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"Could there be a spherical version of the theorem, preserving the geometric spirit in terms of area relationships?"
Drawing as a practical and insightful approach to measurement through drawing of Spherical and Euclidean plane geometry. | 677.169 | 1 |
Let B and C are points of interection of the parabola y=x2 and the circle x2+(y−2)2=8. The area of the triangle OBC, where O is the origin, is
A
2
B
4
C
6
D
8
Video Solution
Text Solution
Verified by Experts
The correct Answer is:D
|
Answer
Step by step video, text & image solution for Let B and C are points of interection of the parabola y=x^(2) and the circle x^(2)+(y-2)^(2)=8. The area of the triangle OBC, where O is the origin, is by Maths experts to help you in doubts & scoring excellent marks in Class 12 exams.
Knowledge Check
Let B and C are the points of intersection of the parabola x=y2 and the circle y2+(x−2)2=8. The perimeter (in units) of the triangle OBC, where O is the origin, is
A8
B4√5
C4√5+2
D4(√5+1)
Question 2 - Select One
Let A(4,−4) and B(9,6) be points on the parabola y2=4x. Let C be chosen on the on the arc AOB of the parabola where O is the origin such that the area of ΔACB is maximum. Then the area (in sq. units) of ΔACB is : | 677.169 | 1 |
Providers
27
Results [8:08] will demonstrate how to use slope, midpoint, and distance …
This lesson [8:08] will demonstrate how to use slope, midpoint, and distance formulas to determine from the coordinates of the vertices if a quadrilateral is a rhombus in a coordinate plane. It is 1 of 4 in the series titled "Coordinate Geometry of Rhombii."
This lesson [8:20] will demonstrate how to use slope, midpoint, and distance …
This lesson [8:20] will demonstrate how to use slope, midpoint, and distance formulas to determine from the coordinates of the vertices if a quadrilateral is a rhombus in a coordinate plane. It is 2 of 4 in the series titled "Coordinate Geometry of Rhombii."
This lesson will demonstrate how to use slope, midpoint, and distance formulas …
This lesson will demonstrate how to use slope, midpoint, and distance formulas to determine from the coordinates of the vertices if a quadrilateral is a rhombus in a coordinate plane. It is 3 of 4 in the series titled "Coordinate Geometry of Rhombii." [16:32]
Learn about the properties of rectangles, rhombuses, and squares, and practice identifying …
Learn about the properties of rectangles, rhombuses, and squares, and practice identifying them. Includes hints for questions. CCSS.Math.Content.3.G.A.1 Understand that shapes in different categoriesQuadrilaterals-parallelograms, rectangles, and trapezoids-are thoroughly introduced in this lesson plan for upper …
Quadrilaterals-parallelograms, rectangles, and trapezoids-are thoroughly introduced in this lesson plan for upper elementary and middle school students. Lots of resources are linked to it to help support the teacher and students. CCSS.Math.Content.3.G.A.1 Understand that shapes in different categories how the diagonals of a rhombus are related. They use interactive sketches to learn about the properties of the angles and diagonals of squares, rectangles, rhombuses, parallelograms, and other quadrilaterals.Key ConceptsThe sum of the measures of the angles of all quadrilaterals is 360°.The alternate angles (nonadjacent angles) of rhombuses and parallelograms have the same measure.The measure of the angles of rectangles and squares is 90°.The consecutive angles of parallelograms and rhombuses are supplementary. This applies to squares and rectangles as well.The diagonals of a parallelogram bisect each other.The diagonals of a rectangle are congruent and bisect each other.The diagonals of a rhombus bisect each other and are perpendicular.Goals and Learning ObjectivesMeasure the angles formed by the intersection of the diagonals of a rhombus.Explore the relationships of the angles of different squares, rectangles, rhombuses, parallelograms, and other quadrilaterals.Explore the relationships of the diagonals of different squares, rectangles, rhombuses, parallelograms, and other quadrilateralsParallelogram to CubeStudents have a chance to review angle measurements in a parallelogram. Building the cube helps students see the transition from two-dimensional shapes and their relationship to three-dimensional figures.QuadrilateralsStudents investigate the possible quadrilaterals that can be made from any four given side lengths, focusing on those that can't make a quadrilateral. Students also look at possible parallelograms with two sides given and possible rhombuses with four sides given.DiagonalsStudents further investigate diagonals in quadrilaterals. If the diagonals are perpendicular, is the figure a rhombus?TrapezoidsHow many right angles can a trapezoid have? How many congruent angles or congruent sides can it have? Can its diagonals be perpendicular or congruent? Students investigate possible trapezoids.More AnglesStudents explore three intersecting lines and the combinations of angles.Diagonals and AnglesThe sides of a parallelogram are extended beyond the vertices, and students explore which angles are congruent and which are supplementary. Students also explore the effect diagonals have on interior angles.Exterior AnglesStudents explore the sum of exterior angles for several polygons and speculate about the results.Angles and SidesStudents explore the relationship between angles and sides in a triangle and discover that the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle (and congruent sides are opposite congruent angles).Ratios and AnglesStudents explore the ratios of the legs of a right triangle to the angles in the triangle. Students see that there is a unique ratio for each angle, and vice versa. This is an informal look at trigonometry.Find the AngleStudents solve equations to find angle measures in polygons.TessellationsStudents explore quadrilateral tessellations and why they tessellate. Students also explore tessellations of pentagons and other polygons. | 677.169 | 1 |
Name: Sarah Hart
Name: Sarah Hart
Date: 04/20/2015
WOODLAND HILLS SECONDARY LESSON PLANS
Content Area: Geometry
Length of Lesson:10 days
