry Regents Prep with Quick Check tasks . Students will figure out, or discuss as a class, how those properties can be proven using the coordinate plane.8 Problems#1 - 4 Students will use either the distance or slope formula, This product contains 8 guided and unguided proofs working with quadrilaterals as parallelograms. There are three proofs that use properties of quadrilaterals. Using playing cards (or any other way of choosing numbers), students are given 4 points and then explore the properties of the quadrilateral they'v. Buy the discounted bundled version for the whole year HERE!Features:**NEW** Turn this PDF into a digital interactive worksheet! REF: 080731b 7 ANS: Parallelogram ANDR with AW and DE bisecting NWD and REA at points W and E (Given).AN ≅RD, AR ≅DN (Opposite sides of a parallelogram are congruent).AE = 1 2 AR, WD = 1 2 DN, so AE ≅WD (Definition quadrilateral proofs with answers category kindle and ebooks pdf''QUADRILATERALS MATHSTEACHER COM AU MAY 2ND, 2018 - PROVE THAT THE ANGLE SUM OF A QUADRILATERAL IS EQUAL TO 360º PROOF HENCE THE ANGLE SUM OF A QUADRILATERAL IS 360º GIVE REASONS FOR YOUR ANSWER SOLUTION ' Quadrilateral Proofs With Answers You can prove this with either a two-column proof or a paragraph proof. In this READY TO GO digital activity, students will use moveable pieces to write proofs about quadrilaterals. These are 12 true/false questions about quadrilaterals. Area Quadrilaterals Worksheet with Answers Luxury New Pics from parallelogram proofs worksheet with answers, source:buddydankradio.com Postulates may be utilized to prove theorems correct. • This product contains some simple geometric proofs that you can use with your students while they begin to explore the process of writing their own two-column proofs. Buy the discounted bundled version for the whole year HERE!Features:**NEW** Turn this PDF into a digital interactive worksheet! Then, they will finish a few proofs t, This set of four activities is perfect for High School Geometry students. Your game plan might go something like this: Look for congruent triangles. angles that have the same measure. Topics covered under this chapter help the students to understand the basics of a geometrical figure named as a quadrilateral, its … One Pair of Opposite Sides are Both Parallel and Congruent. Proof: Opposite sides of a parallelogram. Improve your math knowledge with free questions in "Proofs involving triangles and quadrilaterals" and thousands of other math skills. 7 months ago. In this READY TO PRINT proofs activity, students will write three quadrilaterals geometry proofs. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Edit. All proofs a, Introducing proofs in Geometry can be difficult. Kite: A quadrilateral in which two disjoint pairs of consecutive sides are congruent ("disjoint pairs" … HSG.CO.C.11 - Prove theorems about parallelograms. All pairs of corresponding angles and corresponding sides are congruent No assumptions need to be made. Your students will love this HANDS-ON ACTIVITY.In this activity, students will cut strips of paper and assemble them into two-column proofs. Great for introducing quadrilateral proofs. Show the diagonals of the quadrilateral bisect each other. If a quadrilateral is a parallelogram… Let us discuss some … PERFECT FOR DISTANCE LEARNING!In this digital activity, students will use Google Slides to write quadrilaterals proofs. Make sure your work is neat and organized. Basic Quadrilateral Proofs For each of the following, draw a diagram with labels, create the givens and proof statement to go with your diagram, then write a two-column proof. SIX proofs in all; some require completing the reasons, some require completing both statements and reasons. - 1 sheet of two challenging proofs with higher difficulty level Students complete the proof and use their answers to complete the crossword puzzle, which allows them to se. Performance & security by Cloudflare, Please complete the security check to access. Start studying Quadrilateral Proofs 2.06 FLVS (100%). A full, detailed teacher key is provided with p. This worksheet asks students to use their knowledge of the properties of quadrilaterals to prove or justify the classification of the quadrilateral. Also included in: Geometric Proofs Interactive Note-Taking Materials Bundle, Also included in: Proving Quadrilaterals are Parallelograms Bundle, Also included in: Geometry Proofs Digital Activity Bundle for Distance Learning, Also included in: Geometry Proofs Activities: Cut and Paste Bundle, Also included in: Quadrilateral Proofs Notes and Worksheet Bundle, Also included in: Geometry Proofs Crossword Puzzles Bundle, Also included in: Geometry Items Bundle-Part One (congruent triangles, quadrilaterals, parallel), Also included in: Quadrilaterals Unit (Geometry Unit 5). Proofs involving quadrilaterals. Proofs About Quadrilaterals When Is a Quadrilateral a Parallelogram? Theorems concerning quadrilateral properties. Track stude, This editable quadrilaterals test and study guide covers:Polygon review (names and sum of interior angles)Kites Trapezoids (isosceles, midsegments)Properties of parallelograms (including algebra)Properties of rectangles, rhombi, squaresParallelograms proofsThis test and study guide are EDITABLE, how, Quadrilaterals Proofs - Two Column Proof Practice and Quiz, Interactive Two-Column Proofs with Quadrilaterals using Google Slides, Simple Geometric Proofs with Quadrilaterals - Interactive Note-Taking Materials, Geometric Proofs Interactive Note-Taking Materials Bundle, Introductory Quadrilateral Proofs with Complete Key, Quadrilateral Proofs on the Coordinate Plane, Proofs: Quadrilaterals and Parallelograms, Proving Quadrilaterals are Parallelograms Bundle, Quadrilaterals in the Coordinate Plane Proofs, Quadrilaterals Proofs Digital Activity for Distance Learning, Geometry Proofs Digital Activity Bundle for Distance Learning, Quadrilaterals Proofs Cut and Paste Activity, Geometry Proofs Activities: Cut and Paste Bundle, Quadrilateral Proofs Notes and Worksheet Bundle, Quadrilateral Coordinate Proofs Worksheet, Congruent Triangle Proofs with Quadrilaterals Worksheet, Quadrilateral Proofs Mixed Review Worksheet, Challenging Quadrilateral/Parallelogram Test w/proofs, Parallelograms and Rectangles Geometry Proofs Crossword Puzzle, Geometry Properties of Quadrilaterals Task Cards with QR Codes, Geometry Items Bundle-Part One (congruent triangles, quadrilaterals, parallel), Geometry PROOFS: Congruence, Similarity, Coordinate proofs (Unit 15). Parallelograms are Rectangles, Kite, & Rhombus six proofs in all ; require. Geometry students marketplace where teachers buy and sell original educational materials a 501 ( c ) 3... Graphic organizer that reviews the properties of quadrilaterals fill in, a quadrilaterals... Google Slides to write quadrilaterals proofs students practice the properties of special quadrilaterals you a! Code leads to a happy face, they will finish a few proofs t, set! To write quadrilaterals proofs shape has the identifying properties of quadrilaterals notes part... Given this information, which allows them to se use their answers to complete the crossword puzzle which.: diagonals of the quadrilateral is a parallelogram year bundle that can be difficult Formula and Slope to determine shape! With free questions in `` proofs involving triangles and quadrilaterals '' and of. We might find that the diagonals of the contents that students will write three quadrilaterals Geometry proofs properties regarding.! Are Rectangles, rhombuses, and/or squares purchase will include things like properties regarding transformations Distance, midpoint, more... Justify proofs with quadrilaterals responses using a format of their choosing rhombuses, and/or squares with mixture! Will help students practice the properties of quadrilaterals and Mid-Point theorem them to see the properties! Working with quadrilaterals as parallelograms puzzle, which quadrilateral is a parallelogram ; some require completing both statements reasoning! Strips of paper and assemble them into two-column proofs containing congruent triangles — it six... — it has six pairs of corresponding angles and corresponding sides are congruent c ) ( 3 ) nonprofit.... Will need to know before beginning the proofs and fun, for Algebraic proofs ; or just Algebraic... To prove parallelograms are Rectangles, Kite, & Rhombus each link for notes and practice on quadrilateral proofs. Here and check out the link terms, and more with flashcards games! Fill in, a complete quadrilaterals unit for High School Geometry by,... Proofs, with a mixture of statements and reasons be open-ended in order to student! Html formats cut strips of paper and assemble them into two-column proofs involving triangles quadrilaterals. Shape they 've made of the interior angles of a quadrilateral bisect each other then it designed. Triangles that have exactly the same size and shape Google Slides to write quadrilaterals proofs 501 ( )! Full year bundle that can be purchased here answers Recognizing the mannerism ways to get books! Sell original educational materials worksheets includes notes and practice on coordinate proofs mar 7 2019! Properties of a parallelogram are congruent figure is indeed a parallelogram, its. Send out every week in Our teacher newsletter shape they 've made proofs involving triangles and quadrilaterals '' thousands... And reasoning missing of parallelograms, Rectangles, Kite, & Rhombus quadrilaterals When a. Included, Thank you for checking out my product shape they 've made two triangles that have the... Unit for High School Geometry students proofs with quadrilaterals set of 12 task cards that will students. Class 9 Maths Chapter 8 explains Angle sum Property of a parallelogram, Square, Rectangles, Slope! Site to start getting this info this activity, students will need to download version 2.0 now the... With flashcards, games, and Slope to determine which shape they made... This activity, students will look at each from a different perspective will need know. Free questions in `` proofs involving triangles and quadrilaterals '' and thousands of other math skills which allows them se... — it has six pairs of corresponding angles and corresponding sides are both parallel and congruent use. School Geometry this worksheet generator, you can prove this with either a two-column or. Cake for containing congruent triangles — it has six pairs of them it has six pairs corresponding. Rhombuses, and/or squares takes the cake for containing congruent triangles — it has six pairs of them cut of. Parallelograms, Rectangles, and fun, for Algebraic proofs ; or just identifying Algebraic properties action! Buy and sell original educational materials thousands of other math skills & security by,... Properties of quadrilaterals for Algebraic proofs ; or just identifying Algebraic properties in action answers is additionally useful a review. Distance LEARNING! in this READY to PRINT proofs activity, students will look each... Every week in Our teacher newsletter this assignment is great practice, and more with flashcards games. Steps to 8 steps each has bl, this set of 12 task cards that will help students the... Sum Property of a parallelogram and other study tools this is a?. And check out the link another way to prevent getting this info, and/or.... You be sure unit for High School Geometry students the reasons, some require completing statements! Happy face, they scan that QR Code quadrilaterals Geometry proofs books quadrilateral DRAFT! In general identifying properties of a quadrilateral is 360 the Code leads a! Proofs a, Introducing proofs in Geometry can be purchased here a … Improve your math knowledge with free in! Interpreted differently, as students will look at each from a different perspective of quadrilateral. Practicing proofs with answers link that we offer here and check out the link quadrilateral... Activity.In this activity, students will use moveable pieces to write quadrilaterals.! Mission is to provide a free, world-class education to anyone, anywhere finish!, students will cut strips of paper and assemble them into two-column proofs involving properties a! Think the answer is true, they scan that QR Code included in this READY GO! Size and shape that the information provided will indicate that the information will... Books quadrilateral proofs with answers Recognizing the mannerism ways to get this proofs with quadrilaterals quadrilateral proofs.... Chapter 8 explains Angle sum Property of a full year bundle that can be.. Cake for containing congruent triangles — it has six pairs of corresponding angles and sides., as students will use Google Slides to write proofs About quadrilaterals Google Slides to write proofs About quadrilaterals is! Will allow them to see the various properties in action quadrilateral coordinate proofs can often times be differently... Marketplace where teachers buy and sell original educational materials their responses using a of... Are supplementary and reasoning missing by cloudflare, Please complete the security check to access step proofs will allow to... Is both congruent and parallel type of proof will include a PDF that has both, No binder!, the quadrilateral proofs DRAFT determine which shape proofs with quadrilaterals 've made are 16 simple proofs ranging from 3 to! There is a parallelogram quadrilateral coordinate proofs with this worksheet generator, you proven... The reasons, some require completing the CAPTCHA proves you are a and... Statement-Reason proofs and congruent, anywhere if students think the answer is true, they scan that Code! As students will use moveable pieces to write proofs About quadrilaterals the link & Rhombus six of. With flashcards, games, and fun, for Algebraic proofs ; or just identifying Algebraic proofs with quadrilaterals... Reasoning missing make worksheets for classifying ( identifying, naming ) quadrilaterals, in PDF or html formats, Rhombus... — it has six pairs of them mathematically proving that the information provided will indicate that the of! Of opposite sides of the quadrilateral bisect each other, the quadrilateral bisect each other step proofs proofs with quadrilaterals them! Go digital activity, students will look at each from a different.! World-Class education to anyone, anywhere this diagram takes the cake for containing congruent —... Three proofs that use properties of quadrilaterals facts allow us to prove parallelograms Rectangles! One Pair of opposite sides of a quadrilateral, Types of quadrilaterals! in this digital activity, students LOVE... Leads to a happy face, they scan that QR Code for Distance LEARNING! in lesson! Terms, and special offers we send out every week in Our teacher?!, for Algebraic proofs ; or just identifying Algebraic properties in action inside this type of proof include! Teachers is an online marketplace where teachers buy and sell original educational materials of... Finish a few proofs t, this set of worksheets includes notes and practice on coordinate proofs be... Hands-On ACTIVITY.In this activity, students will LOVE practicing proofs with answers that. Inside this type of proof will include things like properties regarding transformations check out link! Is also included.Note: these notes are part of a quadrilateral bisect each other proof or paragraph! My product proofs About quadrilaterals proofs with quadrilaterals will indicate that the sum of the quadrilateral proofs with is. The same size and shape and quadrilaterals '' and thousands of other math skills resource! Geometry students this info QR Code find that the shape has the identifying properties of parallelograms Rectangles. Proofs will allow them to see the various properties in action to be open-ended in order to student... And gives you temporary access to the web Property educational materials of!! Angle sum Property of a full year bundle that can be difficult have remained in right site start. Quadrilaterals, in PDF or html formats allows them to se they are correct prep notes. This info teachers is an online marketplace where teachers buy and sell original educational materials acquire the quadrilateral proofs.. Quadrilaterals as parallelograms format of their choosing is true, they scan that QR Code both! Has bl, this set of worksheets includes notes and practice on quadrilateral statement-reason!... Congruency proofs with quadrilaterals as parallelograms a parallelogram is congruent with this digital,! That we offer here and check out the link, in PDF or formats... | 677.169 | 1 |
The answer to this question has been given, you answer is very accurate and it can be made very easily, you can see it, along with you are getting the link of the question below, you can know the answer by clicking directly on the question and These five important questions which I am sharing among you are very important.
1. Find the constant of such that the vectors 3i +j-2k ,-i +3j +4k and di-2J-6k are coplanar
2. Find the moment about the point I+2j+3of a force represented by i+j+k acting through the point -2i+3j+k
3. Prove that the four points 4i+5j+k,-j-k,3i+9j+4k and -4i+4j +4k are coplanar
4. Find the volume of parallelopied whose concurrent edges are represented by the vectors i+2j+3k,3i+7j-4k and i-5j+3k | 677.169 | 1 |
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The Euler Characteristic
The Euler characteristic for a surface is given by F-E+V, or the number of faces - the number of edges+the number of vertices. The Euler characteristic is a topological invariant for a surface, so that surfaces with different Euler characteristics cannot be homeomorphic.
Any closed surface with no holes has Euler characteristic 2:
The five Platonic solids are shown above. Each has Euler characteristic 2.
Shape
Number of Vertices, V
Number of Edges, E
Number of Faces, F
Euler Characteristic
F-E+V
Tetrahedron
4
6
4
2
Cube
8
12
6
2
Octahedron
6
12
8
2
Dodecahedron
10
30
12
2
Icosahedron
12
30
20
2
Finding the Euler Characteristic of more complicated shapes is well, more complicated. In general a shape must be cut up and the number of vertices, edges and faces found. It can be a complicated process. | 677.169 | 1 |
Theorem # 2
Conversly, Chords of a circle, equidistant from the centre of the circle are equal.
AB & CD are the equal chords.
They are equidistant from the centre.
(OE = OF = 3)
Theorem # 3
Two equal chords of a circle subtend equal angles at the centre.
AB & CD are equal chords subtending equal angles at the centre.
They are equidistant from the centre.
∠AOB = ∠COD = 55
Theorem # 4
Chords AB & CD of a circle with centre 'O' intersect at E. If OE bisects ∠AED, then the chords AB & CD are equal.
Theorem # 5
The line segment joining the mid-points of two parallel chords of a circle always passes through the centre.
AB & CD are Parallel chords.
E & F are midpoints of AB & CD.
The line FE passes through the centre 'O'
Theorem # 6
Chords cut each other internally
Chords cut each other externally
AB & CD are two chords of a circle intersecting at 'P' internally or externally, then PA x PB=PC x PD. And, if the chords are equal then PA=PC & PB=PD.
Theorem # 7
Two Circles with centres A & B intersect each other at points P & Q. Then the line joining the centres bisect the common chord PQ perpendicularly.
Theorem # 8
Out of two unequal chords of a circle, the longer chord will be closer to the centre than the shorter chord.
AB = 4 is far away from the centre.
CD = 7 is closer to centre.
Theorem # 9
Two circles with centres O & 'O' intersect each other at points P & Q. The straight line APB is parallel to the line joining centres OO' & OO' = (1/2) AB
Theorem # 10
1. If chords of a circle are equal in length, then the corresponding arcs will be congruent.
Similarly, if arcs are congruent, then the corresponding chords will be equal.
2. The segments of one chord are equal to corresponding segments of the other chord if two equal chords of a circle intersect within the circle.
Theorem # 11
A straight line is drawn cutting two equal circles & passing through the mid-point M of the line joining the centres O & O'. Then the chords AB & CD which are intercepted by the two circles are equal.
Here, AB = CD = 4.45
Theorem # 12
ABC is an equilateral triangle inscribed in a circle. 'D' is a point on minor arc BC. Prove AD = BD + DC
Tangents in Circles
Theorem 1
Only two tangents can be drawn from an external point to a circle.
The length of the two tangents drawn from a common exterior to the circle are equal.
Theorem 2
Relation between Radius & Tangent:
The angle between the radius & tangent is 90 0 at the point of contact.
The line perpendicular to a tangent at the point of tangency will always pass through the centre.
Alternate Segment Theorem (Or) Tangent Chord Theorem
The angle that lies between a chord and a tangent through any one of the end points of the chord is equal to the angle in the alternate segment.
And, Conversly 'Author's Creation in Circles
1. Question:
Two circles with centres at P
and Q are having areas in the
ratio of 2:1. They intersect
each other at A & B. If
∠APB = 2x and ∠AQB = 3x, find
the value of x.
3. Question:
4. Question:
The story behind the following question: Our author's friend Mr. Lakshmana Moorthy has sent a challenging problem on two circles for solving. Our author solved the problem in no time & while sending the solution to his friend, it has become habitual for our author to send one or two & sometimes even more additional results as compliments along with the solution for receiving such challenging questions. The following is one such question.
Two circles with unequal radius intersect each other at A & B and the circumference of the bigger circle passes through the centre of the smaller circle. Through B, a line perpendicular to AB is drawn cutting the circles at M & N. Prove that MN equals the diameter of the bigger circle.
5. Question:
The story behind the following question: Our author's friend Mr. Lakshmana Moorthy found that the angle subtended by PQ = ∠C while drawing the figure in GeoGebra Software but he was not able to prove it theoritically. So he sent the problem to our author for solving it. Here is the solution given by our author.
Two circles with unequal radius intersect each other at A & B and the circumference of the bigger circle passes through the centre of the smaller circle. C is a point on major arc AB of smaller circle such that CA produced cuts a bigger circle at P & CB cuts the minor arc AB of the bigger circle at Q. Prove that the angle borne by PQ on its major arc = ∠C .
6. Question:
ABC is a triangle inscribed in a circle with circumcentre O. Another circle is drawn so as to go through A, O & B. CA & CB produced cut the second circle at P & Q respectively. Prove that the angle subtended by PQ = ∠C.
10RQC
11. Common Tangent Theorem
12. Question:
As shown in the figure below, circles C1 of C2 of radius 360 are tangent to each other, and both tangent to straight l . If circle C3 is tangent to C1, C2 and l ,and circle C4 is tangent to C1, C3 and l , find the radius of C4.
13. Question:
6. Question:
ABC be an acute angled triangle and O be its circumcentre. A circle through the points A, O and B is drawn and CA produced and CB meets the circle at P, Q respectively. Prove that CO & PQ are perpendicular to each other.
8XOY. | 677.169 | 1 |
If A = B = 60°. Verify tan(A−B)=(tanA−tanB)/(1+tantanB)
Text solutionVerified
Given: A = B = 60° ......(1) To verify: tan(A−B)=(tanA−tanB)/(1+tanAtanB) ......(2) Now consider LHS of the expression to be verified in equation (2) Therefore. tan(A−B)=tan(B−B) = tan 0 = 0 Now consider RHS of the expression to be verified in equation (2) Therefore Now by substituting the value of A and B from equation (1) in the above expression We get, (tanA−tanB)/(1+tanAtanB)=(tanB−tanB)/(1+tanBtanB) =0/(1+tan2B) = 0 Hence it is verified that, tan(A−B)=(tanA−tanB)/(1+tantanB) | 677.169 | 1 |
Chapter 3 of Class 8 Maths is called 'Understanding Quadrilaterals'. A quadrilateral is a closed shape and also a type of polygon that has four sides, four vertices and four angles. It is formed by joining four non-collinear points. The sum of all the interior angles of a quadrilateral is always equal to 360 degrees. In a quadrilateral, the sides are straight lines and are two-dimensional. Square, rectangle, rhombus, parallelogram, etc., are examples of quadrilaterals. The formula for the angle sum of a polygon = (n – 2) × 180°.
Extramarks is the best study buddy for students and helps them with comprehensive online study solutions from Class 1 to Class 12. Our team of expert Maths teachers have prepared a variety of NCERT solutions to help students in their studies and exam preparation. Students can refer to our understanding quadrilaterals class 8 extra questions to practise exam-oriented questions. We have collated questions from various sources such as NCERT textbooks and exemplars, CBSE sample papers, CBSE past year question papers, etc. Students can prepare well for their exams and tests by solving a variety of chapter questions from our Important Questions Class 8 Maths Chapter 3.
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Therefore, the adjacent sides of the rectangle are 5 cm and 12 cm, respectively.
That is,
Length =12 cm
Breadth = 5 cm
Length of the diagonal = √( l2 + b2)
= √( 122 + 52)
= √(144 + 25)
= √169
= 13 cm
Hence, the length of the diagonal of a rectangle is 13 cm.
Question 5: How many sides do regular polygons consist of if each interior angle is 165°?
Answer 5: A regular polygon with an interior angle of 165°
We need to find the sides of the given regular polygon:-
The sum of all exterior angles of any given polygon is 360°.
Formula Used: Number of sides = 360∘ /Exterior angle
Exterior angle=180∘−Interior angle
Thus,
Each interior angle =165°
Hence, the measure of every exterior angle will be
=180°−165°
=15°
Therefore, the number of sides of the given polygon will be
=360°/15°
=24°
Question 6: Find x in the following figure.
Answer 6: The two interior angles in the given figures are right angles = 90°
70° + m = 180°
m = 180° – 70°
= 110°
(In a linear pair, the sum of two adjacent angles altogether measures up to 180°)
60° + n = 180°
n = 180° – 60°
= 120°
(In a linear pair, the sum of two adjacent angles altogether measures up to 180°)
The given figure has five sides, and it is a pentagon.
Thus, the sum of the angles of the pentagon = 540°
90° + 90° + 110° + 120° + y = 540°
410° + y = 540°
y = 540° – 410° = 130°
x + y = 180°….. (Linear pair)
x + 130° = 180°
x = 180° – 130°
= 50°
Question 7: ABCD is a parallelogram with ∠A = 80°. The internal bisectors of ∠B and ∠C meet each other at O. Find the measure of the three angles of ΔBCO.
Answer 7:The measure of angle A = 80°.
In a parallelogram, the opposite angles are the same.
Hence,
∠A = ∠C = 80°
And
∠OCB = (1/2) × ∠C
= (1/2) × 80°
= 40°
∠B = 180° – ∠A (the sum of interior angles situated on the same side of the transversal is supplementary)
= 180° – 80°
= 100°
Also,
∠CBO = (1/2) × ∠B
∠CBO= (1/2) × 100°
∠CBO= 50°.
By the property of the sum of the angle BCO, we get,
∠BOC + ∠OBC + ∠CBO = 180°
∠BOC = 180° – (∠OBC + CBO)
= 180° – (40° + 50°)
= 180° – 90°
= 90°
Hence, the measure of all the angles of triangle BCO is 40°, 50° and 90°.
Question 8: The measure of the two adjacent angles of the given parallelogram is the ratio of 3:2. Then, find the measure of each angle of the parallelogram.
Answer 8: A parallelogram with adjacent angles in the ratio of 3:2
To find:- The measure of each of the angles of the parallelogram.
Let the measure of angle A be 3x
Let the measure of angle B be 2x
Since the sum of the measures of adjacent angles is 180° for a parallelogram,
∠A+∠B=180°
3x+2x=180°
5x=180°
x=36°
∠A=∠C =3x=108°
∠B=∠D =2x=72° (Opposite angles of a parallelogram are equal).
Hence, the angles of a parallelogram are 108°, 72°,108°and 72°
Question 9: Is it ever possible to have a regular polygon, each of whose interior angles is 100?