STAGE I – DESIRED RESULTS Lesson Topic (Modules, if applicable): Circles / Big Ideas: There are some mathematical relationships that are always true and these relationships are used as the rules of arithmetic and algebra and are useful for writing equivalent forms of expressions and solving equations and inequalities. Numbers, measures, expressions, equations, and inequalities can represent mathematical situations and structures in many equivalent forms. / Understanding Goals (Concepts): Find angle measures of central, inscribed, interior, and exterior angles; find arc measure and length of circles. Special properties of circles are explored including the fact that a circle is a unique geometric shape in which the angles, arcs, and segments intersecting that circle have special relationships. Student Objectives (Competencies/Outcomes): 1.Identify and use parts of circles. Solve problems involving the circumference of a circle. 2.Recognize major arcs, minor arcs, semicircles, and central angles and their measures. Find arc length. 3.Recognize and use relationships between arcs and chords, and between chords and themselves. 4.Find measures of inscribed angles. Find measures of angles of inscribed polygons. 5.Use properties of tangents, solve problems involving circumscribed polygons 6.Find measures of angles formed by lines intersecting on or inside, or outside a circle. 7.Find measures of segments that were in the interior or exterior of a circle. / Essential Questions: How can you use coordinates and algebraic techniques to represent, interpret, and verify geometric relationships? How can a change in one measurement of a 2- or 3-dimensional figure effect other measurements such as perimeter, area, surface area or volume of that figure? / Vocabulary: Chord, circumference, arc, tangent, secant, center, central angle, circumscribed, diameter, inscribed, intercepted, major and minor arcs, point of tangency STAGE II – ASSESSMENT EVIDENCE Performance Task: Students will actively participate in mini-lessons, guided and independent practice, activities (including authentic problem-solving tasks and vocabulary), and group work. Also, students will demonstrate adequate understanding via an end-of-chapter test. / Formative Assessments: Pre-assessments, open-ended higher-order-thinking questions, think-pair-share, graphic organizers, do nows, observation of guided and independent practice, brief in-class writing prompts STAGE III – LEARNING PLAN Materials and Resources: Textbook and notes / Interventions: Flexible grouping, students will be encouraged to attend Math Lab and College and Career Access Center tutoring. Instructional Procedures*: Monday / Tuesday / Wednesday / Thursday / Friday Date: 4/20 / Day: B / Date: 4/21 / Day: A / Date: 4/22 / Day: B / Date: 4/23 / Day: A / Date: 4/24 / Day: B Procedures /
"Do Now"- Find the distance from a chord to the radius of a circle.
"Mini Lesson" – Inscribed Angles
"Guided Practice" – Find the measures of inscribed angles and measures of angles inscribed in polygons.
"Independent Practice"
Students will find the measures of inscribed angles and measures of angles inscribed in polygons. | 677.169 | 1 |
In the given figure, If PQRS is a parallelogram, then find the values of x and y.
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Verified by Experts
The correct Answer is:x=4∘
|
Answer
Step by step video, text & image solution for In the given figure, If PQRS is a parallelogram, then find the values of x and y. by Maths experts to help you in doubts & scoring excellent marks in Class 9 exams. | 677.169 | 1 |
by
Charles, Randall I.
Answer
$D$
Work Step by Step
Let's look at each of the options separately.
Option A: Using the Triangle Inequality Theorem, we need to see if the sum of each of the combinations of two sides is greater than the other side:
$7 + 10 > 25$ --> This statement is false.
$10 + 25 > 7$ --> This statement is true.
$7 + 25 > 10$ --> This statement is true.
A triangle cannot have these lengths for its sides because two of the sides added together are not greater than the third side.
Option B: Using the Triangle Inequality Theorem, we need to see if the sum of each of the combinations of two sides is greater than the other side:
$4 + 6 > 10$ --> This statement is false.
$6 + 10 > 4$ --> This statement is true.
$10 + 4 > 6$ --> This statement is true.
A triangle cannot have these lengths for its sides because two of the sides added together are not greater than the third side.
Option C: Using the Triangle Inequality Theorem, we need to see if the sum of each of the combinations of two sides is greater than the other side:
$1 + 2 > 4$ --> This statement is false.
$2 + 4 > 1$ --> This statement is true.
$1 + 4 > 2$ --> This statement is true.
A triangle cannot have these lengths for its sides because two of the sides added together are not greater than the third side.
Option D: Using the Triangle Inequality Theorem, we need to see if the sum of each of the combinations of two sides is greater than the other side:
$3 + 5 > 7$ --> This statement is true.
$5 + 7 > 3$ --> This statement is true.
$3 + 7 > 5$ --> This statement is true.
These lengths can form a triangle because the sum of two sides is always greater than the third side. | 677.169 | 1 |
Two points, N and Q (not shown), lie to the right of point M on line ℓ
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28 Oct 2015, 04:28
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KudosRe: Two points, N and Q (not shown), lie to the right of point M on line ℓ
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28 Oct 2015, 09:38
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eshan333 wrote:
N and Q , in the question it is given that both lie to the right of M however we do not know whether N lie to the right of Q or Q lie to the right of N , so without knowing the exact position of N and Q
From statement 1 we know that "Twice the length of MN is 3 times the length of MQ" -> 2MN = 3MQ it means that MN>MQ -> hence the position of Q is between M and N. ans:A
Re: Two points, N and Q (not shown), lie to the right of point M on line ℓ
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30 Oct 2015, 14:22
Expert Reply
eshan333 wrote:Hi eshan333,
While you ARE correct that we don't know the exact positions of N and Q, the question does NOT ask us for them (so you have to be careful about when you choose to stop working). The prompt asks for the RATIO of two lengths, NOT the exact measure of either of them. With a bit of 'playing around' and TESTing VALUES, you might find that you change your answer.
Re: Two points, N and Q (not shown), lie to the right of point M on line ℓ
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04 Feb 2016, 18:55
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KudosTwo points, N and Q (not shown), lie to the right of point M on line ℓ. What is the ratio of the length of QN to the length of MQ?
(1) Twice the length of MN is 3 times the length of MQ. (2) Point Q is between points M and N.
------|---------------------------------- line L
M
When you modify the original condition and the question, the que is ratio of QN:MQ, which is for 1) ratio and for 2) number(equation). In a case like this, it is most likely that ratio is the answer.
In 1),
------|--------|--------------------------|----- line L
M Q N
If MQ=2d, 2MN=3*2d, MN=3d and QN=d. Therefore, the que, QN:MQ=d:2d=1:2 is derived, which is unique and sufficient. Therefore, the answer is A.
-> Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.
Re: Two points, N and Q (not shown), lie to the right of point M on line ℓ
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07 Jul 2020, 16:47
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Hi Genoa2000,
DS questions are interesting because they're built to 'test' you on a variety of skills (far more than just your 'math' skills), including organization, accuracy, attention-to-detail, thoroughness and the ability to prove that your answer is correct. DS questions also have no 'safety net' - meaning that if you make a little mistake, then you will convince yourself that one of the wrong answers is correct.
If you choose to select an answer after just reading the information in Fact 2 (and not do any work to PROVE what the correct answer is), then one of two outcomes is likely... You might understand the information perfectly and have the correct answer.... OR.... you have made some type of silly/little mistake and you are about to let some easy points slip away. Doing work "in your head" is the WORST way to approach a GMAT question - so the more often you take that approach, the more likely you lose some 'gettable' points on Test DayThank you for your help.
For your first question, we do get a 1 on the right hand side after we divide each side by x. We can obtain y/x = 1/2 from 2y = x in two steps, like the way you showed; or we can simply divide each side of 2y = x by 2x to obtain y/x = 1/2 in a single step.
For your second question, the division of x does not apply to both 2 and y. You must be thinking of the case where we divide a summation by some number, for instance, (2 + y)/x. In this case, the division indeed applies to both 2 and y, so we can rewrite this expression as 2/x + y/x. However, 2y/x is equal to (2/x) * y or 2 * (y/x), but not (2/x) * (y/x) (which you can verify simply by observing that the product of 2/x and y/x is not equal to 2y/x).
gmatclubot
Re: Two points, N and Q (not shown), lie to the right of point M on line [#permalink] | 677.169 | 1 |
Definition of reflectional symmetry
What is meant by Reflectional symmetry?
Reflection symmetry is a type of symmetry which is with respect to reflections. Reflection symmetry is also known as line symmetry or mirror symmetry. It states that 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.
What is reflective symmetry for kids?
Reflection symmetry is when one half of an object or shape is the reflection of the other half. The two halves are a reflection of each other and the line where you cut is called the line of symmetry, creating the two equal parts.
What is reflection symmetry class 6?
The object and its image are symmetrical with reference to the mirror line. If the paper is folded, the mirror line becomes the line of symmetry. We then say that the image is the reflection of the object in themirror line. (IMAGE REFERENCE: NCERT)
What is Reflectional and rotational symmetry?
Reflectional symmetry means that an object will look exactly the same if it's reflected across a line of symmetry. Rotational symmetry means that an object will look exactly the same if it's rotated the right amount.
How do you teach reflectional symmetry?
Which figure has reflection symmetry?
A rectangle is an example of a shape with reflection symmetry. A line of reflection through the midpoints of opposite sides will always be a line of symmetry. A rectangle has two lines of symmetry.
What is rotational symmetry class 7?
A figure is said to have rotational symmetry if, after a rotation, an object looks exactly the same. The fixed point, about which the rotation turns an object (not changing its shape and size) is called centre of rotation.
What is the difference between rotation and reflection?
Rotation means the shape turns as it moves around a fixed point. Shapes can be rotated clockwise or anticlockwise by a certain number of degrees (90 degrees would be a quarter turn, for example). Reflection means the shape has a mirror image on the other side of the mirror line.
What is rotational symmetry example?
Rotational symmetry is a type of symmetry that is defined as the number of times an object is exactly identical to the original object in a complete 360° rotation. It exists in different geometrical objects such as rhombus, squares, etc.
What is reflective pattern?
Reflective symmetry is where a shape or pattern is reflected in a mirror line or a line of symmetry. The shape that has been reflected will be the same as the original, it should also be the same size and it will be the same distance away from the mirror.
What are examples of symmetry in nature?
From snowflakes to sunflowers, starfish to sharks, symmetry is everywhere in nature. Not just in the body plans which govern shape and form, but right down to the microscopic molecular machines keeping cells alive.
Where can you see symmetry in our daily life?
Symmetry in Real Life
The feathers of a peacock and the wings of butterflies and dragonflies have identical left and right sides.
Hives of honeybees are made of hexagonal shape, which is symmetric in nature.
Snowflakes in winter have all three lines of symmetry.
What are the types of symmetry?
There are four types of symmetry that can be observed in various situations, they are:
Translation Symmetry.
Rotational Symmetry.
Reflection Symmetry.
Glide Symmetry.
What is symmetry in real life?
Symmetry that we see everyday in nature is most often Bilateral Symmetry. This means that the two halves of an object are exactly mirror images of each other. Symmetry in humans the human face has a line of symmetry in some places, but some faces are more symmetrical than others.
What is symmetry in nature called?
The body plans of most animals, including humans, exhibit mirror symmetry, also called bilateral symmetry. They are symmetric about a plane running from head to tail (or toe).
Why is symmetry important?
Symmetry is a fundamental part of geometry, nature, and shapes. It creates patterns that help us organize our world conceptually. We see symmetry every day but often don't realize it. People use concepts of symmetry, including translations, rotations, reflections, and tessellations as part of their careers.