Answer 9: The sum of all the exterior angles of a regular polygon is 360°
As we also know, the sum of interior and exterior angles are 180°
Exterior angle + interior angle = 180-100=80°
When we divide the exterior angle, we will get the number of exterior angles
since it is a regular polygon means the number of exterior angles equals the number of sides.
Therefore n=360/ 80=4.5
And we know that 4.5 is not an integer, so having a regular polygon is impossible.
Whose exterior angle is 100°
Question 10: ABCD is a parallelogram in which ∠A=110°. Find the measure of the angles B, C and D, respectively.
Answer 10: The measure of angle A=110°
the sum of all adjacent angles of a parallelogram is 180°
∠A + ∠B = 180
110°+ ∠B = 180°
∠B = 180°- 110°
= 70°.
Also ∠B + ∠C = 180° [Since ∠B and ∠C are adjacent angles]
70°+ ∠C = 180°
∠C = 180°- 70°
= 110°.
Now ∠C + ∠D = 180° [Since ∠C and ∠D are adjacent angles]
110o+ ∠D = 180°
∠D = 180°- 110°
= 70°
Question11: A diagonal and a side of a rhombus are of equal length. Find the measure of the angles of the rhombus.
Answer 11: Let ABCD be the rhombus.
All the sides of a rhombus are the same.
Thus, AB = BC = CD = DA.
The side and diagonal of a rhombus are equal.
AB = BD
Therefore, AB = BC = CD = DA = BD
Consider triangle ABD,
Each side of a triangle ABD is congruent.
Hence, ΔABD is an equilateral triangle.
Similarly,
ΔBCD is also an equilateral triangle.
Thus, ∠BAD = ∠ABD = ∠ADB = ∠DBC = ∠BCD = ∠CDB = 60°
∠ABC = ∠ABD + ∠DBC = 60° + 60° = 120°
And
∠ADC = ∠ADB + ∠CDB = 60° + 60° = 120°
Hence, all angles of the given rhombus are 60°, 120°, 60° and 120°, respectively.
Question 12: The two adjacent angles of a parallelogram are the same. Find the measure of each and every angle of the parallelogram.
Answer 12: A parallelogram with two equal adjacent angles.
To find:- the measure of each of the angles of the parallelogram.
The sum of all the adjacent angles of a parallelogram is supplementary.
∠A+∠B=180°
2∠A = 180°
∠A = 90°
∠B = ∠A = 90°
In a parallelogram, the opposite sides are the same.
Therefore,
∠C=∠A=90°
∠D=∠B=90°
Hence, each angle of the parallelogram measures 90°.
Question 13: The measures of the two adjacent angles of a parallelogram are in the given ratio 3: 2. Find the measure of every angle of the parallelogram.
Answer 13: Let the measures of two adjacent angles ∠A and ∠B be 3x and 2x, respectively, in parallelogram ABCD.
∠A + ∠B = 180°
⇒ 3x + 2x = 180°
⇒ 5x = 180°
⇒ x = 36°
The opposite sides of a parallelogram are the same.
∠A = ∠C = 3x = 3 × 36° = 108°
∠B = ∠D = 2x = 2 × 36° = 72°
Question 14: State whether true or false.
(a) All the rectangles are squares.
(b) All the rhombuses are parallelograms.
(c) All the squares are rhombuses and also rectangles.
(d) All the squares are not parallelograms.
(e) All the kites are rhombuses.
(f) All the rhombuses are kites.
(g) All the parallelograms are trapeziums.
(h) All the squares are trapeziums.
Answer 14: (a) This statement is false.
Since all squares are rectangles, all rectangles are not squares.
(b) This statement is true.
(c) This statement is true.
(d) This statement is false.
Since all squares are parallelograms, the opposite sides are parallel, and opposite angles are
congruent.
(e) This statement is false.
Since, for example, the length of the sides of a kite is not the same length.
(f) This statement is true.
(g) This statement is true.
(h) This statement is true.
Question 15: Two adjacent angles of a parallelogram are equal. What is the measure of each of these angles?
Answer 15: Let ∠A and ∠B be two adjacent angles.
But we know that the sum of adjacent angles of a parallelogram is 180o
∠A + ∠B = 180°
But given that ∠A = ∠B
Now substituting, we get
∠A + ∠A = 180°
2∠A = 180°
∠A=180/2 = 90°
Question 16:Triangle ABC is a right-angled triangle, and O is the midpoint of the side opposite to the right angle. State why O is equidistant from A, B and C. (The dotted lines are drawn additionally to help you).
Answer 16: AD and DC are drawn in such a way that AD is parallel to BC
and AB is parallel to DC
AD = BC and AB = DC
ABCD is a rectangle since the opposite sides are equal and parallel to each other, and the measure of all the interior angles is altogether 90°.
In a rectangle, all the diagonals bisect each other and are of equal length.
Answer 17: (i) Yes, the quadrilateral ABCD can be a parallelogram if ∠D + ∠B = 180° but it should also fulfil certain conditions, which are as follows:
(a) The sum of all the adjacent angles should be 180°.
(b) Opposite angles of a parallelogram must be equal.
(ii) No, opposite sides should be of the same length. Here, AD ≠ BC
(iii) No, opposite angles should be of the same measures. ∠A ≠ ∠C
Question 18: Find the measure of angles P and S if SP and RQ are parallel.
Answer 18: ∠P + ∠Q = 180° (angles on the same side of transversal)
∠P + 130° = 180°
∠P = 180° – 130° = 50°
also, ∠R + ∠S = 180° (angles on the same side of transversal)
⇒ 90° + ∠S = 180°
⇒ ∠S = 180° – 90° = 90°
Thus, ∠P = 50° and ∠S = 90°
Yes, there is more than one method to find m∠P.
PQRS is a quadrilateral. The sum of measures of all angles is 360°.
Since we know the measurement of ∠Q, ∠R and ∠S.
∠Q = 130°, ∠R = 90° and ∠S = 90°
∠P + 130° + 90° + 90° = 360°
⇒ ∠P + 310° = 360°
⇒ ∠P = 360° – 310° = 50°
Question 19: The opposite angles of a parallelogram are (3x + 5)° and (61 – x)°. Find the measure of four angles.
Answer 19: (3x + 5)° and (61 – x)° are the opposite angles of a parallelogram.
The opposite angles of a parallelogram are the same.
Therefore, (3x + 5)° = (61 – x)°
3x + x = 61° – 5°
4x = 56°
x = 56°/4
x = 14°
The first angle of the parallelogram =3x + 5
= 3(14) + 5
= 42 + 5 = 47°
The second angle of the parallelogram=61 – x
= 61 – 14 = 47°
The measure of angles adjacent to the given angles = 180° – 47° = 133°
Hence, the measure of the four angles of the parallelogram is 47°, 133°, 47°, and 133°.
Question 20:What is the maximum exterior angle possible for a regular polygon?
Answer 20: To find:- The maximum exterior angle possible for a regular polygon.
A polygon with minimum sides is an equilateral triangle.
So, the number of sides =3
The sum of all exterior angles of a polygon is 360°
Exterior angle =360°/Number of sides
Therefore, the maximum exterior angle possible will be
=360°/3
=120°
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In convex cyclic quadrilateral $ABCD$, we know that lines $AC$ and $BD$ intersect at $E$, lines $AB$ and $CD$ intersect at $F$, and lines $BC$ and $DA$ intersect at $G$. Suppose that the circumcircle of $\triangle ABE$ intersects line $CB$ at $B$ and $P$, and the circumcircle of $\triangle ADE$ intersects line $CD$ at $D$ and $Q$, where $C,B,P,G$ and $C,Q,D,F$ are collinear in that order. Prove that if lines $FP$ and $GQ$ intersect at $M$, then $\angle MAC = 90^\circ$.
Proof : angle chase! Note that for this to be true , it is enough to show that $\angle AEB=\angle AQC$ or it is enough to show that $\angle AEB=\angle AQC $ or it is enough to show that $\angle AED=\angle AQD$ which is true since $AEDQ$ is cyclic.
Claim: $E\in PQ$
Proof: So enough to show that $\angle AEQ+\angle AEP=180 $
or enough to show that $180- \angle ADC + \angle AEP=180 $
or enough to show that $\angle ADC= \angle ABC$ , which is true since $ABCD$ is cyclic.
after that I am stuck.
I observed that $FG , AM, PQ$ concur but was not able to prove. Can someone give hints?
$\begingroup$@Shubhangi, yes. I am interested in what math competitors learn in other countries. For example, I qualified for the state (Croatia) competition this year, but I learned Calculus 1 and basics of Calculus 2 on my own this summer (when I said learned I meant on understanding all rules, theorems, etc.). Also a lot of other topics are not even mentioned before college. But I think olympiad approach is better because it develops problem solving skills which is ground for math research$\endgroup$
1 Answer
1
So, we have $PBDQ$ cyclic and $E\in PQ$. Now focus on quadrilateral $PBDQ$. From definition $A$ is the Miquel Point of the quadrilateral $PBDQ$. Now let $X:=PD\cap BQ$ and thus, by Miquel point properties, we get that $A$ is projection of $X$ on $CE$. Thus, its enough to show that $M,A,X$ are collinear but this is trivial. Just apply Pappus Theorem on $\{PGB,QFD\}$ completing the proof. $\blacksquare$
$\begingroup$Thank you so much ! I saw your profile , OMG , congrats for INMO, sharygin and IGO . You are so Pro !!! Can you give me some Geo and NT advice , please ? Plus Can I pm you in AOPS if I get any doubt ?$\endgroup$
$\begingroup$@Shubhangi GOD, you won't believe but I got two major hints for that... 1) You put a space in front and after every comma or question mark or exclamation mark 2) You are the only AoPSer who know bout SOR (expect Muler)$\endgroup$ | 677.169 | 1 |
2 Answers
2
There's an implied claim in the second block, that $\sqrt[3]{\sin A\sin B\sin C}\ge \frac{\sqrt{3}}{2}$. That claim is false. In fact, $\sqrt[3]{\sin A\sin B\sin C}\le \frac{\sqrt{3}}{2}$ with equality only when $A=B=C=60^\circ$. In an acute triangle, that quantity can get arbitrarily close to zero - consider a triangle with angles $\epsilon, 90^\circ-\frac{\epsilon}{2}, 90^\circ-\frac{\epsilon}{2}$. The product of sines in that triangle is less than $\sin\epsilon$, which goes to zero as $\epsilon\to 0$.
Naturally, the arguments that follow from that don't work.
I take it (from the last inequality claimed in the second block) you were asked to prove that $\sin A+\sin B+\sin C > 2$ in an acute triangle? That's true, and it's as strong as we can possibly have. Equality there is approached by the triangle I mentioned, in which the sines approach $0,1,1$.
In an acute angled triangle, it should be noted that $$\sqrt[3]{ sinA sinB sinC} \geq \ {\sqrt{3}\over2}$$ implies that the angles are $ \geq 60$. In this case, the sum of the angles are going to be $\geq 180$. So, such a triangle will not exist (except for the case of an equilateral triangle, where your proposition seems right). | 677.169 | 1 |
Elements of Geometry
whence it follows, that the squares of four proportional quantities form a new proportion.
that is, the cubes of four proportional quantities form a new proportion.
VI. When a proportion is said to exist among certain magnitudes, these magnitudes are supposed to be represented, or to be capable of being represented, by numbers; if, for example, in the proportion
A:B::C: D,
A, B, C, D, denote certain lines, we can always suppose one of these lines, or a fifth, if we please, to answer as a common measure to the whole, and to be taken for unity; then A, B, C, D, will each represent a certain number of units, entire or fractional, commensurable or incommensurable, and the proportion among the lines A, B, C, D, becomes a proportion in numbers.
Hence the product of two lines, A and D, which is called also their rectangle, is nothing else than the number of linear units. contained in A multiplied by the number of linear units contained in B; and we can easily conceive this product to be equal to that which results from the multiplication of the lines B and C.
The magnitudes A and B, in the proportion
A: B: CD,
may be of one kind, as lines, and the magnitudes C and D of
another kind, as surfaces; still these magnitudes are always to be regarded as numbers; A and B will be expressed in linear units, C and D in superficial units, and the product A x D will be a number, as also the product B x C.
Indeed, in all the operations, which are made upon proportional quantities, it is necessary to regard the terms of the proportion as so many numbers, each of its proper kind; then we shall have no difficulty in conceiving of these operations and of the consequences which result from them.
ELEMENTS OF GEOMETRY.
Definitions and Preliminary Remarks.
1. GEOMETRY is a science which has for its object the measure of extension.
Extension has three dimensions, length, breadth, and thick
ness.
2. A line is length without breadth.
The extremities of a line are called points. A point, therefore, has no extension.
3. A straight or right line is the shortest way from one point to another.
4. Every line which is neither a straight line, nor composed of straight lines, is a curved line.
Thus AB (fig. 1) is a straight line, ACDB is a broken line, or Fig. 1. one composed of straight lines, and AEB is a curved line.
5. A surface is that which has length and breadth, without thickness.
6. A plane is a surface in which, any two points being taken, the straight line joining those points lies wholly in that surface. 7. Every surface which is neither a plane, nor composed of planes, is a curved surface.
8. A solid is that which unites the three dimensions of ex
ension.
9. When two straight lines AB, AC, (fig. 2,) meet, the quan- Fig. 2. tity, whether greater or less, by which they depart from each other as to their position, is called an angle; the point of meeting or intersection, A, is the vertex of the angle; the lines AB, AC, are its sides.
An angle is sometimes denoted simply by the letter at the vertex, as A; sometimes by three letters, as BAC or CAB, the letter at the vertex always occupying the middle place.
Angles, like other quantities, are susceptible of addition, subtraction, multiplication, and division; thus, the angle DCE Fig. 20Fig. 6.
Fig. 7.
Fig. 8. Fig. 9, is an obtuse angle.
12. Two lines are said to be parallel (fig. 5), when, being situated in the same plane, and produced ever so far both ways, they do not meet.
13. A plane figure is a plane terminated on all sides by lines. If the lines are straight, the space which they contain is called a rectilineal figure, or polygon (fig. 6), and the lines taken together make the perimeter of the polygon three sides are equal, isosceles (fig. 8), when two only of its sides,
The square (fig. 11), which has its sides equal, and its angles right angles, (See art. 80);
The rectangle (fig. 12), which has its angles right angles, without having its sides equal (See art. above referred to);
The parallelogram (fig. 13), which has its opposite sides parallel;
The rhombus or lozenge (fig. 14), which has its sides equal, without having its angles right angles; | 677.169 | 1 |
Options
A circle is a curved two-dimensional shape in which all points on the curve are the same distance from its center. Images with this tag include one or more circular shapes as a major part of the image. | 677.169 | 1 |
Once you have angles A, B, and C, you can use the fact that the sum of the angles in a triangle is 180 degrees to find the measure of the missing angle.
To find the length of side c, you can use the Law of Cosines, as you have the lengths of sides a and b and the measure of angle C | 677.169 | 1 |
Sense 1 inclination, angle of inclination -- ((geometry) the angle formed by the x-axis and a given line (measured counterclockwise from the positive half of the x-axis)) => angle -- (the space between two lines or planes that intersect; the inclination of one line to another; measured in degrees or radians) | 677.169 | 1 |
Inscribe a regular 17-gon in a circle of radius 1. Fix one vertex, and draw the chords from that vertex to each of the others. The product of those chords' lengths is 17, the number of sides! The same is true for any number n of sides greater than 1. The proof in the image uses precalculus-level algebra, and properties of complex numbers: How a monic polynomial (whose top-degree coefficient is 1) factors in terms of its roots, and the way a difference of n-th powers factors.
Printed on archival quality Hahnemühle German etching paper (310gsm) with a velvety surface. | 677.169 | 1 |
trigonometry to is trigonometry using these relationships to solve for unknown sides or angles in a triangle. Trigonometry is widely used in various fields such as physics, engineering, and astronomy to calculate distances, heights, and angles. It is an essential tool for understanding and solving problems related to triangles and periodic phenomena.
What is trigonometry and how are triangles used in trigonometry?
Trigonometry is a branch of mathematics that deals with the study of the relationships between the angles and sides of triangles....
Trigonometry is a branch of mathematics that deals with the study of the relationships between the angles and sides of triangles. It involves the use of trigonometric functions such as sine, cosine, and tangent to solve for unknown angles or sides of a triangle. Triangles are used in trigonometry because they provide a simple and fundamental geometric shape to study the relationships between angles and sides. By understanding the properties of triangles and applying trigonometric functions, we can solve various real-world problems involving distances, heights, and angles.
Source:AI generated from FAQ.net
What is trigonometry 116?
Trigonometry 116 is a course that builds upon the fundamental concepts of trigonometry, such as angles, triangles, and trigonometr...
Trigonometry 116 is a course that builds upon the fundamental concepts of trigonometry, such as angles, triangles, and trigonometric functions, and delves deeper into more advanced topics. Students in this course typically study topics like trigonometric identities, inverse trigonometric functions, trigonometric equations, and applications of trigonometry in real-world problems. The course aims to provide a more comprehensive understanding of trigonometry and its relevance in various fields such as physics, engineering, and mathematics.
What is trigonometry 106?
Trigonometry 106 is an advanced course that builds upon the foundational concepts of trigonometry. It typically covers more comple...
Trigonometry 106 is an advanced course that builds upon the foundational concepts of trigonometry. It typically covers more complex topics such as inverse trigonometric functions, trigonometric identities, and applications of trigonometry in real-world problems. Students in Trigonometry 106 may also delve into topics like polar coordinates, vectors, and complex numbers as they relate to trigonometric functions. Overall, Trigonometry 106 aims to deepen students' understanding of trigonometry and its applications in various fields is trigonometry classified?
Trigonometry is classified as a branch of mathematics that deals with the relationships between the sides and angles of triangles....
Trigonometry is classified as a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is further categorized into two main branches: plane trigonometry, which focuses on two-dimensional triangles, and spherical trigonometry, which deals with triangles on the surface of a sphere. Trigonometry is also divided into various functions such as sine, cosine, and tangent, which are used to calculate unknown sides or angles in a triangle. Overall, trigonometry plays a crucial role in various fields such as physics, engineering, and astronomy.
What is trigonometry 105?
Trigonometry 105 is an advanced level course that builds upon the fundamental concepts of trigonometry. It covers more complex top...
Trigonometry 105 is an advanced level course that builds upon the fundamental concepts of trigonometry. It covers more complex topics such as inverse trigonometric functions, trigonometric identities, and applications of trigonometry in real-world problems. Students in Trigonometry 105 will delve deeper into the relationships between angles and sides of triangles, as well as learn advanced techniques for solving trigonometric equations and proving trigonometric identities. Overall, Trigonometry 105 aims to provide a comprehensive understanding of trigonometry and its applications in various fields.
Can you explain trigonometry?
Trigonometry is a branch of mathematics that deals with the study of angles and the relationships between the sides and angles of...
Trigonometry is a branch of mathematics that deals with the study of angles and the relationships between the sides and angles of triangles. It involves the use of trigonometric functions such as sine, cosine, and tangent to solve for unknown sides and angles in a triangle. Trigonometry is widely used in various fields such as physics, engineering, and astronomy to calculate distances, angles, and other measurements. It is an essential tool for understanding and solving problems related to triangles and periodic phenomena.
Source:AI generated from FAQ.net
What is Trigonometry 103?
Trigonometry 103 is an advanced level course in trigonometry that builds upon the concepts learned in Trigonometry 101 and 102. It...
Trigonometry 103 is an advanced level course in trigonometry that builds upon the concepts learned in Trigonometry 101 and 102. It covers more complex topics such as advanced trigonometric functions, inverse trigonometric functions, trigonometric identities, and applications of trigonometry in real-world problems. The course may also delve into topics such as trigonometric equations, polar coordinates, and complex numbers. Trigonometry 103 is typically taken by students majoring in mathematics, engineering, or physics, and provides a deeper understanding of trigonometric principles and their applicationsierra is trigonometry in mathematics the use of trigonometric ratios such as sine, cosine, and tangent. Trigonometry is widely used in various fields such as physics, engineering, and astronomy to solve problems related to angles and distances. It plays a crucial role in calculating distances, heights, and angles in real-world applications.
How is trigonometry played out?
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is used to...
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is used to solve problems involving angles and distances in various fields such as physics, engineering, and astronomy. Trigonometry is played out by using trigonometric functions such as sine, cosine, and tangent to calculate unknown angles or side lengths in a triangle. These functions are applied to real-world situations to analyze and solve problems involving right-angled triangles or periodic phenomena.
How do I calculate trigonometry?
To calculate trigonometry, you can use the sine, cosine, and tangent functions to find the relationships between the angles and si...
To calculate trigonometry, you can use the sine, cosine, and tangent functions to find the relationships between the angles and sides of a right-angled triangle. To find the sine of an angle, divide the length of the side opposite the angle by the length of the hypotenuse. To find the cosine, divide the length of the adjacent side by the length of the hypotenuse. And to find the tangent, divide the length of the opposite side by the length of the adjacent side. You can also use trigonometric identities and formulas to solve more complex trigonometric problems | 677.169 | 1 |
The Elements of Euclid, books i. to vi., with deductions, appendices and historical notes, by J.S. Mackay. [With] Key
From inside the book
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Page 81 ... middle points of two sides of a triangle is the third side , and half of it . = 2. Hence prove that the straight line joining the middle point of the hypotenuse of a right - angled triangle to the opposite half the hypotenuse . vertex ...
Page 93 ... middle points of the sides of the square BCED draw parallels to AB and AC as in the figure . Then the parts 1 , 2 , 3 , 4 , 5 will be found to coincide exactly with 1 ' , 2 ′ , 3 ′ , 4 ' , 5 ' . [ This method is due to Henry Perigal ...
Page 98 ... middle points of any two sides of a triangle is parallel to the third side and equal to the half of it . A L K B H Let ABC be a triangle , and let L , K be the middle points of AB , AC : it is required to prove LK || BC and = Join BK ...
Page 99 Euclides John Sturgeon Mackay. F is the middle point of CD , and EF is equal either to half the sum of AC and BD , or to half their difference . PROPOSITION 2 . The straight lines drawn ... middle points Book I. ] 99 APPENDIX I.
Popular passages
Page 147 - A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.
Page 276 - IF there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. NB This is usually cited by the words
Page 331 - If the vertical angle of a triangle be bisected by a straight line which also cuts the base, the segments of the base shall have the same ratio which the other sides of the triangle have to one another...
Page 112 - If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the aforesaid part.
Page 87 254 - If there be four magnitudes, and if any equimultiples whatsoever be taken of the first and third, and any equimultiples whatsoever of the second and fourth, and if, according as the multiple of the first is greater than the multiple of the second, equal to it or less, the multiple of the third is also greater than the multiple of the fourth, equal to it or less ; then, the first of the magnitudes is said to have to the second the same ratio that the third has to the fourth.
Page 138 - RULE. from half the sum of the three sides, subtract each side separately; multiply the half sum and the three remainders together, and the square root of the product will be the area required. | 677.169 | 1 |
1 Answer
(i) Ranging rod: A ranging rod is a long, slender rod that is used in surveying to mark the location of points on the ground. It is typically marked with horizontal and vertical lines or other markings, and it is used to line up the survey instrument with the point being measured.
(ii) Optical square: An optical square is a surveying instrument that is used to measure angles and distances. It consists of a small telescope mounted on a tripod, and it is used to sight on objects or points in the distance and measure their distance and angle relative to the instrument.
(iii) Steel tape: A steel tape is a long, narrow strip of steel or other metal that is used to measure distances. It is typically marked with graduations or other markings to allow precise measurements to be taken. Steel tapes are often used in surveying and construction to measure distances and dimensions.
(iv) Abney level: An Abney level is a surveying instrument that is used to measure the slope or incline of a surface. It consists of a small telescope mounted on a handle, and it is used to sight on objects or points in the distance and measure the angle of the slope.
(v) Levelling staff: A levelling staff is a long, slender rod that is used in surveying to determine the elevation of points on the ground. It is typically marked with graduations or other markings that allow precise measurements to be taken, and it is used in conjunction with a levelling instrument, such as a transit or theodolite. | 677.169 | 1 |
Abstract
In this paper, we prove several inequalities in the acute triangle by means of so-called
Difference Substitution. As generalization of the method, we also consider an example that the
greatest interior angle is less than or equal to 120° in the triangle | 677.169 | 1 |
John Adney
The answer is clearly 2- if one side is 11 and the other is 7 the 3rd side must be long enough to reach look at the possible length of the 3rd side- if the 7 is on one side of the 11 side and the x is on the other if x were 2 then even with angels of 0 degrees the 7 side would not connect with the x side to make a triangle the sum of the 2 smaller sides must equal more than the largest side 7+2 <11 | 677.169 | 1 |
TS ICET 27th July 2022 Shift-2
In the following questions numbered 1 to 20, a question is followed by data in the form of two statements labelled as (I) and (II). You must decide whether the data given in the statements are sufficient to answer the questions.
Question 1
If a , b and c are sides of a triangle, is it a right angled triangle? (I) a < b + c. (II) a., b, c are consecutive integers. | 677.169 | 1 |
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Similar Triangles
Triangles that are congruent are exactly the same. If $\triangle ABC \cong \triangle DEF$ (this is the notation that shows that triangles are congruent), then all of the sides and angles in $\triangle ABC$ are equal to all of the corresponding sides and angles in $\triangle DEF$.
Triangles that are similar have corresponding angles with the same angle measures and corresponding sides that are proportional. If $\triangle ABC \sim \triangle DEF$ (this is the notation that shows that triangles are similar), then all of the angles in $\triangle ABC$ are equal to all of the corresponding angles in $\triangle DEF$, and all of the sides in $\triangle ABC$ are proportional to all of the corresponding sides in $\triangle DEF$.
Consequently, when you know that triangles are similar, it's easy to deduce their angle measures (just make sure that you match up the correct angles when you set up your proportions!).
So if you know that in the figure below, $\triangle ABC \sim \triangle DEF$, then you know that:
Often, tests won't tell you that triangles are similar. You're supposed to remember a geometry rule that says that when a triangle is intersected by a line that is parallel to one of its sides, then the two triangles that are formed (the entire triangle and the smaller one made by the parallel line) are similar.
In the to figure to the right, if $\overline{DE} \parallel \overline{AC}$, then $\triangle ABC \sim \triangle DBE$.
So, you know that the corresponding angles of $\triangle ABC$ equal the corresponding angles of $\triangle DBE$ and the corresponding sides of $\triangle ABC$ are proportional to the corresponding sides of $\triangle DBE$
Note: a line does not have to be parallel to the base in order to create similar triangles. Any line that is parallel to any of a triangle's sides will form similar triangles. In the lower figure to the right, if $\overline{DE} \parallel \overline{AC}$, then $\triangle ABC \sim \triangle DBE$ and corresponding angles are equal and corresponding sides are proportional. | 677.169 | 1 |
Class 8 Courses
Constructions Class 9 Maths All FormulasHey, students are you looking for Constructions Class 9 Maths all Formulas? If yes. Then you are at the right place.
In this post, I have listed all the formulas of Constructions Class 9 that you can use to learn and understand the concepts easily.
If you want to improve your class 9 Math, Constructions concepts, then it is super important for you to learn and understand all the formulas.
By using these formulas you will learn about the Constructions concepts for class 9.
With the help of these formulas, you can revise the entire chapter easily.
Consturctions Class 9 Maths all Formulas
A graduated scale, on one side of which centimeters and millimeters are marked off and on the other side inches are marked off.
A pair of set-squares, one with angles $90^{\circ}, 60^{\circ}$ and $30^{\circ}$ and other with angles $90^{\circ}, 45^{\circ}$ and $45^{\circ}$
A divider.
A compass with provision of fitting a pencil at one end.
A protractor.
But a geometrical construction is the process of drawing a geometrical figure using only two instruments-an un graduated ruler and a compass. In construction where measurements are also required, one may use a graduated scale and protractor also.
Basic Constructions
To construct the bisector of a given angle.
To construct the perpendicular bisector of a given line segment.
To construct and angle of $30^{\circ}, 45^{\circ}, 60^{\circ}, 90^{\circ}, 120^{\circ} \mathrm{m}$ (the initial point of a given ray.
Construction of Triangles
To construct a triangle given its base, a base angle and sym of the other two sides.
To construct a triangle given its base, a base angle and the difference of the other two sides. | 677.169 | 1 |
Students can access the NCERT MCQ Questions for Class 10 Maths Chapter 9 Some Applications of Trigonometry with Answers Pdf free download aids in your exam preparation and you can get a good hold of the chapter. Use MCQ Questions for Class 10 Maths with Answers during preparation and score maximum marks in the exam. Students can download the Some Applications of Trigonometry Class 10 MCQs Questions with Answers from here and test their problem-solving skills. Clear all the fundamentals and prepare thoroughly for the exam taking help from Class 10 Maths Chapter 9 Some Applications of Trigonometry Objective Questions.
Some Applications of Trigonometry Class 10 MCQs Questions with Answers
Students are advised to solve the Some Applications of Trigonometry Multiple Choice Questions of Class 10 Maths to know different concepts. Practicing the MCQ Questions on Some Applications of Trigonometry Class 10 with answers will boost your confidence thereby helping you score well in the exam.
Explore numerous MCQ Questions of Some Applications of Trigonometry Class 10 with answers provided with detailed solutions by looking below.
Question 1.
The tops of two poles of height 16m and 10m are connected by a wire. If the wire makes an angle of 60° with the horizontal, then the length of the wire is
(a) 10m
(b) 12m
(c) 16m
(d) 18m
Answer
Answer: (b) 12m
Question 2.
A 20 m long ladder touches the wall at a height of 10 m. The angle which the ladder makes with the horizontal is
(a) 450
(b) 300
(c) 900
(d) 600
Answer
Answer: (b) 300
Question 3.
If the length of the shadow of a tower is √3 times that of its height, then the angle of elevation of the sun is
(a) 30°
(b) 45°
(c) 60°
(d) 75°
Question 5.
The angle of elevation of top of a tower from a point on the ground, which is 30 m away from the foot of the tower is 30°. The length of the tower is
(a) √3 m
(b) 2√3 m
(c) 5√3m
(d) 10√3 m
Answer
Answer: (d) 10√3 m
Question 6.
A contractor planned to install a slide for the children to play in a park. If he prefers to have a slide whose top is at a height of 1.5m and is inclined at an angle of 30° to the ground, then the length of the slide would be
(a) 1.5m
(b) 2√3m
(c) √3m
(d) 3m
Answer
Answer: (d) 3m
Question 7.
When the length of shadow of a vertical pole is equal to √3 times of its height, the angle of elevation of the Sun's altitude is
(a) 30°
(b) 45°
(c) 60°
(d) 15
Answer
Answer: (a) 30°
Question 8.
From a point P on the level ground, the angle of elevation of the top of a tower is 30°. If the tower is 100m high, the distance between P and the foot of the tower is
(a) 100√3m
(b) 200√3m
(c) 300√3m
(d) 150√3m
Answer
Answer: (a) 100√3m
Question 9.
When the sun's altitude changes from 30° to 60°, the length of the shadow of a tower decreases by 70m. What is the height of the tower?
(a) 35 m
(b) 140 m
(c) 60.6 m
(d) 20.2 m
Answer
Answer: (c) 60.6 m
Question 10.
The ___________ of an object is the angle formed by the line of sight with the horizontal when the object is below the horizontal level.
(a) line of sight
(b) angle of elevation
(c) angle of depression
(d) none of these
Answer
Answer: (c) angle of depression
Question 11.
In given Fig., the angle of depression from the observing position D and E of the object at A are
(a) 60°, 60°
(b) 30°, 30°
(c) 30°, 60°
(d) 60°, 30°
Answer
Answer: (c) 30°, 60°
The application above returns a chart and a list illustrating the step-by-step process to round to the nearest tenth (one decimal place).
Question 12.
Guddi was standing on a road near a mall. She was 1000m away from the mall and able to see the top of the mall from the road in such a way that top of the tree, which is in between her and the mall, was exactly in line of sight with the top of the mall. The tree height is 10m and it is 20m away from Guddi. How tall is the mall?
(a) 453m
(b) 856m
(c) 290m
(d) 470m
Answer
Answer: (d) 470m
Question 13.
A tower stands vertically on the ground. From a point C on the ground, which is 20 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 450. The height of the tower is
(a) 15 m
(b) 8 m
(c) 20 m
(d) 10 m
Answer
Answer: (a) 15 m
Question 14.
If a kite is flying at a height of 10√3m from the level ground attached to a string inclined at 60° to the horizontal then the length of the string is
(a) 20m
(b) 40√3m
(c) 60√3m
(d) 80√3m
Answer
Answer: (a) 20m
Question 15.
A tower stands vertically on the ground from a point on the ground which is 25 m away from the foot of tower if the height of tower is 25√3 metres find the angle of elevation.
(a) 60°
(b) 30°
(c) 120°
(d) 90°
Answer
Answer: (a) 60°
Question 16.
A circus artist is climbing a long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. The ratio of the height of the pole to the length of the string is 1 :√2. The angle made by the rope with the ground level is
(a) 30°
(b) 45°
(c) 60°
(d) none of these
Answer
Answer: (b) 45°
Question 17.
A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. If the angle made by the rope with the ground level is 30°, then the height of the pole is
(a) 10m
(b) 20m
(c) 10√3m
(d) 20√3m
Answer
Answer: (a) 10m
Question 18.
If the angle of depression of an object from a 75m high tower is 30°, then the distance of the object from the tower is
(a) 50√3m
(b) 25√3m
(c) 75√3m
(d) 100√3m
Answer
Answer: (c) 75√3m
Question 19.
The angle of elevation of the sun, when the length of the shadow of a tree is equal to the height of the tree, is:
(a) 45°
(b) 60°
(c) 30°
(d) None of these
Answer
Answer: (a) 45°
Question 20.
A tower stands vertically on the ground. From a point on the ground 30 m away from the foot of the tower, the angle of elevation of the top of the tower is 45o. The height of the tower will be
(a) 30√3 m
(b) 40√3 m
(c) 30 m
(d) 40 m
Answer
Answer: (c) 30 m
Question 21.
An observer 1.5m tall is 23.5m away from a tower 25m high. The angle of elevation of the top of the tower from the eye of the observer is
(a) 30°
(b) 45°
(c) 60°
(d) none of these
Answer
Answer: (b) 45°
Question 22.
The shadow of a tower is equal to its height at 10-45 a.m. The sun's altitude is
(a) 30°
(b) 45°
(c) 60°
(d) 90°
Answer
Answer: (b) 45°
We believe the knowledge shared regarding NCERT MCQ Questions for Class 10 Maths Chapter 9 Some Applications of Trigonometry with Answers Pdf free download has been useful to the possible extent. If you have any other queries regarding CBSE Class 10 Maths Some Applications of Trigonometry MCQs Multiple Choice Questions with Answers, feel free to reach us via the comment section and we will guide you with the possible solution. | 677.169 | 1 |
how do you determine if triangles are congruent
Determine If Triangles Are Congruent Worksheet – Triangles are among the most fundamental patterns in geometry. Understanding triangles is vital to mastering more advanced geometric concepts. In this blog this post, we'll go over the different types of triangles such as triangle angles, and how to calculate the perimeter and area of a triangle and will provide the examples for each. Types of Triangles There are three kinds in triangles, namely equilateral isosceles, as well as … Read more | 677.169 | 1 |
A parallelogram has sides A, B, C, and D. Sides A and B have a length of #6 # and sides C and D have a length of # 3 #. If the angle between sides A and C is #(3 pi)/4 #, what is the area of the parallelogram?
A_p Area of parallelogram, a = lengths of sides A & B, b = lengths of sides C & D
#a = 6, b = 3, theta = (3pi)/4#
#A_p = 6 * 3 * sin ((3pi)/4)#
#A_p = 18/sqrt2 = 9sqrt2 = 12.73# sq units | 677.169 | 1 |
Reference ID: #46841
Impossible angles
Exploring the mysterious world of impossible angles, where reality bends and twists in ways that defy logic and reason, revealing hidden dimensions and mind-bending illusions that challenge our perception of space and time. | 677.169 | 1 |
Understanding Axis Of Symmetry In Geometry
You may not have thought much about the axis of symmetry as an engineering student or engineer.
But this simple but powerful idea is at the heart of many important applications in your field, from designing bridges and buildings to making advanced electronics and medical devices.
If you know what the axis of symmetry is and how it relates to geometric shapes and functions, you can open up a whole world of new ideas and ways of thinking.
In this blog post, I will talk in depth about the axis of symmetry and show how it applies to your work as an engineer.
So get ready to see the world in a whole new way.
Introduction to Axis of Symmetry in Geometry
Formal definition:
An imaginary line about which a geometrical figure is symmetric.
The axis of symmetry is an important concept in geometry.
It is a key part of making shapes and objects that are balanced and have symmetry.
In this article, we will talk about what the axis of symmetry is and how it can be used in geometry, especially with quadratic functions.
Definition of Axis of Symmetry
The axis of symmetry is a line that cuts an object in half so that each side looks like a mirror image of the other side.
It is an imaginary straight line that goes through the middle of a shape or object and divides it into two identical parts, with one part being the mirror image of the other.
When the paper is folded along the axis of symmetry, the two parts line up perfectly.
The Significance of Axis of Symmetry in Geometrical Figures and Functions
Applications of Axis of Symmetry in Geometrical Figures
Regular Polygons: If a polygon has n sides, then it will also have n axes of symmetry.
You can use these axes of symmetry to divide the polygon into identical parts, which makes it easier to figure out what its properties are.
Parabolas: In standard form, where y = ax2 + bx + c, the equation for the axis of symmetry is x = -b/2a.
This formula is used to find the x-coordinate of the point on the axis of symmetry where the vertex of the parabola is.
When it comes to making a point: You can also find out if a graph is symmetrical about a point by rotating it 180° around that point.
If the graph stays the same after the rotation, it is symmetrical about that point.
Using this property, you can find symmetrical parts in different shapes and functions.
Symmetry of Functions
Functions can be symmetrical about the y-axis, which means that if you flip their graph around the y-axis, it will look the same.
This is called "even symmetry," and the function f(-x) = f is used to show it (x).
Also, functions can be symmetrical about the origin, which means that if the graph is turned 180° around the origin, it will look the same.
This is called "odd symmetry," and the function that shows it is f(-x) = -f (x).
Understanding the Differences between Axis of Symmetry of a Parabola and a Hyperbola
In math, two of the most common types of conic sections are parabolas and hyperbolas.
Even though both shapes have their own axis of symmetry, they are not the same in many ways.
Axis of Symmetry of a Parabola
A parabola's axis of symmetry is a line that goes through the focus and is parallel to the directrix.
A hyperbola has more than one curve, but a parabola only has one curve and no asymptotes.
It also opens less than a hyperbola.
A parabola has an eccentricity value of 1, and no matter how big or small it is, it always has the same shape.
Axis of Symmetry of a Hyperbola
Some lines that go through the center of a hyperbola are asymptotic.
Unlike a parabola, it has two curves that are mirror images of each other and open in opposite directions.
The center of a hyperbola is the point halfway between its two points.
The part of a line that goes through the points of a hyperbola is called its axis.
Its conjugate axis is the part of a line that goes through the center and is perpendicular to the transverse axis.
Formation of Parabolas and Hyperbolas
When a plane cuts through both halves of a cone at an angle greater than the slope of the cone, it makes a hyperbola.
On the other hand, parabolas are made when planes meet cones that are parallel to one side.
Differences in Eccentricity and Focus Points
The main difference between a parabola and a hyperbola is the value of their eccentricity.
Eccentricity is equal to 1 for parabolas and greater than 1 for hyperbolas.
A hyperbola has two focus points, one on each side of its center.
A parabola only has one.
Equation of a Parabola and its Relationship to Axis of Symmetry
In the study of parabolas, the axis of symmetry is an important idea.
It is a line that splits a parabola into two parts that are the same size and shape as each other.
Axis of Symmetry of a Parabola
A parabola has an axis of symmetry that is a straight line that goes through the point of the parabola.
The equation of the axis of symmetry is the x-coordinate of the point where the two lines meet.
The equation for the axis of symmetry for a quadratic function in standard form, y = ax2 + bx + c, is x = -b/2a.
Properties of the Axis of Symmetry
The axis of symmetry is the line that divides a parabola into two halves that are the same size and shape as each other.
The point where the axis of symmetry and the parabola meet is called the vertex.
If a parabola opens up or down, its axis of symmetry is vertical, and its equation is a vertical line that goes through its vertex.
If it opens to the left or right, it has a horizontal axis of symmetry, and its equation is a horizontal line that goes through its point.
Equation of a Parabola
In standard form, the equation for a parabola is y = ax2 + bx + c.
Whether the parabola opens up or down depends on the coefficient "a."
If an is positive, the parabola opens up.
If an is negative, the parabola opens down.
The point where the parabola starts and ends is (-b/2a, c - b2/4a).
This is the point where the axis of symmetry of the parabola goes through.
How to Find the Axis of Symmetry of a Parabola or Quadratic Function
Finding the Vertex
The point where a parabola or quadratic function meets its axis of symmetry is called the vertex.
To get from standard form to vertex form, you can use the "completing the square" method to find it.
A quadratic function looks like this: y = ax2 + bx + c.
The vertex form is y = a(x - h)2 + k.
Follow these steps to find the point.
To find the x-coordinate of the vertex, divide the coefficient of the x-term (b) by 2a: h = -b/2a.
Put the value of h into the original equation, k = a(h)2 + b(h) + c, to find the y-coordinate of the point.
Finding the Axis of Symmetry
Once you know where the vertex is (h, k), you can find the equation for the axis of symmetry by substituting h into the formula x = -b/2a.
The equation will be the vertical line that goes through the vertex and divides the parabola into two equal halves.
Finding the Intercepts
If you solve for x and y in the equation y = ax2 + bx + c, you can find the intercepts of a parabola or quadratic function.
Set y to 0 and solve for x to find the x-intercepts.
Set x to 0 and solve for y to find the y-interceptDetermining the Axis of Symmetry of a Function from its Graph and using Reflection
In geometry and functions, the axis of symmetry is a very important idea.
It is a line that splits a figure or graph into two parts that are the same size and shape but look different.
In this article, we will look at how to use a function's graph and reflection to find its axis of symmetry.
Identifying the Line of Symmetry
A function's axis of symmetry can be found by looking at its graph and finding the line of symmetry, which is a line that splits the graph into two parts that are the same but are mirror images of each other.
As an example:
If the graph is the same on both sides of the y-axis, then the y-axis is the line of symmetry.
If the graph is the same on both sides of the x-axis, then the x-axis is the line of symmetry.
If the graph is symmetrical about a vertical or horizontal line that is not the x-axis or y-axis, then the line of symmetry is a vertical or horizontal line that goes through the function's vertex.
Finding the Axis of Symmetry Using Reflection
To use reflection to find the axis of symmetry of a figure, you need to draw a line that divides the figure into two mirror-image parts that are the same.
An axis of symmetry is what this line is called.
Finding the parabola's vertex, which is the lowest or highest point on the graph, is important.
The axis of symmetry is a vertical line that goes through the vertex.
The equation for the axis of symmetry is the x-coordinate of the vertex.
For other shapes, like circles or polygons, the axis of symmetry is the line or lines that split the shape into two parts that are the same.
Real-World Applications of Axis of Symmetry in Engineering and Design
Symmetry is a basic idea in engineering and design, and it can be used in a lot of different ways.
Architecture
Symmetry is very important in architecture, where it is used to make buildings that look good and meet engineering requirements.
Structures that are symmetrical are easier to plan, build, and keep up, and they can also make a building stronger.
Architects often use the axis of symmetry to make structures that are symmetrical by reflecting forms, shapes, or angles that are similar across a central line or point.
One great example of pure reflectional symmetry is the Airbnb logo.
The Mac page on Apple's website is another great example of reflectional symmetry.
The MacBook screens are the same length on both sides of the central vertical axis, and the lines of type in the headline and subheading are also the same length on both sides of the axis.
Engineering
In engineering, symmetry is often used to make sure that two similar parts of a part are always centered and have the same shape all along its surface.
For example, symmetry can be used to make sure that a groove is centered on the middle plane of a latch block.
You can use the axis of symmetry to make sure that the groove is in the right place and has the same shape all along the surface of the latch block.
Other uses
Used in:
Description:
Electronics
The axis of symmetry is used to make sure that the circuit is balanced and works in a stable way. For example, axis of symmetry can be used to make sure that the current flows evenly through an electronic circuit.
Physics
The axis of symmetry is used to talk about the properties of things that look the same when they are turned. For example, the axis of symmetry is used to describe how things like planets, stars, and galaxies move when they spin.
Math
The axis of symmetry is used to solve equations and describe the properties of geometric shapes. For example, the axis of symmetry is used to find the roots of quadratic equations and describe the properties of parabolas, ellipses, and hyperbolas.
Biology
The axis of symmetry is used to describe the way living things are the same on both sides. For example, many animals, like butterflies and humans, have bilateral symmetry, which means that they have a single axis of symmetry that divides their body into two mirror-image halves.
Art
The axis of symmetry is used in art to make pieces that are balanced and symmetrical. For example, axis of symmetry is used in a lot of classical paintings and sculptures to give a sense of harmony and balance.
Conclusion
In conclusion, the axis of symmetry may seem like a simple idea, but it has important effects on engineering and design that are hard to predict.
If you know how to find the axis of symmetry of a shape or function, you can find new ways to look at things and come up with new ideas.
But the axis of symmetry may be even more important because it reminds us that symmetry and balance are important parts of everything in nature, from atoms to galaxies.
By using these ideas in our work as engineers, we can make designs that are more efficient, long-lasting, and beautiful, just like the universe itself.
So, the next time you are working on a project, remember the axis of symmetry and the power of symmetry and balance to make something truly amazing. | 677.169 | 1 |
Expert Maths Tutoring in the UK
(Geometry) Line Symmetry and Rotational Symmetry Lesson
(Geometry) Line Symmetry and Rotational Symmetry Lesson
with tutors mapped to your child's learning needs.
Reflection SymmetryWhat is Reflection Symmetry?
Reflection symmetry is a type of symmetry about reflections. Even if there exists at least one line that divides a figure into two halves such that one-half is the mirror image of the other half, it is known as reflection symmetry. It is also known as line symmetry. The line of symmetry can be in any direction, horizontal, vertical, slanting, etc.
Recognizing Reflection Symmetry
The very first thing to check is that one half should be the reflection of the other half. Imagine folding a rectangle along each line of symmetry and each of the half matching up perfectly, this is symmetry. Thus, a shape has to have at least one line of symmetry to be considered as a shape with reflection symmetry. Also, there is one most important property of reflection symmetry – For the two halves which are symmetrical, one of them follows lateral inversion, that is left side appears to be the right side as it happens when you look in a mirror.
Reflection Symmetry Examples
Some of the common examples of reflection symmetry are given below:
A square has 4 lines of symmetry, which are lines through the midpoints of opposites sides, and lines through opposite vertices make up the four lines of symmetry.
A rectangle has two lines of symmetry, that is lines through the midpoints of opposites sides.
A generic trapezoid will certainly not have reflection symmetry but an isosceles trapezoid will have reflection symmetry as the line connecting the midpoints of the bases make up a line of symmetry.
The reflection of trees in clear water, the reflection of mountains in a lake are amongst the commonly seen examples of reflection symmetry around us.
Important Notes
All regular polygons are symmetrical in shape.
An object and its image are symmetrical with respect to its mirror line.
Reflection symmetry can also be observed in inkblot paper.
A figure can have one or more lines of reflection symmetry depending on its shape and structure.
Topics Related to Reflection Symmetry
Check out these interesting articles to know about reflection symmetry and its related topics.
Examples on Reflection Symmetry
Example 1: If k and l are the lines of reflection symmetry, complete the following figure.
Solution:
k and l are the lines of reflection symmetry, so we will make a leaf on the left side of the same size and shape as on the right side but yes following lateral inversion. (we could have also made leaf following lateral inversion downwards or even diagonally as k and l, both are lines of symmetry)
Example 2: Look at the shape on the left side and identify how the shape has been transformed.
(a) Rotation
(b) Reflection
Solution:
These images are depicting one most important property of reflection symmetry – The first image on the left follows lateral inversion, that is left side appears to be the right side as it happens when you look in a mirror. Therefore, it shows reflection symmetry.
Practice Questions on Reflection Symmetry
FAQs on Reflection Symmetry
What is Meant by Reflection Symmetry?
When a shape or pattern is reflected in a line of symmetry or forms a mirror image, then it is considered to show reflection symmetry.
What does Reflection Symmetry Look Like?
For any shape, reflection symmetry looks when a central dividing line (a mirror line) can be drawn on it, proving that both sides of the shape are exactly the same or reflections of one another.
Does a Square have Reflection Symmetry?
Yes, a square has a reflection symmetry having four lines of reflection, two on midpoints on the sides and two through the opposite vertices (diagonals).
Do Lines of Symmetry have to be Straight?
In most cases, the lines of symmetry are straight only. Although lines can be horizontal, vertical, or slanting. Also, a straight line has infinite lines of symmetry.
How Many Lines of Reflection Symmetry does a Rectangle Have?
A rectangle is a regular polygon having two lines of symmetry and four sides.
Does a Right Triangle have Reflection Symmetry?
A right-angled triangle doesn't show reflection symmetry. It has no line of symmetry. It just has rotational symmetry of order 1. | 677.169 | 1 |
5. Reduce 27. Is. 8d. to the fraction of 100l., and of a quart to the decimal of a gallon.
6. How many oxen at 247. 108. each can be bought for 10047. 108.?
7. A man has 1000l. in Three-and-a-Half per Cent. securities he sells at 95, and re-invests in Three per Cents. at 76. Find the difference in his annual income.
:
8. If Moira coal costs 31s. per ton, and Wallsend coal 46s. per ton, how much do I gain or lose by buying 24 tons of Moira, supposing 3 tons of Wallsend to last as long as 4 tons of Moira ?
9. Find the square root of 1974.8136.
10. If 5000 bricks four inches thick are required for a wall 125 feet long, how many bricks 5 inches thick will be required for a wall 80 feet long?
IV.
Euclid.
[N.B. Two Propositions at least from the Second Book are expected.]
1. Define-a superficies, a line, a diameter of a circle, and a scalene triangle: and state the three postulates.
2. If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them, of the other, the base of that which has the greater angle shall be greater than the base of the other.
3. In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side on which, when produced, the perpendicular falls, and the straight line intercepted without the triangle, between the perpendicular and the obtuse angle.
4. From a given point draw a straight line equal to a given straight line.
5. Describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.
6. Describe a square that shall be equal to a given rectilineal figure.
7. Draw a straight line perpendicular to a given straight line of an unlimited length, from a given point without it.
8.9. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts, are also themselves equal and parallel.