What is a symmetry for kids are the 5 patterns of nature?
Spiral, meander, explosion, packing, and branching are the "Five Patterns in Nature" that we chose to explore. | 677.169 | 1 |
Angles= 45-45-90; sides= 1-1-√(2)
Let's start by understanding the concept of angles and sides in geometry
Let's start by understanding the concept of angles and sides in geometry.
In geometry, an angle is formed by two rays that have a common endpoint, called the vertex. The two rays are called the sides of the angle. Angles are typically measured in degrees or radians. In this case, the given angles are 45-45-90.
A 45-45-90 triangle is a special type of right triangle where the two legs (the sides that form the right angle) are congruent in length, and the hypotenuse (the longest side opposite the right angle) is equal to the length of the legs multiplied by the square root of 2 (√2). So, the sides of a 45-45-90 triangle can be represented as 1-1-√(2).
To visualize this triangle, imagine a right triangle where each of the two legs has a length of 1 unit. Since the angles are both 45 degrees, the triangle will be an isosceles triangle with the two congruent sides forming the right angle. The hypotenuse can be found using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, it becomes 1^2 + 1^2 = √(2)^2, which simplifies to 2 = 2. Therefore, the length of the hypotenuse is √(2).
In summary, a 45-45-90 triangle has two congruent legs, each measuring 1 unit, and a hypotenuse of √(2) units. The angles in this triangle are both 45 | 677.169 | 1 |
Hough transform can be used for pixel linking and curve detection. The straight line represented by y=mx+c can be expressed in polar coordinate system as,
ρ = xcos(θ)+ ysin(θ) …………………..(i)
Where ρ,θ defines a vector from the origin to the nearest pointon the straight line y=mx+c. this vector will be perpendicular from the origin to the nearest point to the line as shown in the below figure.
Any line in the x, y plane corresponds to the point in the 2D space defined by the parameter and θ. This the Hough transform of a straight line in the x,y plane is a single point in the ρ, θ space and these points should satisfy the given equation with x1,y1 as constants. Thus the locus of all such lines in the x, y plane corresponds to the particular sinusoidal curve in the ρ, θ space.
Suppose we have the edge points xi,yi that lie along the straight line having parameters ρ0,θ0. Each edge point plots to a sinusoidal curve in the ρ,θ space, but these curves must intersect at a point ρ0,θ0. Since this is a line they all have in common.
For example considering the equation of a line: y1= ax1+b
Using this equation and varying the values of a and b, infinite lines can pass through this point (x1,y1).
However, if we write the equation as
B= -ax1+y1
And then consider the ab plane instead of xy plane, we get a straight line for a point (xi,yi). This entire line in the ab plane is due to a single point in the xy plane and different values of and b. Now consider another point (x2, y2) in the xy plane. The slope intercept equation of this line is,
Y2= ax2+ b………………….(1)
Writing the equation in terms of the ab plane we get,
B= -ax2+y2………………..(2)
This is another line in the ab plane. These two line will intersect each other somewhere in the ab plane only if they are part of a straight line in the xy plane. The point of intersection in the ab plane is noted as (a',b'). using this (a',b') in the standard slope-intercept form i.e. y=a'x+b', we get a line that passes through the points (x1 , y1) and (x2, y2) in the xy plane. | 677.169 | 1 |
Honors Geometry Companion Book, Volume 1
5.2.3 The Pythagorean Theorem Key Objectives • Use the Pythagorean Theorem and its converse to solve problems. • Use Pythagorean inequalities to classify triangles. Key Terms • A set of three nonzero whole numbers a , b , and c such that a 2 + b 2 = c 2 is called a Pythagorean triple . Theorems, Postulates, Corollaries, and Properties • Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. • Converse of the Pythagorean Theorem If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. • Pythagorean Inequalities Theorem In △ ABC , c is the length of the longest side. If c 2 > a 2 + b 2 , then △ ABC is an obtuse triangle. If c 2 < a 2 + b 2 , then △ ABC is an acute triangle.
The Pythagorean Theorem is one of the most useful relationships in geometry. It states that in a right triangle, the sum of the squared side lengths of the legs of the triangle (the sides that include the right angle) is equal to the square of the length of the other side (the side opposite the right angle). The side opposite the right angle is called the hypotenuse. The Pythagorean Theorem can be used to find the length of one side of a right triangle when the lengths of the other two sides are known. The Converse of the Pythagorean Theorem states that if the sum of the squares of the two shorter sides of a triangle are equal to the square of the third angle, then that triangle is a right triangle. The Converse of the Pythagorean Theorem can be used to determine whether a triangle is a right triangle when the lengths of a triangle's three sides are known. | 677.169 | 1 |
Objective Type Questions Q.28. Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b ∀ a, b ∈ T. Then R is (a) Reflexive but not transitive (b) Transitive but not symmetric (c) Equivalence (d) None of these Ans. (c) Solution. If a ≌ b ∀ a, b ∈ T then a R a ⇒ a ≌ a which is true for all a ∈ T So, R is reflexive. Now, aRb and bRa. i.e., a ≌ b and b ≌ a which is true for all a, b ∈ T So, R is symmetric. Let aRb and bRc. ⇒ a ≌ b and b ≌ a ⇒ a ≌ c ∀ a, b, c ∈ T So, R is transitive. Hence, R is equivalence relation.
Q.29. Consider the non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then R is (a) Symmetric but not transitive (b) Transitive but not symmetric (c) Neither symmetric nor transitive (d) Both symmetric and transitive Ans. (b) Solution. Here, a R b ⇒ a is a brother of b. a R a ⇒ a is a brother of a which is not true. So, R is not reflexive. a R b ⇒ a is a brother of b. b R a ⇒ which is not true because b may be sister of a. ⇒ a R b ≠ b R a So, R is not symmetric. Now, a R b, b R c ⇒ a R c ⇒ a is the brother of b and b is the brother of c. ∴ a is also the brother of c. So, R is transitive.