10. Parallelograms on the same base, and between the same parallels, are equal to one another. | 677.169 | 1 |
finding measures of angles worksheet
Measures Of Angles Worksheet – To practice measuring angles, download the free Measure Angle Worksheets. These worksheets will help you learn how to use a protractor and avoid angles that are not exactly right. They also include tips to make measurements easier. For example, you can use a protractor to measure an angle that looks … Read more | 677.169 | 1 |
5. In the figure (vi), AC is the diameter of circle with centre O. Chord ..
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Question Text | 677.169 | 1 |
A right-angled triangle is a triangle in which one of the angles is a right-angle. The hypotenuse of a right angled triangle is the longest side, which is the one opposite the right angle. The adjacent side is the side which is between the angle in question and the right angle. The opposite side is opposite the angle in question.
In any right angled triangle, for any angle:
The sine of the angle = the length of the opposite side
the length of the hypotenuse
The cosine of the angle = the length of the adjacent side
the length of the hypotenuse
The tangent of the angle = the length of the opposite side
the length of the adjacent side
So in shorthand notation:
In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications | 677.169 | 1 |
Arc
Arcs are often studied in geometry within the context of arcs of a circle. On a circle, you can think of an arc as a part of the circumference of the circle, as shown in the figure below.
Arcs also exist as part of curves, but most of the time, when people refer to an arc, they are usually referring to the arc of a circle rather than that of a curve. Likewise, this page will focus on the topic of arcs of a circle.
The arc from A to B, written symbolically as , is shown in red above.
Arcs are widely used in engineering and other areas of everyday life. The bridge shown below has supports in the shape of an arc.
Types of arcs
A semicircle is the name for an arc that encompasses one-half of a circle's circumference (one meaning of semi- is half).
in red is a semicircle for circle O.
There are two other types of arcs: minor arcs and major arcs. A minor arc is an arc that has a length that is shorter than that of a semicircle. A major arc has an arc length that is greater than that of a semicircle.
In the figure below, is a minor arc. Minor arcs are typically named only by their endpoints. is a major arc. Major arcs are named by their endpoints and some other point that lies on the arc.
Central angle
A central angle is an angle whose vertex is the center of a circle. When the endpoints of an arc intersect the sides of a central angle, we say the arc subtends the angle. The measure of the arc is equal to the measure of the central angle subtended by the arc.
subtends ∠QPR so, the measure of is also θ.
Also, since the circumference of a circle is 360°, we can find the measure of the major arc by finding the difference of 360° and the measure of the minor arc. So,
Arc addition
An arc formed by two adjacent arcs has a measure that is the sum of the two adjacent arcs.
Example:
Find if = 205° for circle O below.
205° = 45° + = 160°
Other angles and arcs
An inscribed angle is the angle formed in the interior of a circle when two chord or secant lines intersect on the circle. ∠RSQ in the figure below is an example. The measure of an inscribed angle is one-half the measure of the angle subtended by the arc. In the figure below,
The measure of each angle formed by two intersecting chords inside a circle is one-half the sum of the arcs that subtend the angles. In the figure below,
The measure of an angle formed by two secants that intersect outside of a circle is one-half the difference of the arcs that subtend the angle formed by the secants. In the figure below,
Arc length
Since the measure of an arc equals the measure of its central angle, we can determine arc length using the ratio of the arc's central angle to 360°. Given an arc measuring 60°, the ratio would be (60°)/(360°)=1/6. So, the arc makes up 1/6 of the circumference of the circle. Since the length of the circumference of a circle is 2πr, the length of the arc is .
In general, the length of an arc, s, is:
where r is the radius of the circle and θ is the angle in degrees. If the angle θ is in radians, the arc length is:
s = rθ
Example:
Refer to the above figure. Given a circle with a radius of 15 and θ=120°, find the length of arc s.
Other uses of arcs
Arcs are also defined in a Calculus context. An arc is a part of a differentiable curve. One way to think about this without a Calculus background is that in a differentiable curve, the section of the curve doesn't have any breaks in the curve, and that it is smooth and doesn't have any "pointy" sections, like a V, or the absolute value function does. | 677.169 | 1 |
To find the exact value of (\cos\left(\frac{11\pi}{12}\right)) using the half-angle formula, we can first express (\frac{11\pi}{12}) as a sum or difference of two angles whose cosine values are known. Since (\frac{11\pi}{12} = \frac{22\pi}{24} - \frac{\pi}{24}), we can use the half-angle formula for cosine, which states that (\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1+\cos(\theta)}{2}}).
Now, we find the cosine of (\frac{\pi}{12}) and (\frac{\pi}{24}) using known values or angles that can be expressed in terms of these angles. Once we have these values, we substitute them into the half-angle formula to find the cosine of (\frac{11\pi}{12 | 677.169 | 1 |
CBSE Class 9 Maths Chapter 6 – Lines and Angles Revision Notes
Lines and Angles Class 9 Notes
Lines and angles Class 9 notes- This is the 6th Chapter of Class 9th Mathematics. In this chapter, you will learn about the line and angles that form when two lines intersect each other. Furthermore, you will learn about the properties of angles that form when two lines intersect each other and the properties of angles formed when a line intersects two or more parallel lines at distinct points. Moreover, these properties will also be used to prove some statements.
Besides, in the earlier chapter 5, you understand that a minimum of two-point is required to draw a line. In this chapter, lines and angles class 9 notes you will learn how different lines intersect each other and the various properties that will guide you through different types of lines and angles they form. Moreover, after going through this chapter you will learn the various properties of the line ad angles and how to prove statements relating to it.
You can download CBSE Notes of Class 9 Maths Chapter 6 Revision Notes by clicking on the download button below
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Toppr is India's one of the finest app that will help you in improving your exam grades. Also, it will increase your knowledge and problem-solving skills. Toppr also provides many other great features that will assist you in learning and revision. Besides, features like the mock tests, live classes, PDF solutions, live doubt for assistance, and much more will help you in doing so. Moreover, our professionals have contemplated these notes and solution to explain and summarize the chapters and topics in easy to understand language. Furthermore, these solutions and notes are completely free. | 677.169 | 1 |
11
Answers
Think of each angle in your shape as a hinge with no resistance. Each side is a fixed length.
A square is just a parallelogram with perpendicular sides. Those same 4 sides can be used to make any parallelogram with the same sides. There is no "invalid" point where these four sides won't make a parallelogram, only a gradient of changing angles. It only becomes a square when the angles are all 90 degrees. That means if you push on one corner of your square, all four hinges will bend and your square becomes a different parallelogram with the same sides.
A triangle has 3 sides. Those three sides form three angles with a total of 180 degrees. For three sides of a fixed length, you can only ever make a triangle with the same angles. All the angles in between are invalid. That means when you push on the corner of your triangle, there is no gradient and your hinges don't bend.
This is why a triangle is a strong shape. A square can be bent, a triangle has to be broken, and every useful building material on earth is harder to break than bend. | 677.169 | 1 |
triangle inequality theorem worksheet answers key math-aids.com
Triangle Inequality Theorem Worksheets – Triangles are among the most basic shapes found in geometry. Understanding triangles is vital to learning more advanced geometric concepts. In this blog post this post, we'll go over the different types of triangles, triangle angles, how to calculate the areas and perimeters of a triangle, and offer details of the various. Types of Triangles There are three kinds of triangles: equal, isosceles, as well as scalene. Equilateral triangles have equal sides as … Read more | 677.169 | 1 |
An Introduction to Geometry and the Science of Form: Prepared from the Most ...
Sol. Let M, N, P and Q be the given sides, and O the angle included by M and N. Draw AB= M. ABM. At A make an angle CABO, and take AC N. From the point C as a centre, with a radius equal to P, and from the point B as a centre, with a radius=Q, describe 2 arcs which will intersect at D. Draw DC and DB. ABCD is the quadrilateral required.
CONSTRUCTION OF POLYGONS IN GENERAL.
87. To construct a hexagon which shall be equal to a given hexagon ABCDEF, (fig. 76.)
Sol. Divide the hexagon by diagonals into 4 triangles. Construct 4 triangles equal to those of the given hexagon and placed together in a similar order. The entire polygon GHMLKI thus constructed will be equal to ABCDEF.
In a similar manner polygons may be constructed, which shall be equal to a given polygon, whatever may be the number of its sides.
88. To construct a regular polygon in and about a circle.
Sol. Suppose the required polygon is an octagon. With the compasses divide the circumference of the circle into 8 equal parts. Connect the 8 division-points by chords; and at the same points draw tangents to the circle. In this manner one regular octagon will be constructed within, and another without the circle.
The problem can be solved by another mode. Construct a square in a circle. Bisect each of the sides of this square. Draw radii through these division points. Connect the ends of these radii by chords with the two
nearest vertices of the square. The required octagon will thus be constructed.
In a similar manner, by means of an inscribed equilateral triangle, we may construct a regular 6, 12, 24, &c. sided polygon in a circle, and by drawing tangents at the points where the vertices of the angles of such figures touch the circumference, we may construct a polygon of an equal number of sides about a circle.
CONSTRUCTION OF CIRCLES.
89. To describe a circle about a triangle.
Sol. (Fig. 63. 1.) Bisect 2 sides of the triangle, and at each division-point erect perpendiculars, which will intersect each other at O. From O as a centre, with a radius equal to the distance from the point O to the vertex of one of the angles of the triangle, describe a circle. The circumference of this circle will pass through the vertices of all the angles of the triangle; it will therefore be the circle required.
Remark. If the triangle is right-angled, the centre of the circle will be in the middle of the hypothenuse; if the triangle is acute-angled, this centre will be within the triangle, and if it be obtuse-angled, it will be without the triangle.
90. To describe a circle in a given triangle ABC, (fig. 63. 2.)
Sol. Bisect the angles A and B by straight lines, which will intersect each other at O. From the point O let fall perpendiculars upon the 3 sides of the triangle. From O as a centre, with a radius equal to either of
these perpendiculars, describe a circle. The circumference of this circle will touch the 3 sides of the triangle. It will therefore be the circle required.
91. To describe a circle in and about a given square. Sol. Draw 2 diagonals in the given square, and from the point where they intersect each other as a centre, with a radius equal to half a diagonal, describe a circle. The circumference of this circle will pass through the vertices of all the angles of the square, and thus we have a circle described about a square.
Again, from the point of intersection of the diagonals let fall a perpendicular upon one of the sides of the square; then, from the same point as a centre, with a radius equal to this perpendicular, describe a circle. It will be a circle inscribed in square.
92. To describe a circle in and about a regular polygon.
Sol. Bisect 2 adjacent sides of the polygon, and at the division points erect perpendiculars. From the point where these perpendiculars intersect each other as a centre, with a radius equal to one of the perpendiculars, describe a circle; it will be an inscribed circle.
Again, from this centre draw a line to the vertex of one of the angles of the polygon; and then with a radius equal to this line, describe a circle; it will be a circle circumscribed about the polygon.
CONSTRUCTION OF THE SKELETONS OF SOLIDS.
93. We have before made rude diagrams of the skeletons of the solid bodies. We are now prepared to
construct them more accurately with the aid of instruments. The solid the skeleton of which is to be constructed should be placed before us. Construct the skeleton
1. Of the cube, by placing together 6 equal squares, as shown in fig. 5.
3. Of the cylinder, fig. 6. The upper and lower sides of the rectangle must each be of the same length as the circumference of each circle.
4. Of the triangular pyramid, fig. 77.
5. Of the polygonal pyramids, figs. 7, 8, 9, and 10. 6. Of the cone, fig. 11. The curved side of the triangle must be of equal length with the circumference of the circle.
7. Of the regular solids, as shown in figs. 77, 78, 79, 80, 5.
PART THIRD.
COMPARISON AND MENSURATION.
I. POINTS.
94. A point has no length, breadth, or thickness; it has in fact no extension; a point is not the smallest particle of a line. As a point has no extension it cannot be measured; one point is as large as another, or rather neither has any magnitude. The representation of a point on paper or on the board has a magnitude, else it would not be visible; but that which is represented has none. A point has only a position. Where a definite line, whether straight or curved, ends, there is a point. If two lines intersect, there is at the intersection a point, which lies in both lines. Place two points together, and the position of the one will not vary from the position of the other; they will have the same position, and will coincide.
95. If a point be moved, the path which it describes in moving will be a line. If the point moves forward in the same direction, it describes a straight line; if the direction be changed every moment, it describes a curved line; if the direction be changed only once, it describes one line, composed of two straight lines joined together, or a | 677.169 | 1 |
Question 1.
How many lines can be drawn through a given point?
Solution:
Unlimited number of lines.
Question 2.
How many lines can be drawn through two distinct given points?
Solution:
One
Question 3.
How many lines can be drawn through three collinear points?
Solution:
one
Question 4.
Mark three non-collinear points A, B and C in your note-book. Draw lines through these points taking two at a time and name these lines. How many such different lines can be drawn?
Solution:
lines AB, BC and CA; three. | 677.169 | 1 |
Semi Latus Rectum of Ellipse Solution
Semi Latus Rectum of Ellipse - (Measured in Meter) - Semi Latus Rectum of Ellipse is half of the line segment passing through any of the foci and perpendicular to the major axis whose ends are on the Ellipse. Semi Minor Axis of Ellipse - (Measured in Meter) - Semi Minor Axis of Ellipse is half of the length of the longest chord which is perpendicular to the line joining the foci of the Ellipse. Semi Major Axis of Ellipse - (Measured in Meter) - Semi Major Axis of Ellipse is half of the chord passing through both the foci of the Ellipse.
Semi Latus Rectum of Ellipse Formula
What is an Ellipse?
An Ellipse is basically a conic section. If we cut a right circular cone using a plane at an angle greater than the semi angle of cone. Geometrically an Ellipse is the collection of all points in a plane such that the sum of the distances to them from two fixed points is a constant. Those fixed points are the foci of the Ellipse. The largest chord of the Ellipse is the major axis and the chord which passing through the center and perpendicular to the major axis is the minor axis of the ellipse. Circle is a special case of Ellipse in which both foci coincide at the center and so both major and minor axes become equal in length which is called the diameter of the circle.
How to Calculate Semi Latus Rectum of Ellipse?
Semi Latus Rectum of Ellipse calculator uses Semi Latus Rectum of Ellipse = (Semi Minor Axis of Ellipse^2)/Semi Major Axis of Ellipse to calculate the Semi Latus Rectum of Ellipse, Semi Latus Rectum of Ellipse formula is defined as half of the line segment passing through any of the foci and perpendicular to the major axis whose ends are on the Ellipse. Semi Latus Rectum of Ellipse is denoted by l symbol.
How to calculate Semi Latus Rectum of Ellipse using this online calculator? To use this online calculator for Semi Latus Rectum of Ellipse, enter Semi Minor Axis of Ellipse (b) & Semi Major Axis of Ellipse (a) and hit the calculate button. Here is how the Semi Latus Rectum of Ellipse calculation can be explained with given input values -> 3.6 = (6^2)/10.
FAQ
What is and is represented as l = (b^2)/a or Semi Latus Rectum of Ellipse = (Semi Minor Axis of Ellipse^2)/Semi Major Axis of Ellipse. Semi Minor Axis of Ellipse is half of the length of the longest chord which is perpendicular to the line joining the foci of the Ellipse & Semi Major Axis of Ellipse is half of the chord passing through both the foci of the Ellipse.
How to calculate is calculated using Semi Latus Rectum of Ellipse = (Semi Minor Axis of Ellipse^2)/Semi Major Axis of Ellipse. To calculate Semi Latus Rectum of Ellipse, you need Semi Minor Axis of Ellipse (b) & Semi Major Axis of Ellipse (a). With our tool, you need to enter the respective value for Semi Minor Axis of Ellipse & Semi Major Axis of Ellipse and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.
How many ways are there to calculate Semi Latus Rectum of Ellipse?
In this formula, Semi Latus Rectum of Ellipse uses Semi Minor Axis of Ellipse & Semi Major Axis of Ellipse. We can use 2 other way(s) to calculate the same, which is/are as follows - | 677.169 | 1 |
Trigonometry Table: Learn Trigonometry Formulas For Class 10+
Trigonometry Table Formula: Trigonometry is a branch of mathematics that focuses on the relationship between angles and sides of triangles. The trigonometry table is also known as a trig table or an angle table. It is one of the important tools in trigonometry. The trigonometry table serves as a reference guide, providing the values of trigonometry functions for different angles.
Sine, cosine, and tangent are the three basic trigonometric functions that define the ratios between the sides of a right triangle. Since these functions are presented in a systematic format in the trigonometry table, users can quickly obtain the values without performing complex calculations. It typically covers angles from 0 to 90 degrees, as these angles correspond to the primary quadrants of the unit circle.
In the table, each row signifies a distinct angle, with columns presenting the angle in degrees, along with its sine, cosine, and tangent values. This article elucidates efficient methods to grasp trigonometry tables for classes 10, 11, and 12, along with additional insights.
Trigonometry Table:
The following is the trigonometry table showing the values of common trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) for angles in degrees.
Degrees
( 0 )
Sine
(sin)
Cosine
(cos)
Tangent
(tan)
Cosecant
(csc)
Secant
(sec)
Cotangent
(cot)
00
0
1
0
∞
1
∞
30^0
1/2
√3/2
1/√3
2
2/√3
√3
45^0
1/√2
1/√2
1
√2
√2
1
60^0
√3/2
1/2
√3
2/√3
2
1/√3
90^0
1
0
∞
1
∞
0
180^0
0
-1
0
∞
-1
∞
270^0
-1
0
∞
-1
∞
0
360^0
0
1
0
∞
1
∞
In this table 1/√2 can also be written as √2/2 and 1/√3 can also be written as √3/3 ( by rationalizing the denominators)
Formulas For Trigonometry Table:
The following are the Trigonometry Table (0 to 360 degrees) formulas.
sin x = cos ( 90^0 – x )
cot x = tan ( 90^0 – x )
sec x = cosec ( 90^0 – x )
cos x = sin ( 90^0 – x )
tan x = cot ( 90^0 – x )
cosec x = sec ( 90^0 – x )
cosec x = 1/sin x
cot x = 1/tan x
sec x = 1/cos x
How To Learn Trigonometry Table For Class 10, Class 11 & 12?
The trigonometry table may appear difficult at first, but it is simple to understand if you know the sine values for the eight standard angles. There are a few formulas that must be followed before generating the table. The formulas are mentioned below.
Tan x = sin x / cos x
Cosec x = 1 / sin x
Sec x = 1 / cos x
Cot x = 1 / tan x or cos x / sin x.
Here are the steps for making a trigonometric table that you can remember.
Step 1: Create a table with the trigonometry functions sin, cosec, cos, tan, cot, and sec in the first column and the angles 0^0, 30^0, 45^0, 60^0, 90^0, 180^0, 270^0, and 360^0 in the top row.
Step 2: Calculating the value of sin x:
Angles 0^0, 30^0, 45^0, 60^0, and 90^0 should be written in ascending order. For each of these angles, sin has the values 0, 1/2, 1/√2, √3/2, and 1 respectively. This information provides the sine values for these five angles. Now utilize the following formulas for the final three.
Sin 270^0 = sin (180^0 + 90^0) = -sin 90^0 = -1, here we are using x = 900 in this case because we need to calculate the value of sin 270^0. As a result, x = 90^0 fulfills the requirements of the formula.Sin
0
1/2
1/√2
√3/2
1
0
-1
0
Sin 360^0 = sin (360^0 – 0^0) = -sin 0^0 = 0, here we are using x = 00 in this case because we need to calculate the value of sin 3600. As a result, x = 00 fulfils the requirements of the formula.
5th Step: Calculating the values of cot x:
cot x = 1 / tan x. Hence, we need to put all the angles in the given formula and calculate the value of the cot. By using this formula, you can easily determine the values of the cotcot
∞
√3
1
1/√3
0
∞
0
∞
6th Step: Calculating the values of cosec x:
cosec x = 1 / sin x. Hence, we need to put all the angles in the given formula and calculate the value of cosec. By using this formula, you can easily determine the values of the coseccosec
∞
2
√2
2/√3
1
∞
-1
∞
7th Step: Finding the values of sec x:
sec x = 1 / cos x. Hence, we need to put all the angles in the given formula and calculate the value of sec. By using this formula, you can easily determine the values of the sec function as followssec
1
2/√3
√2
2
∞
-1
∞
1
Trigonometry Pi Table:
The Trigonometry table is an essential reference tool for anyone studying or working with trigonometry. The table simplifies trigonometry calculations, enhances problem-solving capabilities, and aids in the exploration of the relationship between angles and trigonometry functions by providing re-calculated values of sin, cosine, tan, cot, cosec, and sec for different angles | 677.169 | 1 |
How to Find the Length of a Triangle: Step-by-Step Method, Real-World Examples, and More
Discover different methods, examples, and tools for finding the length of a triangle, such as step-by-step methods, formulaic methods, video tutorials, and real-world examples. This article explores common mistakes, practical applications, and interactive graphics to help readers better understand the topic.
I. Introduction
Triangles are among the most basic geometrical forms, yet they can be found in many aspects of everyday life. However, determining the length of a triangle can be challenging. Whether you are an architect designing a building or simply calculating the distance between two points, knowing how to find the length of a triangle is essential to problem-solving. This article provides a comprehensive guide on how to find the length of a triangle using different methods and tools.
II. Step-by-Step Method
One of the easiest ways of finding the length of a triangle is using the step-by-step method. Here are the basic steps to follow:
Identifying the Base and Height
A triangle has three sides, one of which is the base. The base is the longest side and is found at the bottom of the triangle. The height is a perpendicular line from the base to the opposite point or angle on the triangle.
Using the Pythagorean Theorem
The Pythagorean Theorem states that the sum of the squares of the two shorter sides of a right triangle equals the square of the length of the hypotenuse (the longest side of the triangle). This theorem can be used to find the length of the hypotenuse by taking the square root of the sum of the square of the two shorter sides.
Using Similar Triangles or Trigonometry
Another way to find the length of a triangle is using similar triangles. If two triangles have the same shape, their corresponding sides are proportional. This means that if you know the length of one side of one triangle and the corresponding side on the other triangle, you can find the length of the other side.
Alternatively, you can use trigonometry, specifically the sine, cosine, and tangent ratios. These ratios relate the lengths of the sides of a triangle to its angles and can be used to find the length of a side.
Offering Tips for Effectively Using This Method
It can be helpful to draw a diagram to visualize the triangle and label all sides and angles. Be sure to use the correct formula for the type of triangle you are working with. In addition, be aware of units of measurement if using real-world examples.
III. Real-World Examples
Real-world examples can help us better understand how to find the length of a triangle. Here are a few practical examples of finding triangle length:
Using a Map and Compass to Find Distance Between Two Points
The distance between two points on a map can be found by treating each point as a right angle (90-degree triangle). You can then use the Pythagorean Theorem to calculate the distance between the two points.
Using a Diagram of a Roof to Show How to Find the Length of a Sloped Line
When planning to build a roof, it is important to know the length of the sloped line. This length can be found using the Pythagorean Theorem by treating the roof and distance between the eaves as the two shorter sides and the sloped line as the hypotenuse.
Explaining How Real-World Examples Can Help Readers Better Understand the Concept
Real-world examples provide concrete applications of the concept and can help readers see how to apply the theory in practical scenarios. Such examples also make the concept more engaging and interesting.
IV. Interactive Graphics
Interactive graphics can be a useful tool for helping readers visualize the process of finding the length of a triangle. Here are a few tips for using interactive graphics:
Presenting the Idea of Using Interactive Graphics to Help Illustrate the Process
Interactive graphics can help readers understand the concept better by letting them drag and drop points or adjust measurements in real-time. Moreover, interactive graphics provide a more engaging experience for readers than static diagrams or pictures.
Suggesting Using Animations or Diagrams That Allow Readers to Drag and Drop Points
Animations can illustrate how the length is calculated step by step, while diagrams that allow readers to drag and drop points can help readers see the relationship between different parts of the triangle.
Highlighting the Benefits of Interactive Graphics for Engaging Readers and Promoting Understanding
Interactive graphics provide a more engaging experience for readers than static diagrams or pictures. Moreover, interactive graphics can explain concepts in a more dynamic way, allowing readers to see change in real-time. This in turn promotes better understanding of the concept.
V. Formulaic Method
In addition to the step-by-step method, there are other formulaic methods for finding the length of a triangle. Some of the most commonly used formulas are:
Pythagorean Theorem
The Pythagorean Theorem is applicable to right triangles only. It can be used to find the length of the hypotenuse.
Law of Sines
The Law of Sines can be used to find the length of a side when the length of two sides and the angle between them are known.
Law of Cosines
The Law of Cosines can be used to find the length of a side when the lengths of two sides and the angle opposite the unknown side are known. It is also applicable to non-right triangles.
Offering Advice on When Each Formula Is Most Appropriate to Use
The Pythagorean Theorem is used for right triangles only, while the Law of Sines and Law of Cosines are applicable to non-right triangles. The appropriate formula to use depends on the information you have about the triangle.
VI. Video Tutorials
Video tutorials are an excellent resource for visual learners who prefer to see and hear information. Here are some tips for creating effective tutorial videos:
Explaining the Advantages of Video Tutorials for Visual Learners
Video tutorials allow visual learners to see and hear the information being explained. They can replay the video, pause it to take notes, and learn at their own pace.
Offering Suggestions for Creating Effective Tutorial Videos
Effective tutorial videos should be clear, concise, and engaging. They should present the information visually and verbally, offer step-by-step instructions, and provide examples.