Q.31. If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is (a) Reflexive (b) Transitive (c) Symmetric (d) None of these Ans. (b) Solution: Given that: R = {(1, 2)} a R b and b R c ⇒ a R c which is true. So, R is transitive.
Q.32. Let us define a relation R in R as aRb if a ≥ b. Then R is (a) An equivalence relation (b) Reflexive, transitive but not symmetric (c) Symmetric, transitive but not reflexive (d) Neither transitive nor reflexive but symmetric. Ans. (b) Solution: Here, aRb if a ≥ b ⇒ aRa ⇒ a ≥ a which is true, so it is reflexive. R is not symmetric. Now, a ≥ b, b ≥ c ⇒ a ≥ c which is true. So, R is transitive.
Q.35. If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is (a) 720 (b) 120 (c) 0 (d) None of these Ans. (c) Solution: If A and B sets have m and n elements respectively, then the number of one-one and onto mapping from A to B is n! if m = n and 0 if m ≠ n Here, m = 5 and n = 6 5 ≠ 6 So, number of mapping = 0
Q.36. Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is (a) nP2 (b) 2n – 2 (c) 2n – 1 (d) None of these Ans. (d) Solution: Here, A = {1, 2, 3, ..., n} and B = {a, b} Let m be the number of elements of set A and n be the number of elements of set B ∴ Number of surjections from A to B is nCm × m! as n ≥ m Here, m = 2 (given) ∴ Number of surjections from A to B = nC2 × 2!
Q.56. An integer m is said to be related to another integer n if m is a integral multiple of n. This relation in Z is reflexive, symmetric and transitive. Ans. Here, m = kn (where k is an integer) If k = 1 m = n, so z is reflexive. Clearly z is not symmetric but z is transitive. Hence, the statement is 'False'.
Ans. Relations and functions are mathematical concepts that describe the connections or associations between two sets of elements. A relation is a set of ordered pairs, while a function is a special type of relation in which each input (x-value) is associated with exactly one output (y-value).
2. How do you determine if a relation is a function?
Ans. To determine if a relation is a function, you can use the vertical line test. If a vertical line intersects the graph of the relation at more than one point, then the relation is not a function. However, if every vertical line intersects the graph at most once, then the relation is a function.
3. What is the difference between a one-to-one function and an onto function?
Ans. A one-to-one function is a function in which each input (x-value) is associated with a unique output (y-value). In other words, no two different inputs can have the same output. On the other hand, an onto function (also known as a surjective function) is a function in which every element in the range has a corresponding element in the domain.
4. How can we represent a function algebraically?
Ans. A function can be represented algebraically using an equation or a formula. For example, if f(x) represents a function, we can express it as f(x) = mx + c, where m and c are constants. This equation represents a linear function, where m represents the slope and c represents the y-intercept.
5. What is the domain and range of a function?
Ans. The domain of a function is the set of all possible input values (x-values) for which the function is defined. It represents the values that can be plugged into the function. The range of a function is the set of all possible output values (y-values) that the function can produce. It represents the values that the function can produce as a result.
NCERT Exemplar - Relations and Functions (Part - 2) Free PDF Download
The NCERT Exemplar - Relations and Functions (Part - 2 2) now and kickstart your journey towards success in the JEE exam.
Importance of NCERT Exemplar - Relations and Functions (Part - 2)
The importance of NCERT Exemplar - Relations and Functions (Part - 2 2) JEE Questions
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Students of JEE can study NCERT Exemplar - Relations and Functions (Part - 2) alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the NCERT Exemplar - Relations and Functions 2) is prepared as per the latest JEE syllabus. | 677.169 | 1 |
Math.cos is a built-in JavaScript function that returns the cosine of a specified angle in radians. The cosine function is a trigonometric function that describes the ratio of the adjacent side length to the hypotenuse side length in a right triangle. The Math.cos function takes one argument, the angle in radians, and returns a decimal value between -1 and 1. | 677.169 | 1 |
Angle Sum Theory And Triangle ClassificationMathematics as a subject presents angle sums and triangles topic with various formulas and actually how to solve mathematical problems. Test your knowledge on these sub topics by taking up the quiz below and find out what you can do!
Questions and Answers
1.
Classify this triangle. Choose all names that would apply
A.
Isosceles
B.
Scalene
C.
Right
D.
Obtuse
E.
Acute
Correct Answer(s) A. Isosceles C. Right
Explanation The triangle is classified as isosceles because it has two equal sides. It is also classified as right because it has a right angle.
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1
0
2.
Classify the triangle above
A.
Acute Scalene
B.
Right Scalene
C.
Right Isosceles
D.
Obtuse Sclene
E.
Acute Isosceles
Correct Answer D. Obtuse Sclene
Explanation Based on the given options, the answer "Obtuse Scalene" is the correct classification for the triangle. An obtuse triangle is a triangle that has one angle greater than 90 degrees. A scalene triangle is a triangle that has three unequal sides. Therefore, if a triangle has one angle greater than 90 degrees and three unequal sides, it can be classified as an obtuse scalene triangle.
Rate this question:
3.
Find the missing angle
A.
40
B.
110
C.
80
D.
70
Correct Answer D. 70
Explanation The missing angle can be found by subtracting the sum of the given angles (40, 110, 80) from 180 degrees. This is because the sum of angles in a triangle is always 180 degrees. Therefore, the missing angle is 180 - (40 + 110 + 80) = 70 degrees.
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4.
Classify the triangle.
A.
Scalene Isosceles
B.
Acute Scalene
C.
Acute Isosceles
D.
Obtuse Scalene
Correct Answer C. Acute Isosceles
Explanation The given answer, "Acute Isosceles," suggests that the triangle in question is both acute and isosceles. An acute triangle is a triangle in which all three angles are less than 90 degrees. An isosceles triangle is a triangle that has two sides of equal length. Therefore, an acute isosceles triangle would have two sides of equal length and all three angles less than 90 degrees.
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5.
A triangle has the following angle measurements: 62 and 57. Classify this triangle by its sides and angles.
A.
Obtuse Isosceles
B.
Right Scalene
C.
Obtuse Scalene
D.
Acute Scalene
Correct Answer D. Acute Scalene
Explanation The given triangle has angle measurements of 62 and 57 degrees. Since the sum of the angles in a triangle is always 180 degrees, the third angle can be calculated as 180 - (62 + 57) = 61 degrees.
To classify the triangle by its sides, we need to determine if all sides are equal (equilateral), if two sides are equal (isosceles), or if no sides are equal (scalene). Since no information about the sides is given, we can conclude that the triangle is scalene.
To classify the triangle by its angles, we need to determine if all angles are less than 90 degrees (acute), if one angle is equal to 90 degrees (right), or if one angle is greater than 90 degrees (obtuse). Since all angles in the given triangle are less than 90 degrees, we can conclude that the triangle is acute.
Therefore, the correct classification for the given triangle is "Acute Scalene".
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6.
A triangle has the following angle measurements: 120 and 52. Classify this triangle by its sides and angles.
A.
Obtuse Equilateral
B.
Acute Scalene
C.
Obtuse Scalene
D.
Right Scalene
Correct Answer C. Obtuse Scalene
Explanation The triangle has two angles measuring 120 and 52 degrees, which means that the sum of the angles is greater than 180 degrees, making it an obtuse triangle. Additionally, the triangle has sides of different lengths, making it a scalene triangle. Therefore, the correct classification for this triangle is "Obtuse Scalene."
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7.
A triangle has the following angle measurements: 58 and 73. What is the missing angle measurement?
A.
131
B.
90
C.
27
D.
49
Correct Answer D. 49
Explanation In a triangle, the sum of all three angle measurements is always 180 degrees. To find the missing angle measurement, subtract the sum of the given angles (58 + 73 = 131) from 180. Therefore, the missing angle measurement is 49 degrees.
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2
0
8.
Choose the classifications that fit the triangle above.
A.
Obtuse
B.
Right
C.
Acute
D.
Isosceles
E.
Scalene
Correct Answer(s) A. Obtuse D. Isosceles
Explanation The triangle in the question has one angle that is greater than 90 degrees, making it an obtuse triangle. Additionally, the triangle has two sides that are equal in length, making it an isosceles triangle. Therefore, the classifications that fit the triangle are obtuse and isosceles.
Rate this question:
9.
Find the value of x.
A.
35
B.
86
C.
94
D.
47
Correct Answer B. 86
10.
Find the value of x.
A.
45
B.
40
C.
20
D.
30
Correct Answer D. 30
Explanation The value of x is 30 because it is the last number listed in the given sequence.
Rate this question:
11.
An equilateral triangle can be classified as an isosceles.
A.
True
B.
False
Correct Answer A. True
Explanation In order to be classified as an isosceles, it must have AT LEAST 2 congruent sides. An equilateral triangle has 3 congruent sides. | 677.169 | 1 |
2 ... 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 . XVI . And this point is called the centre of 2 EUCLID'S ELEMENTS .
Page 76 ... line and the circumference . VIII . An angle in a segment is the angle contained by two straight lines drawn from any point in the circumference of the segment , to the extre- mities of the straight line which is the base of the segment ...
Page 79 ... line standing upon another straight line makes the adjacent angles equal to one another , each of them is a right angle ; ( 1. def . 10. ) therefore each of the angles AFE , BFE , is a right angle : wherefore the straight line CD , drawn ...
Page 83 ... drawn only two equal straight lines from the point D to the circumference , one upon each side of the line through the centre . At the point M , in the straight line MD , make the angle DMB equal to the angle DMK , ( 1. 23. ) and join ...
Page 96 ... lines drawn from their centres are equal : ( III . def . 1. ) therefore the two sides BG , GC , are equal to the two EH , HF , each to each : and the angle at G is equal to the angle at H ; ( hyp . ) therefore the base BC is equal to | 677.169 | 1 |
Students who are looking for DAV Books Solutions you are in right place, we have discussed the solution of Science class 6 book Chapter 3 Locating Places on the Earth which is followed in all DAV School Solutions are given below with proper Explanation please bookmark our website for further updates!! All the Best !!
Class- 8 DAV Social Science Locating Places on the Earth Question and Answer
A. Tick the correct Option:
1. The equator does not pass through which one of the following continents?
Answer: Europe
2. The heat zone lying between 231/2% N and 661/2% N is-
Answer: Temperate Zone
3. The longest circle drawn midway between two poles is-
Answer: The Equator
4. When the time is 12 noon at 0° longitude, the time at 75 E longitude will be-
Answer: 5 p.m.
5. Which one of the following is a correct statement about longitudes?
Answer: All of them have equal lengths.
B.Fill in the blanks
1.The earth rotates from ___________ to ___________.
2.All the places on the same meridian will have the ___________ local time.
3.The distance between the two lines of latitudes is always ___________.
4. The ___________ are the imaginary lines that connect the north and south pales.
5.Each degree of longitude corresponds to a time difference of ___________ minutes.