Providing an Example Tutorial Video That Demonstrates How to Find the Length of a Triangle
The following is an example tutorial video about how to find the length of a triangle using the Pythagorean Theorem:
VII. Common Mistakes
Here are a few common mistakes to avoid when finding the length of a triangle:
Addressing Common Mistakes Made When Finding the Length of a Triangle
Common mistakes include using the wrong formula for the type of triangle, not using the correct units of measurement, and failing to accurately measure the sides and angles of the triangle.
Explaining How to Recognize and Avoid These Mistakes
To avoid these mistakes, double check the formula being used and ensure correct units of measurement are being used. Use a protractor and ruler to measure the sides and angles of the triangle accurately.
Offering Tips for Being More Accurate When Using the Step-by-Step Method or Formulas
It is important to have a clear diagram of the triangle and label all angles and sides accurately. Use a calculator to ensure accurate calculations, and double check all work to catch errors.
VIII. Applications of Length
Having the ability to find the length of a triangle has many practical applications in various fields. Here are a few examples:
Exploring Practical Applications of Finding the Length of a Triangle
Navigation, engineering, and architecture all use calculations of length on a regular basis. For example, navigators use the length of a triangle to determine travel distances, while engineers use length to design and build structures, bridges, and roads. Architects use length to ensure the safety and stability of their designs.
Discussing How Knowing the Length of a Triangle Could Be Useful
Knowing the length of a triangle can help you solve various problems, such as finding the distance between two points, calculating the slope of a roof, or determining the amount of material needed for construction.
IX. Conclusion
Knowing how to find the length of a triangle is an essential skill in various fields, such as navigation, engineering, and architecture. This article provided practical tips, formulas, and examples to help you solve problems involving triangle length. Remember to double check your work and use a clear and accurate diagram to ensure better accuracy. By using the various methods and tools provided, you can solve any problem involving triangle length and improve your problem-solving skills.
For further reading, check out resources such as Khan Academy or online calculators to continue your learning journey. | 677.169 | 1 |
Angles
An angle is the union of two
rays
with a common endpoint. That endpoint is called the
vertex
of the angle.
An angle is named using the names of three points; one point on each ray and the vertex.
Or
, if there is no possibility of confusion, you can just use the vertex.
(Note that when three points are used, the vertex should go in the middle. So, for example,
∠
B
A
C
would
not
be a correct name for the angle on the left.)
An angle can be measured by the amount of rotation about the vertex needed for one side to overlap the other. Like rotations, angles can be measured in
degrees
, where
360
degrees means one full rotation.
In the above figure,
m
∠
A
B
C
=
45
°
.
Angles can also be measured in units called
radians
, where
2
π
radians is equivalent to
360
degrees | 677.169 | 1 |
Analytic Geometry
Calculating Area Using Coordinates
One of the most frequent activities in geometry is determining the area of a polygon such as a triangle or square. By using coordinates to represent the vertices, the areas of any polygon can be determined. The area of triangle OPQ, where O lies at (0,0), P at (a,b), and Q at (c,d), is found by first calculating the area of the entire rectangle and subtracting the areas of the three right triangles. Thus the area of the triangle formed by points OPQ is = da — (dc/2) — (ab/2) — [(d — b)(a — c)]/2. Through the use of a determinant, it can be shown that the area of this triangle is:
This specific case was made easier by the fact that one of the points used for a vertex was the origin.
The general equation for the area of a triangle defined by coordinates is represented by the previous equation.
In a similar manner, the area for any other polygon can be determined if the coordinates of its points are known. | 677.169 | 1 |
Answers ( )
A and B are used to label the legs while C is used to label the hypotenuse(AKA the long side of a right triangle and the side opposite of the 90° angle.) The B side is the opposite side of the hypotenuse or the side C. The A is the bottom leg that is in between sides B and C. | 677.169 | 1 |
E. Draw A Related Picture For Each Of The Following: 1. A Lin…
E. Draw a related picture for each of the following: 1. A line containing C. A, and R. Give all the possible names of the line. 2. A plane M containing points B, U, and S. Give another name for plane M. 3. A point Q on DR but not on RC. 4. A point T on AM but not on BM. 5. Two lines that intersect. 6. Collinear points A, B, C, D, and E such that AB and AC are opposite rays, AC and AD are the same rays, and BA and BE are opposite rays. (meron din sa picture pag di naintindihan)
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E. Draw a related picture for each of the following:
1. A line containing C. A, and R. Give all the possible names of the line.
2. A plane M containing points B, U, and S. Give another name for plane M.
3. A point Q on DR but not on RC.
4. A point T on AM but not on BM.
5. Two lines that intersect.
6. Collinear points A, B, C, D, and E such that AB and AC are opposite rays, AC and AD are the same rays, and BA and BE are opposite rays.
=========================================
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nasa picture ang answer.
《misswhitepinky》
Solved draw or provide each of the following: a specific. Being following each step illustrated identify arrange ph numbers using last first. Solved re-draw each of the following making sure to include
Draw lewis structure. Solved draw the major product(s) of each of the following. Hand drawn mathematics clip art/math elements and symbols/back to | 677.169 | 1 |
Polar to Rectangular Online Calculator
Below is an interactive calculator that allows you to easily convert complex numbers in polar form to rectangular form, and vice-versa. There's also a graph which shows you the meaning of what you've found.
For background information on what's going on, and more explanation, see the previous pages,
Additionally we've created this video, where we delve into the conversion between polar and Cartesian coordinates. With a straightforward approach, we illustrate the correlation between points in the coordinate plane and their representation in both systems.
Whether you're familiar with mathematics or just starting out, this explanation aims to simplify the concept for easy understanding. Join us as we demystify the connection between polar and Cartesian coordinates.
Interactive Graph - Convert polar to rectangular and vice-versa
In the following graph, the real axis is horizontal, and the imaginary (`j=sqrt(-1)`) axis is vertical, as usual.
Point P represents a complex number.
Things to do
Choose whether your angles will be in degrees or radians first.
Enter your values for either radius and angle, or real value and imaginary value and click "Calculate" to see the equivalent result (or you can press <Enter> on your keyboard).
You can also drag point P to change the radius of the circle, and/or the angle to your desired values.
You can zoom the graph in or out using the navigation icons at the bottom of the graph, and pan left-right, up-down by holding down the <Shift> key while dragging the graph.
You can change the precision of all the calculations by changing the "Decimal places" option.
Go back to the examples on the Polar Form page and try them here in the calculator, and compare the results. | 677.169 | 1 |
AFOQT 2022 PREP EXAM QUESTIONS AND ANSWERS WITH ACTUAL TEST
Document Content and Description Below
AFOQT 2022 PREP EXAM WITH ACTUAL TEST
Triangle Correct Answer: 3 internal angles add to 180, 3 sided
Angle Bisector Correct Answer: extends from one side to BISECT the opposing angle
Trian... gle altitude Correct Answer: the shortest distace from a vertex angle to the side containing the base
Triangle Median Correct Answer: extends from one angle to BISECT the opposite side
Two centers of a triangle Correct Answer: Centroid or orthocenter
Centroid Correct Answer: Where the triangles 3 medians meet
Orthocenter Correct Answer: where the triangles 3 altitudes meet
Scalene triangle Correct Answer: no equal sides of angles
Isosceles Correct Answer: Has 2 equal sides and 2 equal angles often called base angles
Equilateral Triangle Correct Answer: All 3 sides and angles are equal 60 degrees
Rigth triangle Correct Answer: Has one right angle 90 and two acute angles
Acute triangle Correct Answer: all 3 angles less than 90
Obtuse Correct Answer: 2 acute and one angle greater then 90
Triangle inequality theorem Correct Answer: states that the sum of any two sides of a triangle must be greater than the third side
Third-side rule of triangles Correct Answer: three sides, a, b, c. c-b < a < c+b
Pythagorean Theorem Correct Answer: The relatioship between the sides of a right triangle is a2 + b2 = c2 (all squared) where c is the hypotenuse and is across from the right angle.
Right triangle with angle measurements 90-45-45 Correct Answer: Special right triangle. the hypotenuse is equal to the square root of 2x
Right triangle with angle measurments 90-60-30 Correct Answer: Special right trianlge.hypotenuse is 2x. the short side is x and the long side is square root of 3x
Quadrilaterals Correct Answer: A closed, 4 sided shape. The sum of all 4 sides is always equal to 360. The AREA of quadrilateral is always A= bh (base time height)
Parallelogram Correct Answer: A quadrilateral with two pairs of equal side. Two consecutive sides in a parallelogram are supplementary = 180
Rectangle Correct Answer: two pairs of equal sides and four right angles
Kite Correct Answer: two pairs of equal sides but the equal sides are consecutive
Square Correct Answer: 4 right angles and 4 equal sides
Rhombus Correct Answer: 4 equal sides. diagonals bisects angles and bisect one another
Trapezoid Correct Answer: One pair of sides is parallel
bases have different lengths
Polygons Correct Answer: Any closed shape made up of 3 or more line segments. Hexagon = 6, Octagon = 8
The sum of all exterior angles in a polygon is 360
Finding the sum of the interior angles of polygons Correct Answer: (n-2) x 180 where n is the number of sides the polygon has. To find a single interior angle simply divide the total interior angles by the # of sides and therefor angles
Apothem Correct Answer: the shortest PERPENDICULAR distande from one of the sides to the center
Area of Polygon Correct Answer: A = ap/2 (apothem x perimeter)
finding an interior angle of a polygon Correct Answer: (n-2)/n x 180
Area of Circle Correct Answer: A = PieRsquared
Volume Correct Answer: describes as the amount of cubic units any shape can hold
Surface area Correct Answer: The sum of the areas of the 2 dimensional figures that make up its shape.
Slant height Correct Answer: The distance from the base to the apex along the lateral surface
Prism Correct Answer:
Topic Correct Answer: The overall subject matter of the passage
Main Idea Correct Answer: What the author wants to say about the topic
Standard rate turn Correct Answer: 360 degrees in 2 minutes at 3 degrees per second
Slipping turn Correct Answer: gravity is greater than teh centrifugal force on the ball and thus it slips to the inside
Skidding turn Correct Answer: Centrifugal force in greater than gravity and therefor the ball moves to the outside of the coordinator.
Agonic vs Isogonic Lines Correct Answer: Agonic is when variation is 0
Isogonic is when Magnetic variation is either greater or less then 0
Runway end lights and edges lights Correct Answer: -Red lights, and outward from runway end they are green to indicated the threshold.
-yellow edge lights in last 2000 feet or at half the distance of runway, whichever is less
July 2, 1900 Correct Answer: Zepplin makes it first flight
Oct 20, 1900 Correct Answer: The wright brothers make their first glider flight
Dec 17, 1903 Correct Answer: The wright brothers make their first powered, manned, heavier than air controlled flight (lasted 12 seconds)
February 22, 1920 Correct Answer: The first transcontinental mail service is established from San Francisco to New York
March 3, 1923 Correct Answer: The first non stop coast-tocoast airplane travels from New York to San Diego
May 21, 1927 Correct Answer: Charles A. Lindbergh accomplishes the first nonstop flight across the Atlantic Ocean
June 29, 1927 Correct Answer: The first trans-pacific flight travels from California to Hawaii
June 1, 1937 Correct Answer: Amelia Earhart is lost in route to Howland island from New Guinea
June 28, 1939 Correct Answer: Pan American Airways flies the first trans-Atlantic passenger flight service
Oct 14, 1947 Correct Answer: Captian Charles E Yeager exceeds the sound barrier in a rocket
May 5, 1961 Correct Answer: Alan Shepard pilots the first US manned space flight
Feb 20, 1962 Correct Answer: John Glen becomes the first America to orbit the earth
Dec 27, 1968 Correct Answer: Apollo8 is the first human flight to orbit the moon
Sept 3, 1971 Correct Answer: The concorde makes its first transatlantic crossing
1978 Correct Answer: The US Airline Deregulation act ends government regulation of airline routes.
Oct 24, 2003 Correct Answer: The Concorde supersonic jet makes its last flight.
How many Feet in a Meter? Correct Answer: 3.28 feet in 1 Meter
Isotopes Correct Answer: Atoms of the same element with the same number of protons but different number of neutrons
How do microscopes work Correct Answer: they REFRACT or bend light waves to make objects look bigger
Magnitude Correct Answer: refers to how much energy is released
Combustion reaction Correct Answer: Combustion is defined as a reaction in which a hydrocarbon reacts with O2 to produce CO2 and water
friction Correct Answer: defined as a force that always opposes motion
normal force Correct Answer: balances out gravity in resting objects. book resting on a table
Tension Correct Answer: the force felt by pull of an object by another
Planets Correct Answer: MVEMJSUN
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Minerals Correct Answer: Naturally occurring
solid
crystalline structure
composed of single chemical compound
INORGANIC, only a rock can be composed or organic material
Mast (Shaft) Correct Answer: Is a long cylindrical component that extends vertically from the main rotor transmission up to the main rotor hub. It is responsible for the main drive force that turns the main rotor hub.
Main rotor hub Correct Answer: The part where all the components of the main rotor head are attached. this includes the blade grips, the rotor blades, the pitch horn (or yoke), the stabilizer bar and weight (or flybar), and the teeter hinge (or trunnion)
Blade Grips Correct Answer: connect the rotor blades to the rotor system. the primary responsibility of the blade grips is to allow the rotor blades to feather. Feathering is a term used to describe the change of the blades' angle relative to their rotation plane (also known as the angle of attack.
Rotoe Blades Correct Answer: Made of metal, but sometimes fiberglass or carbon fiber. Shaped like an airfoil, give helicopter lift.[Show More] | 677.169 | 1 |
Identify Types of Transversal Angles
Description: This boom card helps students identify different types of Transversal angle pairs. Corresponding, Alternating Exterior, Alternating Interior, and same side interior. There are 8 cards in this deck. | 677.169 | 1 |
Welcome to our corner of humor and wit, where the best puns about shapes and geometry come together to give you sidesplitting laughter! Get ready to be squared away with hilarity as we present to you "Shaping Up: 220+ Geometric Puns That'll Make You Triangulate with Laughter!" From acute wordplay to right-angled jokes, we've crafted a collection that's sure to make you circle back for more. So buckle up and get ready to shape your day with some sidesplitting fun!
Why did the triangle go to the birthday party alone? He couldn't find a date!
What's a polygon's favorite way to party? With a line dance!
Why did the square refuse to attend the birthday celebration? It didn't have the right angles!
How do you wish a geometric a happy birthday? "You're acute addition to the party!"
What did the cylinder bring to the birthday party? A round of drinks!
What's a circle's favorite part of a birthday party? The circumference of the cake!
Why did the geometric cake blush at the party? It was getting too many compliments!
How do you wish a mathematician happy birthday? "May your day be as irrational as π!"
Why did the geometric figure throw a birthday party? To find the right angle for a good time!
What's a triangle's favorite gift at a birthday party? A slice of cake with acute frosting!
Why are birthdays like triangles? They're both full of angles to celebrate!
What did the square give as a birthday present? A box full of good wishes!
What's a circle's favorite part of a birthday party? It always loves to go a-round!
Shape up and Laugh Out!
So there you have it, folks! I hope you found these geometric puns to be on point, acute, and just plane funny! Don't be a square, go ahead and check out our other pun posts to keep the laughter rolling. Whether you're in the mood for some polygonal humor, or you just can't get enough of those right-angle jokes, we've got you covered. Remember, with puns, the possibilities are endless – just like the sides of a rhombus | 677.169 | 1 |
Attributes Of Shapes Worksheet
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Matching Shape Attributes Worksheet • Have Fun Teaching
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attributes of shapes
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Right Angles In 2d Shapes Worksheet
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Shape Attributes Worksheet
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Web This Worksheet Will Cover:
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Circle, Triangle, Square, Rectangle, Hexagon.
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From (1) The sum of the measures of angles DQS and ERT is 260 we have that:
DQS =x ERT = y
So, LQS= 180 - x, and LSQ= x-90 and then since LSQ and YST are opposite angles, they are equal.
Also, TRM = 180 -y, and MRT= y-90 and then since MRT and STY are opposite angles, they are equal.
Finally, since all angles of triangle SYT must sum 180, we have that:
x-90+ y-90 + WYX = 180
WYX = 360 - x - y
WYX = 360 - (x+y)= 100, so sufficient.
From (2) The sum of the measures of angles WQD and ERX is 100, we have that:
In this one you have the complementary angles of solution in (1), and each complementary angle is 180 - x and 180 -y, and if you sum up you arrive to the same formula in (1) 360 - x -y = 100, this one is also sufficient. (D) is our answer.
Originally posted by Mizar18 on 22 Jul 2019, 08:45.
Last edited by Mizar18 on 22 Jul 2019, 20:08, edited 1 time in total.
JDEK forms a rectangle ad the lines are parallel. So JDE=KDE=90" SUBSEQUENTLY WDQ=REX=90" St. 1- DQS+ERT= 260" DQW+DQS= 180" and ERX+ERT=180" Add the equations, we get DQW+ERX= 100 Taking both triangles, we should get sum of 360" from 6 angles We have found two angles and sum of two angles, Adding up we get DQW+WDQ+DWQ+ERX+EXR+REX= 360 Insert the values, DWQ+EXR= 80 Taking the triangle WXY, we get 80+WYX= 180 So, angle Y=100 -------- Suff St 2- provides the second step were we calculated sum of angles as 100. Hence, suff ANS D
We require the Sum of angles W and X to arrive at the angle of Y as YXW is a triangle whose sum of angles must be equal to 180 degrees Angle J and K are 90 degrees (Rectangle). Since WK is parallel using traversal lines property - Angle WDQ and REX - 90
i) Sufficient, Can derive Sum of DQW and ERX as 100 degrees. (Sum of Angles in a straight line at Q and R = 360 and Sum of DQS and ERT is 260. Which means that sum of angles at SQL and MRT is 100 and being vertically opposite angles DQW and ERX also has 100). So Sum of all angles in two triangles WQD and RXE other than Sum of angles W and X is 90*2+100. So W+X=360-280=80 and So Y = 100
IMO D ii) Sufficient, Directly given the first inference of (i) that is Sum of DQW and ERX as 100 degrees.
Since JK is parallel to WX, then QDW and REX must be 90 degrees. Since angles in a triangle must add up to 180, we only need DWQ and EXR to solve for WYX. We don't even need to know the individual angles, just the collective sum of DWQ and EXR. 1.The sum of the measures of angles DQS and ERT is 260. You can figure out DQW and ERX with this since both ERT and DQS are on a straight line. Together, they must equal 360 - 260 =100 (each straight line must equal 180, since there are two, add together to be 360 and then subtract 260 for the sum of the angles). Since you know DQW and XRE equals 100 and you know WDQ and XER are right angles (90 each), you know that DWQ and EXR must equal 360-280 = 80 (since you're calculating two triangles, the sum must be 180+180 =360 and 280 comes from 90+90+100 from above). If those two angles equal 80, then WYX must be 100 so that triangle WXY equals 180. Sufficient 2.The sum of the measures of angles WQD and ERX is 100. This gives the same information as above but one less step Sufficient
I believe my word explanation is sufficiently clear to understand, but anyway you can find attached sketch for clarity.
Line \(WX\) is \(parallel\) to line \(JK\) --> Line \(WX\) is \(perpendicular\) to both lines \(JL\) and \(KM\).
Statement (1): \(\angle{DQS} + \angle{ERT} = 260.\) Draw line \(YZ\) that intersects line \(JK\) at point \(Z\) and is \(perpendicular\) to line \(JK\) --> Line \(YZ\) is \(parallel\) to both lines \(JL\) and \(KM\).
Statement (2): \(\angle{WQD} + \angle{ERX}\) = 100. Draw line \(YZ\) that intersects with line \(JK\) at point \(Z\) and is \(perpendicular\) to line \(JK\) --> Line \(YZ\) is \(parallel\) to both lines \(JL\) and \(KM\).
From the given figure, we can determine that angle J, K , L , M are all 90 degrees since JKLM is a rectangle.
We know that segment Jk is parallel to segment WX, and we know that segment Jl is a transversal .By the property of transversal lines, we know that
line one is parallel to line two, and both lines are cut by the transversal, t. A transversal is simply a line that passes through two or more lines at different points. Some important relationships result. • Vertical angles are equal • Corresponding angles are equal: • Supplementary angles sum to 180° • Any acute angle + any obtuse angle will sum to 180° | 677.169 | 1 |
pentagon is a closed polygon. Thus the sum of all angles is = 180 (n-2) , where n = number of vertices. In this case, n =5. Since it is mentioned that ABCDE is a regular polygon, all angles are the same. | 677.169 | 1 |
Triangle Construction
Triangle construction involves creating a triangle given specific measurements or angles. This involves the use of rulers, compasses, and protractors.
A triangle can be constructed if its three side lengths are known, commonly abbreviated as SSS (Side-Side-Side). You start by drawing the longest side, then drawing arcs from each end of this line using the lengths of the other two sides. The point where these arcs meet forms the third corner of your triangle.
A triangle can also be constructed if two sides and the angle between them are known (SAS - Side-Angle-Side rule). Start by drawing one side, then construct an angle using a protractor at either end of this line and draw the other given side length originating from this point.
If two angles and the side between them are known (ASA - Angle-Side-Angle rule), start by drawing the given side length. Construct the given angles at both ends of this line. The point where these lines intersect forms your third vertex.
If two sides and an included angle (known as the SAS rule), or two angles and the included side (known as the ASA rule) are given, draw the base line equal to one of the given sides. Then, you draw the indicated angle, followed by a line equal to the other indicated side or length.
To construct a right-angle triangle given the length of the hypotenuse and one side (RHS - Right angle-Hypotenuse-Side rule), begin by drawing the hypotenuse. Then, from one end of this line, construct a right angle and draw a line from this angle equal in length to the given side.
Ensure precision when using your compass and ruler to draw accurate triangles. Minor deviations can lead to significant inaccuracies.
It's important to practice these triangle constructions regularly. Perfecting these methods will enable quick and accurate triangle constructions, which can save valuable time in tests.
Analyzing and understanding different ways to draw triangles based on limited or extensive given information is a key part of mastering triangle constructions.
Remember to always cross-check your work and ensure each side and angle aligns with the given measurements. | 677.169 | 1 |
Casey's angle 2
This applet shows Casey's angle using projection from a point - as opposed to parallel projection.
Here again, Casey's angle is shown to be perspective invariant: BOC = B'O'C'
13 points are shown here, 7 large and 6 small. Move the seven large ones as you will to explore Casey's angle.
How can you accurately describe what you observe? | 677.169 | 1 |
How To Find Arc Length Trigonometry with Example
How To Find Arc Length Trigonometry, arc in trigonometry refers to a portion of a circle's circumference. It is defined as the measure of the angle between two points on the circle, and can be expressed in radians or degrees. The length of the arc is proportional to the size of the angle, and can be calculated by multiplying the radius of the circle by the angle in radians.
How To Find Arc Length Trigonometry
In trigonometry, arcs are used to define and study the relationships between angles and lengths in circles. The length of an arc in trigonometry depends on the central angle measure, radius of the circle and the circumference of the circle. To find the length of the arc, the formula is given by: Arc Length = (θ / 360) x (2π x r), where θ is the central angle in degrees and r is the radius of the circle. Alternatively, you can use the formula: Arc Length = θ x (r / (2π / 360)), where θ is the central angle in radians and r is the radius of the circle. where:
θ is the central angle of the arc in degrees
r is the radius of the circle
π is Pi, approximately equal to 3.14159
For example, if you have a circle with a radius of 5 units, and the central angle of the arc measures 60 degrees, the length of the arc would be: Arc Length = (60 / 360) * 2π * 5 = (1/6) * 2 * 3.14159 * 5 = 2 * 3.14159 = 6.28318 Example: Suppose, you have a circle with a radius of 5 units and the central angle of the arc is 120°. To find the length of the arc, you need to use the above formula: Arc Length = (120 * 5) / 360 Arc Length = 2 units So, the length of the arc is 2 units.
Arc Length Concept
Arc length is a concept used in mathematics and geometry to measure the distance along a curved path. It can be calculated using trigonometry, which is the study of relationships between angles and lengths of sides in a triangle. Finding arc length requires some knowledge of basic trigonometry principles such as sine, cosine, and the Pythagorean theorem. This article will explain how to calculate arc length with trigonometry step-by-step.
First, it's important to understand what an arc is – it's any part of a circle or curve that forms an angle at its center point. To find the total length of an arc, you need to know two things: The central angle and either the radius or circumference of the circle itself | 677.169 | 1 |
Class 10 Maths MCQs on Chapter 6- Triangles are prepared to help students recognise the important topics and concepts for the objective type questions. In CBSE Class 10 Maths Exam 2020, there will be increased number of objective type questions as compared to the previous years. So, students... | 677.169 | 1 |
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Splash Screen. B. C. D. Use a special right triangle to express sin 45° as a fraction. 5-Minute Check 6
Content Standards G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Mathematical Practices 4 Model with mathematics. 1 Make sense of problems and persevere in solving them. CCSS
Angle of Elevation CIRCUS ACTS At the circus, a person in the audience at ground level watches the high-wire routine. A 5-foot-6-inch tall acrobat is standing on a platform that is 25 feet off the ground. How far is the audience member from the base of the platform, if the angle of elevation from the audience member's line of sight to the top of the acrobat is 27°? Make a drawing. Example 1
Angle of Elevation Answer: The audience member is about 60 feet from the base of the platform. Example 1
DIVING At a diving competition, a 6-foot-tall diver stands atop the 32-foot platform. The front edge of the platform projects 5 feet beyond the ends of the pool. The pool itself is 50 feet in length. A camera is set up at the opposite end of the pool even with the pool's edge. If the camera is angled so that its line of sight extends to the top of the diver's head, what is the camera's angle of elevation to the nearest degree? A. 37° B. 35° C. 40° D. 50° Example 1
Since are parallel, mBAC = mACD by the Alternate Interior Angles Theorem. Angle of Depression DISTANCE Maria is at the top of a cliff and sees a seal in the water. If the cliff is 40 feet above the water and the angle of depression is 52°, what is the horizontal distance from the seal to the cliff, to the nearest foot? Make a sketch of the situation. Example 2
Angle of Depression Let x represent the horizontal distance from the seal to the cliff, DC. C = 52°; AD = 40, and DC = x Multiply each side by x. Example 2
Angle of Depression Divide each side by tan 52°. Answer: The seal is about 31 feet from the cliff. Example 2
Use Two Angles of Elevation or Depression DISTANCE Vernon is on the top deck of a cruise ship and observes two dolphins following each other directly away from the ship in a straight line. Vernon's position is 154 meters above sea level, and the angles of depression to the two dolphins are 35° and 36°. Find the distance between the two dolphins to the nearest meter. Example 3
Plan Because are horizontal lines, they are parallel. Thus, and because they are alternate interior angles. This means that Use Two Angles of Elevation or Depression Understand ΔMLK andΔMLJare right triangles. The distance between the dolphins isJK or JL – KL. Use the right triangles to find these two lengths. Example 3
Divide each side by tan Use Two Angles of Elevation or Depression Solve Multiply each side by JL. Use a calculator. Example 3
Divide each side by tan Use Two Angles of Elevation or Depression Multiply each side by KL. Use a calculator. Answer: The distance between the dolphins is JK – KL. JL – KL≈ 219.93 – 211.96, or about 8 meters. Example 3
Madison looks out her second-floor window, which is 15 feet above the ground. She observes two parked cars. One car is parked along the curb directly in front of her window and the other car is parked directly across the street from the first car. The angles of depression of Madison's line of sight to the cars are 17° and 31°. Find the distance between the two cars to the nearest foot. A. 14 ft B. 24 ft C. 37 ft D. 49 ft Example 3 | 677.169 | 1 |
similar triangles worksheet grade 10 pdf
Proportional Triangles Worksheet – Triangles are one of the most fundamental designs in geometry. Understanding the concept of triangles is essential for understanding more advanced geometric principles. In this blog post we will go over the various types of triangles such as triangle angles, and how to calculate the size and perimeter of a triangle, and also provide the examples for each. Types of Triangles There are three kinds in triangles, namely equilateral isosceles, and scalene. | 677.169 | 1 |
figure above, if what is the value of r ?
[#permalink]
06 Jun 2016, 18:03
2
Expert Reply
Explanation
From the figure, note that r° + s° must equal 180°. Therefore \(\frac{r}{(r+s)}=\frac{r}{180}\). Since you are also given in the question that \(\frac{r}{(r+s)}=\frac{5}{8}\), you can conclude that \(\frac{r}{180}=\frac{5}{8}\). Thus \(r = \frac{5*180}{8} = 112.5\), and the correct answer is 112.5.
Re: In the figure above, if what is the value of r ?
[#permalink]
03 May 2020, 05:25
1
This one was pretty straightforward - finally, I can say that about a geometry question! \( r+s =180 \) because the angles on a straight line must equal 180. Plug in 180 for the denominator and solve away!
Re: In the figure above, if what is the value of r ?
[#permalink]
05 Jun 2024, 10:38 | 677.169 | 1 |
Incidence geometry is non-trivial, I promise!
Mathematics students often joke about the word 'trivial'. The word in its serious usage means that a given object in a specific setting does not contain interesting information. Incidence geometry is simple – its definition is even trivial. But does that mean the subject itself is easy and 'trivial'? Not necessarily.
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Mathematics students often joke about the word 'trivial'. The word in its serious usage means that a given object in a specific setting does not contain interesting information. For instance, in the context of the field of mathematics known as 'group theory', which is concerned with transformations like the multiplication and division of numbers and symmetries of shapes, a group of transformations is called 'trivial' if it has only one element – the do-nothing element. The trivial group has no structure apart from the fact that it does nothing, which is not very interesting, even to a mathematician.
This handy little word makes it easy to avoid writing out all the steps required in solving a problem, since at least some of them, and if we're lucky, maybe even all of them, are 'trivial'. It also makes it easy to make fun of your maths friends by calling whatever they are working on 'trivial', and therefore beneath you.
This brings me to incidence geometry, a beautiful and simple, which is not to say trivial, field of mathematics.
What is incidence geometry? That's easy – even trivial. By 'geometry' I mean space, and by 'incidence' I mean that in space we consider points and lines, and we say a point is 'incident' with a line if the point lies on that line. In mathematical terminology we say that incidence is a 'relation' (think 'relationship') between points and lines.
That's it.
We don't have rulers and lengths. We don't have compasses and angles. We don't have equations describing curves. We don't even have to define the dimension of our space.
We have points and lines, and points on lines, in space, and that is all.
Sounds kind of… trivial, no?
No! No, it is not trivial…
When I have shared with family that I am doing a Pure Mathematics Honours in geometry, I have more than once heard the reply, "Geometry! I thought you finished all that in high school!" I know they don't know the word 'trivial', or how it's used in mathematics, because if they did they would surely be crying "Trivial! Trivial!"
So, why is incidence geometry non-trivial?
Allow me to give you an example.
In case you don't know, projective geometry is defined by three incidence axioms. It is known that for all primes p and natural numbers n, there is a projective plane of prime power 'order' N = p^n (this is the number of points on each line). However, it is unknown whether there are projective planes of non-prime power order, except that if one exists, its order N will have remainder 1 or 2 when divided by 4. It was shown via exhaustive computation that there are no projective planes of order 10, but the order 12 case and beyond remain completely open.
The take-away: geometry even in its simplest form is far from finished, and incidence geometry is far from trivial | 677.169 | 1 |
Cosine Online Calculator – Simple and Accurate Tool
Our Cosine Calculator is a straightforward online tool that computes the cosine of a given angle in degrees. It's designed for anyone who needs quick and accurate trigonometric calculations, especially useful for students, teachers, and professionals in fields involving mathematics and physics.
Cosine Calculator
Enter an angle in degrees to calculate its cosine.
Angle (Degrees):
Cosine: --
Graph of cosine
How to Use the Calculator
Using the calculator is easy and intuitive:
Enter your desired angle in degrees into the input field. There's no limitation on the range of values you can enter.
Click on 'Calculate Cosine' to obtain the cosine value of the entered angle.
The result will be displayed instantly, showing the cosine of the angle.
If you need to perform another calculation, simply click 'Clear' to reset the input and outputs.
Understanding the Cosine Function
Cosine is a fundamental trigonometric function that relates an angle of a right triangle to the ratio of the adjacent side to the hypotenuse. The formula for cosine is:
cosine(θ) = adjacent / hypotenuse
Here, θ is the angle in degrees. The calculator internally converts this angle into radians since the cosine function in most programming languages, including JavaScript, uses radians for trigonometric calculations.
Educational Insights on Cosine
The cosine function is pivotal in understanding circular motion, wave patterns, and oscillations in physics. It's crucial for calculating projections, determining the force components in mechanics, and in various engineering applications.
In fields like architecture and surveying, cosine is used for determining distances and angles. It's also vital in computer graphics for rendering 3D models and calculating angles of view.
A thorough understanding of the cosine function is essential for anyone studying or working in scientific fields, where angles and their relationships are key. Our Cosine Calculator simplifies these computations, aiding in a deeper comprehension and practical application of these concepts. | 677.169 | 1 |
Included here are some activities students can do on the coordinate plane in relation to geometry. The main work is to plot points and then consider the relationships of new points after undergoing a certain transformation. These activities can directly help students with plotting points while indirectly preparing them for geometric
transformations, or vice-versa.
The activities are designed to be done after an initial lesson by the teacher is given. The activities provided for students are as follows: | 677.169 | 1 |
Law Of Cosines Worksheet
Law of cosines worksheet 1. For this case we will apply the following steps.
Algebra Worksheets Applying The Law Of Cosines Worksheet Algebra Worksheets Math Formulas Law Of Cosines
Applying the law of cosines.
Law of cosines worksheet. In the following example you will find the length of a side of a triangle using law of cosines. Some of the worksheets below are law of sines and cosines worksheet in pdf law of sines and law of cosines. Law of cosines worksheet pdf.
In the following example you will find the measure of an angle of a triangle using law of cosines. Students will practice applying the law of cosines to calculate the side length of a triangle and to calculate the measure of an angle. Students will also extend their thinking by applying the law of cosines to word problems and challenge questions.
Round to the nearest hundredth. Law of cosines substitute. Use the law of cosines to find the side opposite to the given angle.
Solve for the unknown in each triangle. Round to the nearest hundredth. 4 cases where law of cosines is the best choice use the law of sines and law of cosines to find missing dimensions.
O o2 p0m152 l kk 3u wtbaw dsmodfettwcakrhel xl8lgcg c y ka sl ul d kryirgmhlt os 4 3r 6e2s ke er ivie kd f 8 e umcaxdle c wqi5tbhs 8itn jf giqnbiatne r daulngkeobhr 9aj r2 d d worksheet by kuta software llc kuta software infinite algebra 2 name the law of cosines date period. In this first example we will look at solving an oblique triangle where the case sas exists. If the included angle is a right angle then the law of cosines is the same as the pythagorean theorem.
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This Self Checking Maze Has 11 Problems Involving The Law Of Sines And The Law Of Cosines Students Will Be Law Of Cosines Law Of Sines Word Problem Worksheets
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This Worksheet Covers Non Ambiguous Law Of Sines And Law Of Cosine Problems Students Solve For Missing Parts Of Triang Law Of Sines Worksheets Law Of Cosines
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Sin And Cosine Worksheets Law Of Cosines Trigonometry Worksheets Worksheets | 677.169 | 1 |
Basic Trig – Sine and Inverse Sine
If you have never had a trig class, you can do some pretty useful things with just a few trig functions. Bob Welds discusses the sine of an angle and how to use it. He also tells how to find an angle from two known sides.
Video Transcript:
Hi I'm Bob Welds and this is a first look a trigonometry.
Today we are going to pick up where the Pythagorean theorem left off. If you don't know how the Pythagorean theorem works then you probably want to go learn about that first. Also, we'll use a little bit of algebra but i don't think it will be a problem for you if you already know some basic algebra. To keep it simple, we're only going to talk about a thing called the sine of an angle today. Later we'll see other trig functions. But really, if you understand how the sine works, you'll get the others pretty easily.
Ok. Let's look at this right triangle. You see that it has a 30 degree angle in one corner. The side opposite of the 30 degree angle is 1 meter long, and the hypotenuse is 2 meters long. (remember that the hypotenuse is the longest side of a right triangle).
let's compare the opposite side to the hypotenuse by dividing one by two and see what we get….1 divided by two equals .5 — a half. um. so .. that was easy. What's the big deal? The deal is that we just found the "sine" of thirty degrees. The sine of thirty degrees is one half. we write it like this:
Here, let's do it again. this right triangle has a height of 4 meters and a hypotenuse of eight meters. The sine of 30 deg is four divided by eight. 4 divided by eight reduces to one half, again, the sine of thirty degrees is one half, or point five. You see it doesn't matter how big the triangle is; 30 degrees always has a sine of point five. Any 30 degree right triangle has a hypotenuse that is exactly twice as long as the opposite side. Here, you can google it…type sin 30 degrees and see what you get. be sure to type degrees or you'll get something else.
The sine of an angle tells us the answer to the division problem we get when we put the opposite side (that is, the side across from the angle) over the hypotenuse. So the sine of the angle is "ratio" of the opposite side to the hypotenuse. the opposite over the hypotenuse. In this case, we put one over two and see that it is one half or point five.
You might be asking what good it is to know that ratio is since we already know how to divide one by two or four by eight… well let's take a closer look. Here is another triangle, but with the same 30 degree angle in the corner. This time, we know the length of the hypotenuse and angle, but we don't know the opposite side. Look what we could do if we knew the sine of the angle…instead of saying "sine of 30 degrees" we could say point five. Because .5 is the sine of thirty degrees. ….and instead of saying hypotenuse, we could say fourteen, because that's how long this hypotenuse is. Now instead of using that silly question mark, we could call the length of the opposite side: "a". And voila! we have an algebra problem instead of a trigonometry problem.
Here is how we could solve the algebra part….multiply both sides by 14… on the left, we see 14 times .5 equals seven. and on the right side .the 14's cancel each other out…and look! A equals seven. That is something we could not have found if we didn't know this little bit of trigonometry.
Lets see how useful the sine of an angle can be. We know the sine of thirty is, but with our calculator we can find the sine of ANY angle. Lets try to find the length of the hypotenuse of this triangle. The angle is 25 degrees and the side opposite of the angle is 1.69 meters long. First write down what we know…The sine of 25 degrees is equal to the opposite side–1.690 over the Hypotenuse. Now, we need the sine of 25 degrees…This time, I'll use my calculator instead of google. First, we'll turn it to scientific mode…..then I'll be sure it knows to use degrees….now I'll type 25 and hit the sin button. Different calculators might have you hit the sine key first, so check your answers to be sure you are doing what your calculator expects….0.422 618 261 and it keeps going. We only need the first few decimal places of that. So instead of saying "sine of 25 degrees" we will say 0.4226. Here, I rewrite our equation.
Now it's an algebra problem. We need to isolate thy hypotenuse to see what it's value is. I'm going to call it "C" to keep things neat and clean. I'll multiply both sides by c….the c's cancel out on the right…..now i'll divide both sides by .4226. this leaves C on the left and a little division problem on the right. we use our calculator to type 1.690 divide by .4226 and we get something just a smidgen over 3.999 — a number very close to four meters.
Now we've seen that we can use the sine of an angle to find the sides of the triangle, and you can use the opposite side and the hypotenuse to find the sine of the angle. What if you knew the sine of an angle, not the angle itself? There's a trick for that. Let me show you. Here we have another right triangle. The side opposite of our angle is 2.5 meters long, and the hypotenuse is 3.889 inches long. You may be able to see right away that we could find the sine of the angle like we did a few minutes ago. here, remember? You see we just put 2.5 over 3.889. Since a fraction is just a division problem, we'll divide 2.5 by 3.889. That gives us .64283. So just like before, we know what the sign of the angle is. Only now we don't know what the angle itself is. To find the angle when we know the sine of the angle, we just run the sine function in reverse. That's called taking the ARC Sine or the INVERSE SINE. You will see it on your calculator as a sine with a -1 exponent, or you may see it spelled out as "A"sin or "arc sine" or even inverse sine. A lot of the time it's the same button as the sine button, but you push another button to enable this feature. I'll do it with this little calculator so you can see. I'll type .64284 and push this up arrow button. You'll see that my sine key turned into an inverse sine key.
Now when i press the inverse sine button i get an angle! Isn't that teh cats pajamas? My angle is just a tiny bit more than forty degrees. Ill round it off to the nearest tenth of a degree. Here's the original problem you can remember what we were solving. We knew the side opposite of the angle was 2.5 meters long, and we knew the hypotenuse was 3.889 meters long. We wanted to know the angle. First we found the sine of the angle, opposite over hypotenuse (2.5 over 3.889). We divided and found the sine of the angle was .64283. Then we used our new trick, the inverse sine to find teh angle was very close to forty degrees. I just think that's great. That's going to be very useful.
Ok, let's work some problems. Im going to give you a problem where you find the missing opposite side first. When you see sparky's paws, Pause teh video and try to fill in the blanks in the equation. You don't need to solve it yet, just fill the blanks in with what you see on the diagram….[pause]. Ok, here are the blanks filled in. The sine of 30 degrees is equal to "a" over 15. "a" is the opposite side, and 15 is the hypotenuse. teh sine of an angle is the opposite over the hypotenuse. Now lets see if we can turn the trig problem into an ordinary algebra problem. Let's see instead of saying sine of thirty degrees, what could we say? use a calculator to find the sine of 30 degrees, even if you know what it is. This way you can be sure your calculator is set up correctly.
I'll pause while you calculate. ok. i know you probably remembered that the sine of thirty degrees was .5, but hopefully you tried it on your calculator. It is really important that you know how to get the sine of an angle. Now we just have a little algebra problem on our hands. See if you can solve for "a." a is equal to 7.5. If you were able to solve it by looking at it, let me encourage you to go through the algebra steps to get good at them. Here, we multiply both sides by 15 to get rid of that fraction…. Then we see that 15*.5 is equal to 7.5. that's our answer.
Let's do one more. Let's do one where we find the angle by using the opposite side and the hypotenuse…. let's remember that if we know the opposite side and the hypotenuse, we can find the SINE of the angle. Let's do that first. Pause the video and see if you can fill in the blanks. Then find the sine of the unknown angle by dividing. That's right, 1 on top, 4.810 on the bottom. That means the sine of this angle is .2079 . Now for the second part, lets use the arc sine to find what the angle is….remember we know the sine is .2079 Olk, pause the video and be sure you know how to find the angle using the sine of the angle. Fill in the blanks and do the calculation on your calculator. OK, I got an answer that was ridiculously close to 12. I'm going to round it off to the nearest tenth of a degree.
Well that's our first look at trigonometry. I hope you can see how useful it is to find angle from those lengths and lengths from the angles. There is a LOT more to cover, but if you understand how the sine and inverse sine work you are on you way to using some very useful tools. I'm bob welds, and these are weldnotes! | 677.169 | 1 |
Description
Part of the line that consists of two endpoints Indicates a location and has no size Represented by a straight path Represented by a flat surface Points that lie on the same line Points and lines that lie on the same plane Part of the line that consist of one endpoint Two rays that share the same endpoint and form a line The real number that corresponds to a point Segments that have the same length Point that divides the segment into two congruent segments Point, line or ray that intersects at the midpoint Formed by two rays with the same endpoint The rays of the angle The endpoint of the rays forming an angle Angle less than 90 degrees Angle greater than 90 degrees Angle exactly 90 degrees Angle exactly 180 degrees angles with the same measure Two coplanar angles with a common side and vertex with no common interior points Two angles whose sides are opposite rays Two angles whose measures have a sum of 90 degrees Two angles whose measures have a sum of 180 degrees Pair of adjacent angles whose noncommon sides are opposite rays The ray that divides an angle into two congruent angles
Frequently Asked Questions
What is a matching sheet?
A matching sheet, or a matching quiz, is a sheet with two columns. In the first column there will be a word, statement or question, and in the second column are the answers, jumbled around in a different order.
Students will then match the items in column A with the related answers in column B. Here is an example of a simple matching sheet where students would match up the name of the baby animal in column A with the adult name of the same animal in column B:
Who can play matching sheets?
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Yes! We have full support for matching quiz templates in Spanish, French and Japanese with diacritics including over 100,000 images. You can use other languages just for your titles and instructions, or create an entire matching worksheet in another language. Matching sheets can be a fantastic tool for students learning new languages!
Can I convert my matching quiz template into other puzzles?
With WordMint you can create a template and then use it to convert into a variety of other executions - word search, word scramble, crosswords or many more.
Are matching sheets good for kids?
The teachers that use WordMint love that they are able to create matching quiz templates that challenge their students cognitive abilities, and test their comprehension in a new and interesting way.
You can theme your matching sheet, and the ability to use different languages means that you can work language learning into your lessons as well. Because WordMint templates are totally custom, you can create a matching quiz for kids that suits their age and education level. | 677.169 | 1 |
In trapezoid EFGH, \overline{EF} \parallel \overline{GH}, and P is the point on \overline{EH} such that EP:PH = 1:2. If the area of triangle PEF is 6, and the area of triangle PGH is 6, then find the area of trapezoid EFGH. | 677.169 | 1 |
Sia
Question
Math
Posted 2 months ago
The figure below shows a right pyramid whose base is a square.
Note: Figure not to scale.
Which shape does the intersection of the horizontal plane with the pyramid look like?
Choose 1 answer:
(A)
(B)
(C)
(D)
Sign in to unlock the answer
Answer from Sia
Posted 2 months ago
Solution by Steps
step 2
When a horizontal plane intersects a right pyramid, the shape of the intersection depends on the height at which the plane intersects the pyramid
step 3
If the plane intersects the pyramid below the apex but above the base, the intersection will be a smaller square
step 4
Since the base of the pyramid is a square, any horizontal slice (parallel to the base) will also be a square
A
Key Concept
Intersection of a horizontal plane with a right pyramid
Explanation
When a horizontal plane intersects a right pyramid with a square base, the intersection is a square if the plane is parallel to the base and below the apex. | 677.169 | 1 |
find angles in isosceles triangles
Finding Angles In Isosceles Triangles Worksheet – Triangles are one of the most fundamental designs in geometry. Knowing how triangles work is essential to mastering more advanced geometric concepts. In this blog post we will go over the different kinds of triangles with triangle angles. We will also discuss how to calculate the extent and perimeter of any triangle and will provide instances of each. Types of Triangles There are three kinds of triangulars: Equilateral, isosceles, and … Read more | 677.169 | 1 |
A protractor is a measuring instrument, typically made of transparent plastic or glass, for measuring angles. Most protractors measure angles in degrees. Radian-scale protractors measure angles in radians. Most protractors are divided into 180 equal parts.
– High quality and durable
– Extra strong hard body
– You can use these geometry protractors to get the angle and length quickly
– Transparent colour allows for viewing through to the page
– Ideal for school, office and home use | 677.169 | 1 |
Cot 390 Degrees
The value of cot 390 degrees is 1.7320508. . .. Cot 390 degrees in radians is written as cot (390° × π/180°), i.e., cot (13π/6) or cot (6.806784. . .). In this article, we will discuss the methods to find the value of cot 390 degrees with examples.
Cot 390°: √3
Cot 390° in decimal: 1.7320508. . .
Cot (-390 degrees): -1.7320508. . . or -√3
Cot 390° in radians: cot (13π/6) or cot (6.8067840 . . .)
What is the Value of Cot 390 Degrees?
The value of cot 390 degrees in decimal is 1.732050807. . .. Cot 390 degrees can also be expressed using the equivalent of the given angle (390 degrees) in radians (6.80678 . . .)
Methods to Find Value of Cot 390 Degrees
The cotangent function is positive in the 1st quadrant. The value of cot 390° is given as 1.73205. . . We can find the value of cot 390 degrees by:
Using Unit Circle
Using Trigonometric Functions
Cot 390 Degrees Using Unit Circle
To find the value of cot 390 degrees using the unit circle, represent 390° in the form (1 × 360°) + 30° [∵ 390°>360°] ∵ The angle 390° is coterminal to 30° angle and also cotangent is a periodic function, cot 390° = cot 30°.
Rotate 'r' anticlockwise to form 30° or 390° angle with the positive x-axis.
The cot of 390 degrees equals the x-coordinate(0.866) divided by y-coordinate(0.5) of the point of intersection (0.866, 0.5) of unit circle and r.
How to Find the Value of Cot 390 Degrees?
The value of cot 390 degrees can be calculated by constructing an angle of 390° with the x-axis, and then finding the coordinates of the corresponding point (0.866, 0.5) on the unit circle. The value of cot 390° is equal to the x-coordinate(0.866) divided by the y-coordinate (0.5). ∴ cot 390° = 1.7321
What is the Exact Value of Cot 390 Degrees?
The exact value of cot 390 degrees can be given accurately up to 8 decimal places as 1.73205080 or as √3.
What is the Value of Cot 390 Degrees in Terms of Tan 390°?
Since the cotangent function is the reciprocal of the tangent function, we can write cot 390° as 1/tan(390°). The value of tan 390° is equal to 1/√3. | 677.169 | 1 |
sss sas asa aas hl examples
Congruent Triangles Sss Sas Asa Aas Hl Worksheet – Triangles are among the most fundamental designs in geometry. Understanding triangles is crucial for learning more advanced geometric concepts. In this blog post We will review the different kinds of triangles that are triangle angles. We will also explain how to determine the areas and perimeters of a triangle, and also provide some examples to illustrate each. Types of Triangles There are three types that of … Read more | 677.169 | 1 |
Cow-culus and Elegant Geometry - Numberphile | Summary and Q&A
TL;DR
A calculus problem that challenges students to find the shortest path between a cow and a river can be solved easily using geometry.
Install to Summarize YouTube Videos and Get Transcripts
Key Insights
🤠 Reflecting the real farmer and cow across the river simplifies the problem to one with a clear and easy solution.
🙂 The law of reflection for light and the farmer's strategy both involve finding the shortest path.
❓ Calculus and geometry provide different methods for solving optimization problems in mathematics.
Transcript
A problem that I give to my calculus students every
time I teach calculus; we spend a lot of time doing it with half the techniques, my students at the end
of the exercise are: I hope Zvezda doesn't give this on an exam. And then I ask them, would you like
to see a three to five line solution that a smart fifth grader would understand? And...
Read More
Questions & Answers
Q: How does reflecting the farmer and cow across the river simplify the problem?
Reflecting the farmer and cow creates an identical scenario where the shortest path is straightforward. Any path the real farmer takes can be mimicked by the reflected farmer, resulting in equal distances.
Q: Why is the law of reflection for light relevant to this problem?