Answer: (1) West, East (2) same (3) equal (4) longitude (5) four
C.Match the following
Ans:
1. Two division of earth
Hemispheres
2.Latitude are measured in
Degree
3. Tropic of cancer
23 Degree 30'N
4.British Royal Observatory
Greenwich
5.The place through which standard meridian of india passes
Mirzapur
D. Answer the following questions in brief.
1. Which two basic points on the earth serve as the reference points?
Ans: (i) North Pole (ii) the South Pole
2. Mention the latitudinal location of the heat zones of the earth.
Ans: The latitudinal location of the heat zones of the earth are:
Torrid Zone
Tropic of Cancer- Tropic of Capricorn 23 1/2oN to 23 1/2oS
Hottest
North Temperate Zone
Tropic of Cancer- Arctic Circle 23 1.2oN to 66 1/2oN
Moderate
South Temperate Zone
Tropic of Capricorn-Antarctic Circle 23 1/2oS to 66 1/2oS
Moderate
North Frigid Zone
Arctic circle 66 1/2o to 90oN (North Pole)
Coldest
South Frigid Zone
Antarctic Circle 66 1/2oS to 90oS(South Pole)
Coldest
3. Why does the Torrid zone have the maximum temperature?
Ans: The torrid zone receives the maximum amount of heat throughout the year because the rays of the sun fall vertcally on this regon. This is located between the tropic of Cancer and Tropic of Capricorn in 23 1/2N to 23 1/2 S.
4. What is the significance of Greenwich Mean Time?
Ans: Greenwich Mean Time helps to know the international time and local time of a country.
5. Why is the Standard Meridian selected by a country a multiple of 7.5°?
Ans: The Standard Meridian selected by a country is a multiple of 7.5° because 15 degrees of longitude is equivalent to 1 hour in time (that is 360 / 24) it is convenient to adopt time zones at 7.5 degrees or 15 degrees multiples.In other words time zones are separated by multiples of 1 hour or 1/2 hour.
E. Answer the following questions.
1. State three main characteristics of parallels of latitudes.
Ans: Three main characteristics of parallels of latitude are:
Each degree of latitude is divided into 60 minutes and a minute is further sub-divided into 60 seconds.
These lines are parallel to each other and are of equal distance to each other and hence are called parallels of latitude.
Only the Equator is a Great Circle, all the rest of the latitudes are Small Circles.
2. Why do we use standard time? Explain with an example from India.
Answer: We know that places located on different meridians have different local times. It would create a lot of difficulties if a country adopts different local times.
For example, India lies between 68° 7'E and 97° 25'E longitudes. Thus, the people of India have to face problems of different local times. So, to avoid this problem and maintain its uniformity all over. a country, the standard time is used.
3. Which heat zone is most suitable for us to live and why?
Ans: Temperate zone is most suitable for us to live because the sun rays in this zone are never directly overhead. The angle of the sunrays decreases as we go towards the poles. That is why this zone is neither very hot nor very cold. They have moderate temperature.
4. Why is the time difference between each meridian of longitude 4 minutes? Explain.
Ans:
The Earth rotates once a day, so that's 360 degrees (the number of degrees in a circle) in 24 hours. If you do the math, that's 360/24 = 15 degrees per hour. That is why, the time difference between each meridian of longitude is 4 minutes. | 677.169 | 1 |
A regular polygon is a closed shape with straight edges and equal sides and angles. Based on the size of its interior angles, it is classified as either convex or concave. A convex polygon has interior angles less than 180° each, while a concave polygon has at least one interior angle greater than 180°. Regular polygons are primarily convex and possess unique properties that will be explored in this article.
Defining Regular Polygons
A regular polygon has equal side lengths and equal interior angles. Examples of regular polygons include equilateral triangles, squares, and rhombuses. In addition, a regular polygon's diagonals are all of equal length. While most regular polygons are convex, there are also some concave regular polygons that have a star shape. Now, we will take a closer look at the properties of regular convex polygons.
Properties of Regular Convex Polygons
Two Important Circles: A regular convex polygon has two significant circles associated with it. The circumcircle lies outside the polygon and passes through all of its vertices, while the incircle passes through the midpoint of each of its sides. The circumcircle's radius is the distance from the polygon's center to any of its vertices, and the incircle's radius is the distance between the center and the midpoint of any side, also known as the apothem.
Calculating Area: One interesting application of a regular polygon's properties is the ability to estimate its area using the apothem. By breaking the polygon down into triangles and utilizing the apothem, we can determine the area of any regular polygon with N sides.
An Example Calculation:
To calculate the area of a regular hexagon with side length s and apothem l, we can divide it into six triangles. The area of one triangle is equal to 0.5 x s x l, and to find the area of the entire hexagon we multiply it by 6 (the number of sides). Therefore, the formula for the area of a regular hexagon is 6 x 0.5 x s x l, or 3 x s x l.
Examples of Regular Polygons
A regular polygon with three sides is known as an equilateral triangle, and one with four sides is called a square. For polygons with more than four sides, the term "regular" is added before the name of the polygon, such as a regular pentagon. Some examples of regular (equiangular) convex polygons include regular hexagons, octagons, and decagons.
Formulas for Regular Polygons
Exterior Angles: In a regular convex polygon, the sum of all exterior angles is always 360° or 360/N, where N is the number of sides and ∅ is the exterior angle.
Interior Angles: The sum of the interior angles in a regular polygon depends on the number of sides it has. For example, a triangle has a sum of 180°, while a quadrilateral has a sum of 360°. A general equation for finding the sum of interior angles in any regular polygon is 180(N-2), where N is the number of sides.
Regular polygons possess unique properties and formulas related to their side lengths, angles, and area. Understanding these properties can assist in solving various problems involving regular polygons.
Understanding Interior and Exterior Angles in Polygons
In a polygon, the interior angle is the angle formed by two consecutive sides on the inside of the figure. On the other hand, the exterior angle is the angle formed by one side and the extension of the adjacent side. Let's examine how we can find these angles for a regular decagon.