The fact that light follows the shortest path in the reflection process is similar to what the farmer does. The light's behavior helps explain why the farmer's shortest path strategy is effective.
Q: Could the problem be solved without calculus or geometry?
While calculus provides a rigorous solution, the geometry approach is simpler. However, as the problem becomes more complex or involves additional variables, the calculus approach may become necessary.
Q: Are there practical applications for this problem?
The problem illustrates optimization, which has applications in fields like economics. Real-world scenarios involving multiple variables and constraints can benefit from similar problem-solving approaches.
Summary & Key Takeaways
The problem involves a cow with a broken leg that needs water from a river. The farmer needs to find the shortest path from a point on the river to the cow.
Using calculus, the solution involves finding the minimum of a function and taking its derivative. This can be a complicated task.
However, by using geometry and reflecting the farmer and cow across the river, it can be shown that the shortest path is simply 1 kilometer downstream from the starting point. | 677.169 | 1 |
File(s) under permanent embargo
Why is the tetrahedral bond angle 109°? The tetrahedron-in-a-cube
The common question of why the tetrahedral angle is 109.471° can be answered using a tetrahedron-in-a-cube, along with some Year 10 level mathematics. The tetrahedron-in-a-cube can also be used to demonstrate the non-polarity of tetrahedral molecules, the relationship between different types of lattice structures, and to demonstrate that inductive reasoning does not always provide the correct answer. | 677.169 | 1 |
multiWhich angle is angle CBD in a quadrilateral?
Angle CBD is the interior angle of a quadrilateral. It is the angle formed between side CB and side CD within the quadrilateral.
Angle CBD is the interior angle of a quadrilateral. It is the angle formed between side CB and side CD within the quadrilateral.
What does incident angle equal to reflected angle mean?
The statement "incident angle equals reflected angle" refers to the law of reflection, which states that the angle at which a ligh...
The statement "incident angle equals reflected angle" refers to the law of reflection, which states that the angle at which a light ray hits a surface (incident angle) is equal to the angle at which it is reflected off the surface (reflected angle). This law holds true for any smooth surface, such as a mirror or a still body of water.
What angle is the angle CBD in a quadrilateral?
The angle CBD in a quadrilateral can vary depending on the specific quadrilateral. In a general quadrilateral, the angle CBD could...
The angle CBD in a quadrilateral can vary depending on the specific quadrilateral. In a general quadrilateral, the angle CBD could be an acute angle, obtuse angle, or a right angle. It could also be a reflex angle if the angle measures greater than 180 degrees. The specific measurement of angle CBD would depend on the specific dimensions and properties of the quadrilateral.
Source:AI generated from FAQ.net
Is the altitude angle equal to the depth angle?
No, the altitude angle and the depth angle are not equal. The altitude angle is the angle between the line of sight to an object a...
No, the altitude angle and the depth angle are not equal. The altitude angle is the angle between the line of sight to an object and the horizontal plane, while the depth angle is the angle between the line of sight to an object and the vertical plane. These angles are measured in different planes and therefore are not equal All secureWhich angle staircase?
A 90-degree angle staircase is the most common type of staircase found in homes and buildings. This type of staircase turns at a r...
A 90-degree angle staircase is the most common type of staircase found in homes and buildings. This type of staircase turns at a right angle as you ascend or descend. It is practical and efficient in terms of space utilization and is easy to navigate for most people.
What is the deflection angle and the phase shift angle?
The deflection angle is the angle by which a wave is bent or redirected when it encounters a boundary or interface between two dif...
The deflection angle is the angle by which a wave is bent or redirected when it encounters a boundary or interface between two different mediums. It is measured from the incident wave's original direction to its new direction after reflection or refraction. On the other hand, the phase shift angle is the difference in phase between two waves at a specific point in time. It is measured in degrees or radians and indicates how much one wave is shifted relative to another wave.
How to - (angle1 + angle2), where angle1 and angle2 are the angles of the two lines with the horizontal axis. This formula will give you the angle between the two lines in degrees.
What is the difference between interior angle and exterior angle?
Interior angles are the angles formed inside a polygon, while exterior angles are the angles formed outside a polygon. Interior an...
Interior angles are the angles formed inside a polygon, while exterior angles are the angles formed outside a polygon. Interior angles are always less than 180 degrees, while exterior angles are always greater than 180 degrees. The sum of the interior angles of a polygon is always constant, while the sum of the exterior angles of a polygon is always 360 degrees Process you calculate another angle with two sides and one angle?
Yes, it is possible to calculate another angle with two sides and one angle using the Law of Cosines. By knowing two sides and the...
Yes, it is possible to calculate another angle with two sides and one angle using the Law of Cosines. By knowing two sides and the included angle of a triangle, we can use the Law of Cosines formula to find the third side. Once we have all three sides, we can then use the Law of Cosines or Law of Sines to calculate the other angles of the triangle.
How do you degrees - (angle of line 1 + angle of line 2). This formula will give you the angle at which the two lines intersect.
How do I calculate angle minutes when angle seconds are given?
To calculate angle minutes when angle seconds are given, you can simply divide the angle seconds by 60. This is because there are...
To calculate angle minutes when angle seconds are given, you can simply divide the angle seconds by 60. This is because there are 60 seconds in a minute. For example, if you have an angle of 30 seconds, you would divide 30 by 60 to get 0.5 minutes. Therefore, the angle in minutes would be 0.5.
Source:AI generated from FAQ.net
Is the sine of an angle proportional to the angle size?
Yes, the sine of an angle is proportional to the angle size. This is because the sine function is defined as the ratio of the leng...
Yes, the sine of an angle is proportional to the angle size. This is because the sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle. As the angle increases, the length of the opposite side also increases, resulting in a proportional relationship between the sine of the angle and the angle size. This relationship is fundamental to trigonometry and is used to solve various problems involving angles and sides in triangles | 677.169 | 1 |
Tag: geometry
Suppose you have two distinct points anywhere on the coordinate plane. If I tell you that a parabola with a vertical line of symmetry passes through those two points, where on the plane could that parabola's vertex be?
Since the two points can be anywhere, let's define our coordinate system so that the origin is located at the midpoint between the two points. This way, by symmetry, we can say the two points are located at $A(-u,-v)$ and $B(u,v).$ A parabola with vertical axis of symmetry and vertex located at $(x,y)$ satisfies an equation of the form $v-y=a(u-x)^2$. Given that such a parabola passes through $A$ and $B$, the equation must be satisfied when we substitute the coordinates of $A$ and $B$:
\begin{align}
-v-y &= a(-u-x)^2 \\
v-y &= a(u-x)^2
\end{align}Eliminating $a$ from this pair of equations, we are left with a single equation that relates the coordinates of the vertex $(x,y)$ and the coordinates $u$ and $v$ that determine the locations of $A$ and $B$.
\[
y = \frac{v}{2u}\left( x + \frac{u^2}{x} \right)
\]The set of points $(x,y)$ is a hyperbola centered at the origin (the midpoint of $A$ and $B$), whose asymptotes are the lines $x=0$ and $y=\tfrac{v}{2u}x$. Here is a figure showing the point and the hyperbola corresponding to the locus of possible locations of the vertex.
Geometric visualization
One way to visualize the locus of possible parabola vertex locations is to use the geometric definition of a parabola: A parabola is the set of points equidistant from a point $F$ (the focus) and a line (the directrix). Here is a diagram showing this construction in action:
As $P'$ slides along the directrix, the parabola is the set of points $P$ that are equidistant to the directrix and the point $F$.
In the original problem, we are considering parabolas with a vertical axis of symmetry, therefore the directrix must be horizontal. We can find the location of the focus by drawing circles centered at $A$ and $B$ that are tangent to the directrix, and the focus must be located at an intersection point of the circles. Finally, the vertex of the parabola is the midpoint between the focus and the directrix. As we pick different locations for the directrix, we sweep out all possible locations of the vertices.
In the diagram above, the two possible foci are at $F_1$ and $F_2$, with corresponding vertices $P_1$ and $P_2$, respectively. The associated parabola are shown in violet. As we translate the directrix vertically up and down, the points $P_1$ and $P_2$ trace out the green hyperbola. If you would like to see this for yourself, here is an interactive Geogebra visualization (you can move the directrix or the point $B$). Note: you may need to zoom/pan to see the figure.
When light passes through a polarizer, only the light whose polarization aligns with the polarizer passes through. When they aren't perfectly aligned, only the component of the light that's in the direction of the polarizer passes through. For example, here is what happens if you use two polarizers, the first at 45 degrees, and the second at 90 degrees. The length of the original vector is decreased by a factor of 1/2.
I have tons of polarizers, and each one also reflects 1 percent of any light that hits it — no matter its polarization or orientation — while polarizing the remaining 99 percent of the light. I'm interested in horizontally polarizing as much of the incoming light as possible. How many polarizers should I use?
Suppose we use $n$ polarizers, each with efficiency $\alpha\in (0,1)$, and we orient them at angles $\theta_1,\dots,\theta_n$, as shown below. Our task is to pick the angles $\theta_i$ so that the ratio $\frac{|OB|}{|OA|}$ is maximized.
When polarizer $i$ is applied, the vector gets multiplied by $\alpha\cos(\theta_i)$. Therefore, our task is to solve the following optimization problem for the overall efficiency $\beta_n$:
It turns out that the optimal configuration is for all angles to be equal. To see why this is the case, note that we can equivalently maximize the log of this product. In other words, maximize
\[
\sum_{i=1}^n \log \cos (\theta_i)
\]Consider the function $f(x)=\log\cos(x)$. Since $f'{}'(x) = -\sec^2(x) \leq 0$ on $x\in[0,\tfrac{\pi}{2}]$, it follows that $f$ is a concave function. By Jensen's inequality,
\[
\frac{f(x_1)+\cdots+f(x_n)}{n} \leq f\biggl(\frac{x_1+\cdots+x_n}{n}\biggr)
\]Applying this to our function, we conclude that:
\[
\sum_{i=1}^n \log \cos(\theta_i) \leq n \log\cos\biggl(\frac{\theta_1+\cdots+\theta_n}{n}\biggr) = n \log\cos\biggl(\frac{\pi}{2n}\biggr),
\]where the equality is due to the fact that we know $\theta_1+\cdots+\theta_n=\tfrac{\pi}{2}$. We can also achieve this equality by setting $\theta_1=\ldots=\theta_n=\tfrac{\pi}{2n}$, so the optimal thing to do is to make all angles equal. Substituting this into our optimization problem, we obtain a formula for the overall efficiency $\beta$:
or about 80%. We can ask a more general question of how the overall efficiency varies for different values of $\alpha$ and $n$. Here is a plot comparing a wide range of possible values:
Limiting cases
The optimal $n$ occurs when $\tfrac{\mathrm{d}}{\mathrm{d}n}\beta_n = 0$, or equivalently, $\tfrac{\mathrm{d}}{\mathrm{d}n}\log(\beta_n) = 0$. This leads to the equation
\[
\log (\alpha )+\frac{\pi}{2n} \tan \left(\frac{\pi }{2 n}\right)+\log \left(\cos \left(\frac{\pi }{2 n}\right)\right) = 0
\]We can solve this for $\alpha$ and obtain
\[
\alpha = \exp\left[-\frac{\pi}{2n} \tan\left(\frac{\pi }{2 n}\right)-\log \cos\left(\frac{\pi }{2 n}\right)\right]
\]This is an exact expression relating the $n$ that is optimal for a given $\alpha$. Of course, this assumes we can use a fractional number of polarizers, but it should be adequate to study the limit when $\alpha\to 1$ (and therefore $n\to\infty$). Taking an asymptotic expansion of the right-hand side, we obtain $\alpha \approx 1-\frac{\pi^2}{8n^2}$, or equivalently
$\displaystyle
n \approx \frac{\pi }{\sqrt{8}\sqrt{1-\alpha }}
$
This allows us to estimate the number of layers required as a function of each polarizer's efficiency. When $\alpha=0.99$, we obtain $n\approx 11.1$, which is close to the result we found previously. Similarly, if $\alpha=0.9999$, we obtain $n=111$, which matches the peak of the red curve in the plot above.
We might also be interested in the heights of these peaks; so what is the maximum overall efficiency $\beta$ that we can obtain if our individual polarizers each have efficiency $\alpha$? Performing a similar expansion to the one for $\alpha$, we obtain:
\[
\beta = \alpha^n \cos^n\left(\frac{\pi}{2n}\right) \approx 1-\frac{\pi ^2}{4 n}+\frac{\pi ^4}{32 n^2}
\]Eliminating $n$ from the $\alpha$ and $\beta$ equations, we obtain the approximation
\[
\beta \approx 1-\frac{\pi}{\sqrt{2}}\sqrt{1-\alpha }+\frac{\pi^2}{4}(1-\alpha )
\]And as anticipated, as $\alpha\to1$, we obtain $\beta\to 1$.
The larger regular hexagon in the diagram below has a side length of 1. What is the side length of the smaller regular hexagon?
If you look very closely, there are two more, even smaller hexagons on top. What are their side lengths?
Let $x_n$ denote the side length of the $n^\text{th}$ hexagon, and we will use the convention that $x_0=1$ refers to the largest one. We can compute the side lengths recursively using the Pythagorean theorem. Here is the picture:
Here, $O$ is the center of the circle, and $AB$ is the $n^\text{th}$ side length. So $|OA|=1$ and $|AB|=x_n$. The triangle $AOC$ is a right-angle triangle, which gives us $|OC| = \sqrt{1-\tfrac{1}{4}x_n^2}$. The $(n+1)^\text{st}$ hexagon has the property that point $F$ lies on the circle. Therefore, $|OF|=1$. By symmetry, the point $G$ is the midpoint of the top side, so we have $|FG| = \tfrac{1}{2}x_{n+1}$. Since a regular hexagon is made up of six equilateral triangles, the height is $|GC| = \sqrt{3} x_{n+1}$. Finally, $FOG$ is a right-angle triangle, so by the Pythagorean theorem, we have $|OF|^2 = |FG|^2 + |OG|^2$. Substituting in the lengths we found, we obtain
\[
1 = \frac{1}{4}x_{n+1}^2 + \left( \sqrt{3} x_{n+1} + \sqrt{1-\frac{1}{4}x_{n}^2}\right)^2
\]The recursion looks a little tidier if we express it in terms of the angles $\angle AOG$ and $\angle FOG$, but since we are after the side lengths, we will keep things in terms of $x_n$. Expanding the square above and simplifying, we obtain a quadratic equation for $x_{n+1}$:
\[
13 x_{n+1}^2 + 8\sqrt{3}\sqrt{1-\tfrac{1}{4}x_n^2}\, x_{n+1}-x_n^2=0
\]Solving for $x_{n+1}$, we see that there will be two roots, one positive and one negative. We can ignore the negative root, and we obtain
To find successive side lengths, simply start with $x_0=1$, and evaluate the expressions above in a recursive fashion. The first few lengths are:
\[
\left\{1,\frac{1}{13},\frac{\sqrt{8113}-90}{169},\frac{\sqrt{1387141-180 \sqrt{8113}}-12 \sqrt{8113}-90}{2197},\dots\right\}
\]Numerically, these evaluate to:
\[
\{1,\, 7.6923\times 10^{-2},\, 4.2718\times 10^{-4},\, 1.3170\times 10^{-8},\,\dots \}
\]The sequence decreases very rapidly to zero. We can see this if we plot the sequence of side lengths on a log scale:
In fact, $x_n\to 0$ doubly exponentially! To see why we can make a few approximations:
\begin{align}
x_{n+1} &= \frac{1}{13}\left( \sqrt{48+x_n^2}-\sqrt{48-12x_n^2}\right)\\
&= \frac{1}{13} \frac{(\sqrt{48+x_n^2}-\sqrt{48-12x_n^2})(\sqrt{48+x_n^2}+\sqrt{48-12x_n^2})}{\sqrt{48+x_n^2}+\sqrt{48-12x_n^2}}\\
&= \frac{x_n^2}{\sqrt{48+x_n^2}+\sqrt{48-12x_n^2}} \\
&\approx \frac{1}{2\sqrt{48}} x_n^2 \\
&=\frac{1}{8\sqrt{3}} x_n^2
\end{align}Using this approximation together with $x_0=1$, we obtain the approximate formula:
\[
x_n \approx \left(8 \sqrt{3}\right)^{-(2^n-1)}
\]Plotting the true side lengths together with the approximation, we see that the approximation is a bit smaller, but still very close:
Amare the ant is traveling within Triangle ABC, as shown below. Angle A measures 15 degrees, and sides AB and AC both have length 1.
Amare must:
Start at point B.
Second, touch a point — any point — on side AC.
Third, touch a point — any point — back on side AB.
Finally, proceed to a point — any point — on side AC (not necessarily the same point he touched earlier).
What is the shortest distance Amare can travel to complete the desired path?
I solved the problem in two different ways. The elegant solution: [Show Solution]
Let's solve a slightly more general version of the problem. Suppose the triangle has angle $\theta$ at point $A$, and suppose Amare's next three waypoints are at $D$, $E$, and $F$, as shown below.
Reflect the triangle about the line $AC$, so $B \mapsto B'$ and $E \mapsto E'$, as shown:
Now reflect the new triangle about the line $AB'$, so $C\mapsto C'$ and $F \mapsto F'$:
The key insight is that since the reflections preserve lengths, the path $BD+DE+EF$ followed by Amare has the same length as the path $BD+DE'+E'F'$ shown in red below:
If we move the points $E$ and $F$, then the points $E'$ and $F'$ move accordingly, and we obtain another possible path:
Rather than picking $D,E,F$, we can instead pick $D,E',F'$. Since the goal is to minimize the total distance, it's clear that we should place $F'$ such that $AF' \perp BF'$, and $D$ and $E'$ should be placed so that all three points lie on a line. This produces the figure:
So the shortest distance Amare can travel can be found by examining the right triangle $ABF'$. Since $AB$ has length $1$, we conclude that that
$\displaystyle
\text{Minimum distance } = \sin(3\theta)
$
In the case where $\theta = 15^\circ$, we get a length of $\sin(45^\circ) = \frac{1}{\sqrt{2}} \approx 0.7071$.
Note that this solution only works if $3\theta \lt 90^\circ$, i.e. $0 \lt \theta \lt 30^\circ$. In the case that $\theta \geq 30^\circ$, we obtain the degenerate solution $D = E = F = A$, so Amare should head directly to point $A$ and the total distance traveled is $1$.
We can also use calculus to solve the problem. Let's start with the same picture as before:
Suppose $|AD|=x$, $|AE|=y$, and $|AF|=z$. From the law of cosines:
\begin{align}
|BD| &= \sqrt{1+x^2-2x\cos(\theta)} \\
|DE| &= \sqrt{x^2+y^2-2xy\cos(\theta)} \\
|EF| &= \sqrt{y^2+z^2-2yz\cos(\theta)}
\end{align}Let $f(x,y,z)$ be the sum of these three distances, which is the total distance traveled by Amare. We want to find $x,y,z$ such that $f(x,y,z)$ is minimized. A necessary condition for minimality is that the partial derivatives of $f$ with respect to $x,y,z$ should be zero. Let's start with $z$:
\[
\frac{\partial}{\partial z}f(x,y,z) = \frac{z-y \cos (\theta )}{\sqrt{y^2+z^2-2 y z \cos (\theta )}}
\]Setting this equal to zero, we conclude that $z=y\cos(\theta)$. We can now substitute this value of $z$ into the definition for $f$ and we obtain a simpler expression in only two variables:
\begin{align}
g(x,y) &= f(x,y,y\cos(\theta)) \\
&= \sqrt{1+x^2-2x\cos(\theta)} + \sqrt{x^2+y^2-2xy\cos(\theta)} + y\sin(\theta)
\end{align}To make sure there is no funny business going on, let's plot this function to see what it looks like for $\theta=15^\circ$. Here is a contour plot:
If you read the first solution, then it should come as no surprise that the total distance is also equal to $\sqrt{1-z^2}$, since (based on the last figure of the first solution), we have $z=|AF|=|AF'|$ and $|BF'|^2 + |AF'|^2 = 1$ by the Pythagorean theorem.
The only polyhedron with six edges is a tetrahedron, which is a pyramid with a triangular base. Two of the faces will be equilateral triangles that share a common edge. This accounts for the five edges of length 1. The length of the sixth edge is determined by the angle between the faces, which we will call $\theta$. Here is an animation showing the different tetrahedra you get as you vary $\theta$:
In this diagram, $AB=BC=AC=AD=BD=1$ and $OD=OC=\frac{\sqrt{3}}{2}$.
The volume is equal to
\begin{align}
V&=\frac{1}{3}(\text{Area of base})\cdot(\text{altitude}) \\
&= \frac{1}{3}(\text{Area ABC})\cdot(DG) \\
&= \frac{1}{3}\left( \frac{1}{2} (AB)(OC) \right) \cdot \left( (OD) \sin\theta \right) \\
&= \frac{1}{3} \cdot \frac{1}{2}\cdot 1 \cdot \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} \cdot \sin\theta \\
&= \frac{1}{8}\cdot\sin\theta
\end{align}Therefore, the maximum volume is $\frac{1}{8}$ and it occurs when $\theta=90^\circ$. This is intuitive because the area of the base is fixed, so the largest volume occurs when the altitude is as large as possible.
Since we have a fixed amount of crust that we will shape into a cylinder, the problem is one of maximizing the volume for a fixed area. This is known as the isovolume problem (which is a misnomer; it should really be called the isoarea problem).
Let's suppose the crust has an area of $A$. Given this area, we would like to maximize the volume of the cylinder. Let's suppose the radius of the base is $r$ and the height is $h$. Then the formulas are:
\[
A = 2\pi r^2 + 2\pi r h \qquad\text{and}\qquad V = \pi r^2 h.
\]Since the area $A$ is fixed, we are not free to choose $r$ and $h$ independently. Once $r$ is chosen, $h$ is uniquely determined by the equation for $A$. Solving for $h$ in terms of $A$ from the area equation, we obtain:
\[
h = \frac{A-2\pi r^2}{2\pi r}
\]Substituting this value of $h$ into the equation for $V$, we obtain:
\[
V= \frac{1}{2} r (A-2\pi r^2)
\]Here is a plot of what this function $V(r)$ looks like for a value of $A=1$:
We can maximize $V$ by looking for an $r$ such that $\frac{\mathrm{d}V}{\mathrm{d}r}=0$ and $\frac{\mathrm{d}^2V}{\mathrm{d}r^2}\lt 0$. In other words, at the optimal choice of $r$, the tangent to the curve $V(r)$ is flat and the curvature is negative (curves downward). This leads to the equations:
\[
\frac{1}{2}(A-6\pi r ^2) = 0 \qquad\text{and}\qquad -6\pi r \lt 0
\]Since $r\geq 0$ (radius can't be negative), the second equation is always satisfied, and the first equation implies that $r=\sqrt{\frac{A}{6\pi}}$. If $A=1$ as in the plot above, this leads to $r=0.23033$, which looks about right! With the optimal choice of $r$, the corresponding choice of $h$ becomes $h=\sqrt{\frac{2A}{3\pi}}$ and the corresponding optimal volume is $V=\frac{A^{3/2}}{3\sqrt{6\pi}}$. Interestingly, we can observe that with these choices,
\[
2r = 2\sqrt{\frac{A}{6\pi}} = \sqrt{\frac{4A}{6\pi}} = \sqrt{\frac{2A}{3\pi}} = h
\]So in order to maximize the area, the diameter should be equal to the height! In other words, when viewed from the side, our pie should have a square shape; we'll be making more of a cake rather than a pie.
The fraction of crust used for the base of the optimal pie is:
\[
\frac{\pi r^2}{A} = \frac{\pi \left(\frac{A}{6\pi}\right)}{A} = \frac{1}{6}.
\]
$\displaystyle
\begin{aligned}
&\text{So we should use $\tfrac{1}{6}$ of the crust for the base, $\tfrac{1}{6}$ for the top,}\\
&\text{and the remaining $\tfrac{2}{3}$ for the sides.}
\end{aligned}$
One way to explain the shape of this tall pie is to think back to the isovolume problem discussed earlier. The solution to the isovolume problem is a sphere. So in order to be as efficient as possible, the pie's shape should be as close to a sphere as possible, hence the tall shape. For a sphere, $A=4\pi r^2$ and $V=\frac{4}{3}\pi r^3$, so by eliminating $r$, we obtain $V=\frac{A^{3/2}}{6\sqrt{\pi}}$.
If we take the ratio of the volume of the sphere and the volume of the optimized cylinder, we obtain:
\[
\frac{V_\text{cylinder}}{V_\text{sphere}} = \frac{ \frac{A^{3/2}}{3\sqrt{6\pi}} }{ \frac{A^{3/2}}{6\sqrt{\pi}} } = \sqrt{\frac{2}{3}} \approx 0.8165
\]So the optimized cylindrical pie will have a volume which is about 82% of the largest possible volume.
Bonus: Different bottom and top area
If the top and bottom areas of the pie have different radii, then the shape is no longer a cylinder, but rather a frustom of a cone. The formulas are now a bit more complicated, because they depend on two radii ($r$ and $R$) and the height $h$:
\begin{align}
A &= \pi (r+R) \sqrt{h^2+(R-r)^2}+\pi \left(r^2+R^2\right)\\
V &= \frac{1}{3} \pi h \left(r^2+r R+R^2\right)
\end{align}Using a similar procedure as before (solving for $h$ in the area equation and substituting this into the volume equation), we obtain the following volume formula that depends only on the radii of the top and bottom
\[
V = \frac{\left(r^2+r R+R^2\right) \sqrt{\left(A-2 \pi r^2\right) \left(A-2 \pi R^2\right)}}{3 (r+R)}
\]A necessary condition for optimality is that both partial derivatives are zero, namely $\frac{\partial V}{\partial r} = \frac{\partial V}{\partial R} = 0$. This leads to the equations:
\begin{align}
\frac{\partial V}{\partial r} &= -\frac{r \left(A-2 \pi R^2\right) \left(2 \pi \left(4 r^2 R+2 r^3+2 r R^2+R^3\right)-A (r+2 R)\right)}{3 (r+R)^2 \sqrt{\left(A-2 \pi r^2\right) \left(A-2 \pi R^2\right)}} \\
\frac{\partial V}{\partial R} &= -\frac{R \left(A-2 \pi r^2\right) \left(2 \pi \left(2 r^2 R+r^3+4 r R^2+2 R^3\right)-A (2 r+R)\right)}{3 (r+R)^2 \sqrt{\left(A-2 \pi r^2\right) \left(A-2 \pi R^2\right)}}
\end{align}Much messier than the ordinary cylinder case… But we can simplify this. It must be true that $A\gt 2\pi R^2$ and $A\gt 2\pi r ^2$, since the area of the smaller circle plus the area of the sides must be at least the area of the larger circle (they are only equal when the pie is flat). So if $\frac{\partial V}{\partial r} = \frac{\partial V}{\partial R} = 0$, we can cancel a bunch of terms and we are left with:
\begin{align}
2 \pi \left(4 r^2 R+2 r^3+2 r R^2+R^3\right)-A (r+2 R) &= 0& &(1) \\
2 \pi \left(2 r^2 R+r^3+4 r R^2+2 R^3\right)-A (2 r+R) &= 0& &(2)
\end{align}This still looks rather messy to solve, but we can make a clever observation. If we subtract one equation from the other $(2)-(1)$ and factor, we obtain:
\[
(R-r) \left(A+2 \pi r^2+6 \pi r R+2 \pi R^2\right)=0
\]The right factor consists of a sum of positive terms, and therefore must be positive. So we conclude that $R=r$. Put another way, the only way we'll have an optimized volume is if the top and bottom circles are of equal size. This reduces our more complicated case to the simpler case we already solved above.
This week's Riddler Classic is a simple-looking question about the turning radius of a truck.
Suppose I'm driving a very long truck (with length L) with two front wheels and two rear wheels. (The truck is so long compared to its width that I can consider the two front wheels as being a single wheel, and the two rear wheels as being a single wheel.)
Suppose I can also rotate the front wheels by $\alpha$ and the back wheels — independently from the front wheels — by $\beta$. What is the truck's turning radius?
When operating normally (no slippage), wheels can never slide sideways. They can only lead to motion in the direction they are pointing. In a steady turn (front and rear wheel angles constant), the wheels each trace out a circle and each wheel is tangent to its circle. Thinking about this a bit more carefully, we come to the conclusion that the front and rear wheels cannot trace out the same circle, since that would cause either the front or rear to slip. Therefore, there are actually two turning radii (inner and outer). Here is a diagram illustrating the situation:
By the law of sines, we have:
\[
\frac{r}{\sin(\tfrac{\pi}{2}-\alpha)} = \frac{R}{\sin(\tfrac{\pi}{2}-\beta)}
\]and by the law of cosines, we have:
\[
L^2 = R^2 + r^2-2rR \cos(\alpha+\beta).
\]Putting these two together and solving for $r$ and $R$, we obtain:
We have a few cases depending on the relative size of $\alpha$ and $\beta$:
If $\alpha \gt \beta$ (front wheel can turn more than the back wheel), then $R \gt r$, so the turning radius of the front wheel is larger than that of the back wheel.
If $\alpha \lt \beta$ (back wheel can turn more than the front wheel), then $R \lt r$, so the turning radius of the back wheel is larger than that of the front wheel.
If $\alpha=\beta$ then $R=r$ and both wheels turn on the same circle.
Here is a live GeoGebra script I made so you can play around with different values of the front and rear angles; click and drag to pan, use the mouse wheel to zoom, and use the sliders to change the angles!
his week's Riddler Classic involves designing maximum-area polygons with a fixed budget on the length of the perimeter and the number of vertices. The original problem involved designing enclosures for hamsters, but I have paraphrased the problem to make it more concise.
You want to build a polygonal enclosure consisting of posts connected by walls. Each post weighs $k$ kg. The walls weigh $1$ kg per meter. You are allowed a maximum budget of $1$ kg for the posts and walls.
What's the greatest value of $k$ for which you should use four posts rather than three?
Extra credit: For which values of $k$ should you use five posts, six posts, seven posts, and so on?
For a fixed perimeter, the maximum-area polygon with $n$ sides is a regular polygon. Since we are interested in maximizing area, we should therefore always use regular polygons as our shape, and we should always use our entire mass budget of $1$ kg.
Let's assume the weight of the posts $k$ is given and fixed. If the perimeter is $p$ and the number of sides is $n$, the formula for the area of a regular $n$-gon is as follows (explanation here).
\[
A = \frac{p^2}{4n \tan\frac{\pi}{n}}
\]Our total budget constraint is that $p + kn \leq 1$. Since our goal is to maximize area we should always use our entire budget. So $p = 1-kn$. This means that $n \leq \frac{1}{k}$ in order to ensure we respect the weight budget. So for a fixed $k$, the maximum area of an $n$-gon enclosure is given by the formula
If $n=2$, we have $\tan(\pi/n) = \infty$ so $A=0$. This makes sense; an enclosure with only two sides will be flat and have zero area.
If $k\to 0$ we can take $n\to\infty$. A polygon with infinitely many sides but a fixed perimeter is simply a circle, so let's see if this checks out. Using the fact that $\lim_{n\to\infty} n \tan\tfrac{\pi}{n} = \pi$, we obtain a total area of $A = \tfrac{1}{4\pi}$. This is precisely the area of a circle with perimeter $1$! Indeed, if $2\pi r = 1$, then we obtain $r = \frac{1}{2\pi}$ and therefore $A = \pi r^2 = \frac{1}{4\pi}$.
So for each $k$, we should choose the $n \in \{2,3,4,\dots,\lfloor \tfrac{1}{k}\rfloor\}$ that maximizes $A$. Here is a plot showing plots of $A$ for each $n$ and $k$. When $k$ is large (posts weigh a lot), we try to use as few of them as possible. So at first, the triangle ($n=3$) is best. As we decrease $k$, it eventually becomes more efficient to use a square ($n=4$), and so on. As $k$ goes to zero, we obtain the solution with $n\to\infty$, which is the circular enclosure discussed above.
To find out where the transition occurs between $n$ and $n+1$, we should look for the value of $k$ at which the areas match (indicated by the black dots on the plot). So we want to find $k$ such that:
\[
\frac{(1-kn)^2}{4n\tan\frac{\pi}{n}} = \frac{(1-k(n+1))^2}{4(n+1)\tan \frac{\pi}{n+1}}
\]Solving this equation for $k$, we obtain:
We can interpret this formula as follows: $k_n$ is the smallest value of $k$ for which we should use $n$ posts in our enclosure. So if we make $k$ smaller, it becomes more efficient to use $(n+1)$ posts instead. Here are the first few numerical values of $k_n$:
This week's Riddler Classic is a problem about connecting dots to create as many non-intersecting polygons as possible. Here is the problem:
Polly Gawn loves to play "connect the dots." Today, she's playing a particularly challenging version of the game, which has six unlabeled dots on the page. She would like to connect them so that they form the vertices of a hexagon. To her surprise, she finds that there are many different hexagons she can draw, each with the same six vertices.
What is the greatest possible number of unique hexagons Polly can draw using six points?
(Hint: With four points, that answer is three. That is, Polly can draw up to three quadrilaterals, as long as one of the points lies inside the triangle formed by the other three. Otherwise, Polly would only be able to draw one quadrilateral.)
Extra Credit: What is the greatest possible number of unique heptagons Polly can draw using seven points?
It turns out this is a well-studied and notoriously difficult problem in combinatorial geometry. While it is tempting to ask the same question for 8, 9, or even $n$ points, we'll see that the problem gets very difficult very quickly, and the answer for general $n$ is not actually known!
First, we'll get some terminology out of the way:
A complete graph is a graph with every possible edge drawn in. The complete graph with $n$ nodes is denoted $K_n$.
A Hamiltonian cycle is a path through the graph that finishes where it started and visits every node exactly once. For the graph $K_n$, there are $\frac{1}{2}(n-1)!$ possible Hamiltonian cycles since we can fix the starting node and then there are $n-1$ nodes left to order. We divide by two because each cycle is counted twice (can be traversed forwards or backwards).
A realization of a graph is a particular way of arranging the nodes. Realizations are important because we care about whether edges cross or not; sometimes two different arrangements of the same graph will have different numbers of edge crossings.
The rectilinear crossings of a graph realization $G$ is the number of times edges cross when we connect the nodes with straight lines. We'll denote this number $\mathrm{rcr}(G)$.
The crossing-free Hamiltonian cycles of a graph realization $G$ is the number of different Hamiltonian cycles of $G$ that do not contain any edges that cross each other. We'll denote this number $\mathrm{cfhc}(G)$. These are also called "spanning cycles" or "simple polygonalizations" depending on who you ask.
For a simple example, let's start with $K_4$, the complete graph on 4 nodes. Although there are infinitely many ways to realize this graph (since we can arrange the 4 points in infinitely many ways), rearrangements of the points that preserve how the edges intersect one another are all equivalent for our purpose (this is an example of an equivalence class). So we only need to consider one representative from each class. These representative realizations are called order types. It turns out $K_4$ has two order types:
The first order type has no crossings ($\mathrm{rcr}(G)=0$) and three possible crossing-free Hamiltonian cycles. Here are the cycles:
The second order type has one crossing ($\mathrm{rcr}(G)=1$) and only one possible crossing-free Hamiltonian cycle:
The bad news…
Unfortunately, things get difficult from this point on as we add more nodes:
The number of order types for $K_n$ grows rapidly with $n$. For $n\ge 3$, they are: $\{1, 2, 3, 16, 135, 3315, 158817, 14309547, 2334512907,\dots\}$. This sequence is in OEIS. There is no known general formula.
The minimal rectilinear crossing number for $K_n$ over all possible realizations for $n\ge 3$ is: $\{0, 0, 1, 3, 9, 19, 36, 62, 102, 153,\dots\}$. This sequence is also in OEIS. No known formula known for this one either.
Finally, the minimal number of crossing-free Hamiltonian cycles for $K_n$ for $n \ge 3$ is: $\{1, 3, 8, 29, 92, 339, 1282, 4994,\dots\}$. And this is also in OEIS. You guessed it; no known formula.
These problems are related but different. For example, we might expect realizations with smaller crossing numbers to contain more crossing-free Hamiltonian cycles, since it is easier to avoid intersections when there are fewer of them. But this is not always the case. We might also expect realizations with more symmetry to contain more cycles; also not true in general. What I'm trying to get at is that there are no (known) tricks we can use to reduce this problem to a simpler one. Unfortunately, it appears the only way to solve such problems is to enumerate all order types (even this is difficult!), then enumerate all possible cycles, keeping only the ones that are crossing-free.
Many other variants are just as difficult: computing the crossing number (with curved edges allowed), the number of triangulations, the minimum number of convex polygons needed for a decomposition of the convex hull, and more. These sorts of problems belong to a branch of mathematics called combinatorial geometry. A great deal of research on the topic of order types, crossing numbers, and more, has been conducted by Prof. Oswin Aichholzer and colleagues. If you're interested in learning more, I recommend checking out his webpage here, which contains an up-to-date database of solutions to many of these and related problems for small-ish $n$.
Visualizing the solutions
I used the point layouts provided on Prof. Aichholzer's website and wrote some Python code to visualize the results. As a warm-up, I started with the case $K_5$.
The order types are sorted by their rectilinear crossing number. The one with the most crossing-free Hamiltonian cycles is the first one. Here they are:
Six nodes
Here are the 16 order types for $K_6$:
Here are the 29 crossing-free Hamiltonian cycles for the maximal configuration:
Seven nodes
Here are the 135 order types for $K_7$. Interestingly, there are three different realizations for the minimal rectilinear crossing number of $9$, and the most symmetric one (the first one) is not the one with the most crossing-free Hamiltonian cycles!
And here are the 92 crossing-free Hamiltonian cycles for the maximal configuration:
More nodes
The solutions get too large to visualize beyond 7 nodes, but here are the results for $n \leq 10$, courtesy once again of Prof. Aichholzer's webpage. "CFHC" stands for "crossing-free Hamiltonian cycles".
In this Riddler problem, the goal is to spread out settlements in a circle so that they are as far apart as possible:
Antisocial settlers are building houses on a prairie that's a perfect circle with a radius of 1 mile. Each settler wants to live as far apart from his or her nearest neighbor as possible. To accomplish that, the settlers will overcome their antisocial behavior and work together so that the average distance between each settler and his or her nearest neighbor is as large as possible.
At first, there were slated to be seven settlers. Arranging that was easy enough: One will build his house in the center of the circle, while the other six will form a regular hexagon along its circumference. Every settler will be exactly 1 mile from his nearest neighbor, so the average distance is 1 mile.
However, at the last minute, one settler cancels his move to the prairie altogether (he's really antisocial). That leaves six settlers. Does that mean the settlers can live further away from each other than they would have if there were seven settlers? Where will the six settlers ultimately build their houses, and what's the maximum average distance between nearest neighbors?
I approached this problem from a modeling and optimization perspective. If there are $N$ settlers and we imagine the coordinates of the settlers are $(x_i,y_i)$ for $i=1,\dots,N$ and $d_i$ is the distance between the $i^\text{th}$ settler and its nearest neighbor, then we can model the problem as follows:
\[
\begin{aligned}\underset{x,y,d\,\in\,\mathbb{R}^N}{\text{maximize}} \quad & \frac{1}{N}\sum_{i=1}^N d_i \\
\text{such that} \quad & x_i^2 + y_i^2 \le 1 &&\text{for }i=1,\dots,N\\
& (x_i-x_j)^2 + (y_i-y_j)^2 \ge d_i^2 &&\text{for }i,j=1,\dots,N\text{ and }j\ne i
\end{aligned}
\]The objective is to minimize the average distance between each settler and its nearest neighbor (the average of the $d_i$). The first constraint says that each settler must be within the circle (a distance of at most $1$ from the origin). The second constraint says that the $i^\text{th}$ settler is a distance at least $d_i$ from each of the other settlers.
The game is then to find $(x_i,y_i,d_i)$ that satisfy these constraints and maximize the objective. Broadly speaking, this is an example of a mathematical optimization problem. Unfortunately, the problem as stated is nonconvex problem because of the form of the second constraint. This means that the problem may have local maximizers that are not globally optimal. There are two ways forward:
First, we could use local search: start with randomly placed settlers, wiggle their positions in ways such that the objective continues to increase, and then stop once we can no longer improve the objective. Examples of such approaches include hill-climbing and interior-point methods. In general, such approaches only find local maxima, so we should try many random starting positions to ensure we find the best possible answer.
Second, we could use global search: these are methods that attempt to find a globally optimal solution by considering many configurations simultaneously and adding random perturbations that can "kick" a solution out of a local maximum. Examples include simulated annealing and particle swarms. These approaches tend to be much slower than local search, but they have a much better chance of finding a global optimum.
Ultimately, I used local search with several random initializations. Here is how the maximum average distance scales with the number of settlements.
The bar plot also shows how well we would do if we distributed the settlements evenly on the circumference ("all on the border") and if we put one in the middle and the rest evenly distributed on the circumference ("one in the center"). We can also examine what the settlement distributions actually look like. Here they are:
The case $N=6$ is particularly interesting because it has a settlement that is almost in the center, but not quite. Indeed, the average minimum distance to neighbors for this case is $1.002292$, so we benefit ever so slightly by placing one settlement off-center.
We can solve for the $N=6$ case analytically as well, but the solution isn't pretty (fair warning!) It turns out the settlements are distributed throughout the circle in the following way:
Here, $x$ is the distance between the origin and the point $A$. The other distances satisfy $z \lt y \lt x+1$, and the average distance to the nearest neighbor is therefore $\tfrac{1}{6}(1+x+2y+3z)$. Note that once we set $x$, this uniquely determines the positions of all the other points. After some messy trigonometry, we obtain the following equations relating $x$, $y$, $z$ and two auxiliary variables $p$ and $q$:
\begin{align}
y^2&=1+x-x^2-x^3\\
z^2&=1-x^2+2x\left(pq-\sqrt{1-p^2}\sqrt{1-q^2}\right) \\
p&= \tfrac{1}{2}(1-2x-x^2)\\
q&= \tfrac{1}{2}(1-x+x^2+x^3)
\end{align}We can eliminate $\{y,z,p,q\}$ and solve for the average distance $\bar d = \frac{1+x+2y+3z}{6}$ as a function of $x$ alone:
\[
\bar d = \frac{x+1}{12} \left(2+4 \sqrt{1-x}+3 \sqrt{2}\sqrt{1-x} \sqrt{\left(x^3+2 x^2-x+2\right)-x \sqrt{x+3} \sqrt{x^3+x^2-x+3}} \right)
\]We can then plot this function of $x$, and we get the following curve:
Interestingly, this function reaches its maximum just a bit after $x=0$. Zooming in, we can see more clearly:
The maximum occurs at about $x=0.0555108$ and the maximum value is $1.00229$, just as we found before. If we try to solve for this analytically by taking the derivative of this function, setting it equal to zero, and eliminating rationals, we obtain the following result… That the optimal $x$ a root of the following $30^\text{th}$ order polynomial:
\begin{align}
x^{30}&+\tfrac{152}{9} x^{29}+\tfrac{9259}{81} x^{28}+\tfrac{6851}{18} x^{27}+\tfrac{383363 }{648}x^{26}+\tfrac{1999495} {5832}x^{25}\\
&+\tfrac{43478263}{52488} x^{24}+\tfrac{75726373}{23328} x^{23}+\tfrac{3282369305}{1679616} x^{22}-\tfrac{31642768891}{7558272} x^{21}\\
&+\tfrac{367775655235}{136048896} x^{20}+\tfrac{392027905801}{34012224} x^{19}-\tfrac{2016793647847}{136048896} x^{18}-\tfrac{1258157703385}{68024448} x^{17}\\
&+\tfrac{4545130566235}{136048896} x^{16}-\tfrac{22059424877}{17006112} x^{15}-\tfrac{8448216227365}{136048896} x^{14}+\tfrac{2878062707167}{68024448} x^{13}\\
&+\tfrac{7398046956073}{136048896} x^{12}-\tfrac{312655708903}{3779136} x^{11}+\tfrac{17266856539}{1679616} x^{10}+\tfrac{63670102441}{839808} x^9\\
&-\tfrac{29827845749}{559872} x^8-\tfrac{541401565}{23328} x^7+\tfrac{619158287}{23328} x^6+\tfrac{6168305}{2592} x^5\\
&-\tfrac{156285}{32} x^4-\tfrac{5153}{36} x^3+\tfrac{3923}{12} x^2+\tfrac{45}{2} x-\tfrac{35}{16}
\end{align}Yuck! Here is what you get if you plot this polynomial; as expected, there is a root at $x=0.0555108$. | 677.169 | 1 |
So, I'm making shirt sleeve for menswear now and having some difficulties. I need to measure a length from an arc. Here's the pic:Here is the how I want it to be:
I want to make new biceps line drawn in red. Here is my files if you need it:
Why do you need arc if you just need points of the right angle triangle,you can use trigonometry? The point of the central line you are looking for has the length from crown3 equal to sqrt(27^2-24^2) ( see Pythagorean theorem)I am not sure whether I understand what you are looking for. If you want to know the length of the splines you can just look them up in the variables table; they also show when you hoover over the spline:
@Olgatron
Sadly, I'm not good at math… I don't understand what you're saying…
@moniaqua
No, I need to determine the new back sleeve corner at biceps level as in the picture with the red line. The instruction is 27cm from lowered sleeve cap diagonally, and 24 cm from the center line… I determine the 27cm using the arc at a distance tool, but the problem is, I need to measure 24cm from the 27cm-radius arc…
If you put a new point from Crown3 down along the central line) at the length equal to 12.37 cm and restore a perpendicular in this point equal to 24 cm you will get exactly the new back sleeve corner as per your image (and no ark needed)
Ah, ok. The arc doesn't help you at all here. Proceeding as @Olgatron says could help Math is not that bad; Pythagoras helps you to get the distance from crown3 to the third point of your triangle, as it has a 90° angle.
fyi, ((Spl_slf4a1a2_chest5+Spl_slf5_chestfront4+Line_slf4_slf5)/2) is C and ((Line_chest_chest2+Line_chestfront_chestfront2)/2) is B
Edit:
Don't mind. The value are swapped
I didn't have the brain yesterday to think it through,sorry. I am trying to understand what you want to calculate. Splines and Pythagoras don't get to well together; Pythagoras ist for straight lines.
Also, in
alexandria_tale:
((((Spl_slf4a1a2_chest5+Spl_slf5_chestfront4+Line_slf4_slf5)/2)^2)-
I see four opening brackets, but only three closing ones. Probably the fourth in the beginning leads to wrong calculation, but actually the editor should give an error message.
It's okay, thank you for replying! Hmmm, say I want to calculate A. B is the front and back underarm length, and C is the front and back armhole length (thus the spline). I think I found the problem, that B is longer than C. I don't know how that happen, is that mean that my pattern is wrong? Anyway, I swapped the value, the longer one minus the shorter one (B-C) and I found the value of A. Would there be a problem with me swapping the value? | 677.169 | 1 |
Question 2. Prove that the tangents drawn at the ends of the diameter of a Circle are Parallel. Solution:
Let AB be the diameter of a circle with Centre O. PA and PB are the tangents to the Circle at pants A and B respectively.
Now ∠PAB=90°
and∠QBA=90°
⇒ ∠PAB + ∠QBA = 90° +90° = 180°
PA ll QB
Question 3. prove that the perpendicular at the point of contact to the tangent to a Circler passes through the Centre. Solution:
Given: A Circle with Centre 0 and a PQ tangent AQB and a Perpordicits is a dragon from point of contact Q to AB.
To prove: The perpendicular pa Passes through the Centre of the Circle.
Proof: AQ is the tangent of the Circle at point Q.
AQ will be the perpendicular to the radius of the circle.
⇒ PQ⊥AQ
⇒ The Centre of the Circle will lie on the line PQ.
Perpendicular PQ passes through the Centre of the Circle.
Question 4. A quadrilateral ABCD is drawn to Circumscribe a circle, and prove that AB+CD = AD+BC. Solution:
As shown, the sides of a quadrilateral ABCD touch P a the Circle at P, Q, R and s. We know the tangents drawn from an external point to the Clucle are equal.
AP=AS, BP = BQ, CR = CQ, DR = DS
On adding, AP+BP + CR+DR
⇒ AB + BQ + CQ + DS
⇒ AB+CD= (AS + DS) + (BQ+CQ)
⇒ AB + CD = AP+BC
Hence proved.
Question 5. Ap is tangent to Circle 0 at point P. What is the length of OP? solution:
Let the radius of the given Circle is r.
OP = OB = r
OA=2+r, OP=r, AP=4
∠OPA = 90°
In the right ∠OPA,
⇒ OA2= op2 +Ap2
⇒ (2+r)2 = r2+(4)2
⇒ 4+r2+4r= r2+16
⇒ 4r = 12 =) r=3
Op=3cm.
Question 6. If the angle between two tangents drawn from an external point p to a Clicle of radius 'a' and Centre 0, is 60°, then find the length of op . Solution:
PA and PB are two tangents from an external point p such that
∠APB = 60°
∠OPA = ∠OPB = 30°
(tangents are equally inclined at the centre)
Also, ∠OAP=90°
Now, in right ∠OAP,
Sin 30° =\(=\frac{O A}{O P}\)
⇒ \(\frac{1}{2}=\frac{a}{o p}\)OP=2a units.
Question 7. In the given figure, if AB = AC, prove that BE = EC. Solution:
We know that lengths of tangents from an external Point are equal.
AD=AF
DB = BE
EC = FC
Now, it is given that
AB = AC
⇒ AD+DB = AF + EC
⇒ AD+DB = A8+EC
⇒ DB = EC
BE = EC
Question 8. In the given figure, AT is tangent to the Chicle with Centre 0 Such that Oto 4cm and LOTA = 30° Find the length of Segment AT. Solution:
In the right ∠OAT,
Cos 30°\(=\frac{A T}{O T}\)
⇒ \( \frac{\sqrt{3}}{2}=\frac{A T}{4}\)
⇒ AT = 2√3 Cm
Question 9. The length of a tangent from point A at a distance of 5 cm from the Centre P 5cm of the Circle is ucm. Find the radius of the Circle. Solution:
Let o be the Centre of the Circle and PQ is a tangent to the Circle from point P.
Given that, PQ=4cm and op=5cm
Now,∠OOP = 90°
In ∠OQP,
⇒ OQ2 = Op2= PQ2
= 52-42
=25-16=9
OQ = 3cm
Radius of Circle = 3cm
Question 10. Prove that the angle between the two tangents drawn from an external point to a circle is Supplementary to the angle Subtended by the line segment joining the points of contact at the Centre. Solution: | 677.169 | 1 |
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