Interior Angle: To determine the interior angle of a regular decagon, we can use the formula 180° - 360°/N, where N is the number of sides. For a decagon, N=10, so the interior angle would be 180° - (360°/10) = 144°. This means that each interior angle in a regular decagon measures 144°.
Exterior Angle: To find the exterior angle of a regular decagon, we can divide 360° by the number of sides, or N. For a decagon, this would be 360°/10 = 36°.
Sum of Interior Angles: The sum of all interior angles in a regular polygon can be calculated using the formula (N-2) x 180°, where N is the number of sides. For a regular decagon, this would be (10-2) x 180° = 1440°.
The Beauty of Convex Polygons: Understanding Diagonals
A convex polygon is defined as a shape where all its interior angles are less than 180 degrees, and its edges never cross. Unlike concave polygons, where diagonals can extend to the exterior of the shape, the diagonals in a convex polygon will always stay within the figure, adding to its visual appeal and symmetry. A diagonal is a line segment connecting any two non-consecutive vertices of a polygon with more than 3 sides. The number of diagonals in a convex polygon with 'N' sides can be easily calculated using the formula N(N-3)/2.
For example, a heptagon, a convex polygon with 7 sides, has 14 diagonals. This can be obtained by substituting N=7 in the formula, giving us N(N-3)/2 = 7(7-3)/2 = 14 diagonals. In the figure shown, all 14 diagonals of a regular heptagon can be seen.
The Special Properties of Regular Polygons
A regular polygon is a shape with all its sides and interior angles being equal. It exhibits unique properties that make it stand out among other polygons. Some of these properties include:
All sides and interior angles are equal.
The diagonals are also equal in length.
The circumcircle, a circle passing through all the vertices, has a radius known as the circumradius.
The incircle, a circle passing through the midpoints of each side, has a radius called the apothem.
The area of a regular polygon can be calculated by dividing it into smaller triangles.
All the vertices are equidistant from the center of the polygon.
The sum of all exterior angles is always equal to 360°.
Frequently Asked Questions
What Defines a Regular Polygon? A regular polygon is a shape with equal sides and interior angles, creating a perfect symmetry.
What is the Minimum Number of Sides a Regular Polygon Can Have? A regular polygon must have at least 3 sides, but it can have an unlimited number of sides.
What are Some Examples of Regular Polygons? Regular polygons come in various shapes, including equilateral triangles, squares, and rhombuses.
How Do You Find the Area of a Regular Polygon? The area of a regular polygon can be calculated by dividing it into triangles and adding up their individual areas.
What Shapes Do Regular Polygons Have? Regular polygons can take on various shapes depending on the number of sides, but all sides and angles are equal, creating a perfect symmetry. | 677.169 | 1 |
The Coordinate Plane is really called the Cartesian Plane, which
was developed by a french Mathematician, named Renee Descartes. He
described the slope as "climbing" up or down a line (from left to
right). The french word for "climb" is "monte" (with an accent over
the 'e'). That's why he used 'm' for slope. | 677.169 | 1 |
Question Video: Understanding that Orientation is a Non-Defining Attribute of Shapes
Mathematics • First Year of Primary School
If I turn this shape so it stands on its corner, will it still be a square?
00:26
Video Transcript
If I turn this shape so it stands on its corner, will it still be a square?
We can see that if we draw this shape so that it's standing on its corner, it will still be a square. So the answer is yes. If we turn the shape so that it stands on its corner, it will still be a square. | 677.169 | 1 |
The circumference of a circle makes an angle of 360° or 2π rad at its center. If the circumference is divided into 360 equal parts, each segment will make an angle of 1°. The angle is the ratio of circumference to the radius of the circle. Each degree is divided into 60 min of an arc and each minute is divided into 60 s of an arc. Therefore, angle measurement is linked to the length and hence is traceable to the standard meter through various kinds of techniques and instrumentation. Angle gauges with different tolerances are used to transfer secondary length standards to the laboratories and industry. We will discuss optical methods to measure angles. These include autocollimator, goniometer, ...
Get Introduction to Optical Metrology now with the O'Reilly learning platform.
O'Reilly members experience books, live events, courses curated by job role, and more from O'Reilly and nearly 200 top publishers. | 677.169 | 1 |
What line represents zero degrees latitude?
The equator is at zero degrees latitude. It is the starting point for measuring latitude. The equator is an imaginary line that runs around the middle of the Earth halfway between the North Pole and the South Pole. The equator runs through the top of South America, through the middle of Africa, and then Indonesia and north of New Guinea. | 677.169 | 1 |
Ans. The standard equation of a hyperbola with eccentricity is given by:
$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ for a hyperbola with horizontal transverse axis, and
$\dfrac{y^2}{a^2} - \dfrac{x^2}{b^2} = 1$ for a hyperbola with vertical transverse axis.
2. How do you find the eccentricity of a hyperbola?
Ans. The eccentricity of a hyperbola can be found using the formula $e = \sqrt{1 + \dfrac{b^2}{a^2}}$, where $a$ is the distance from the center to a vertex, and $b$ is the distance from the center to a co-vertex.
3. What is the significance of eccentricity in a hyperbola?
Ans. The eccentricity of a hyperbola determines its shape and characteristics. It represents how "stretched out" the hyperbola is, with higher eccentricity values indicating a more elongated shape. The eccentricity also affects the distance between the foci and the vertices of the hyperbola.
4. What are the latus rectum of a hyperbola?
Ans. The latus rectum of a hyperbola is a line segment that passes through the foci and is parallel to the transverse axis. It is the segment connecting the points on the hyperbola that are closest to the foci. The length of the latus rectum can be found using the formula $L.R. = \dfrac{2b^2}{a}$, where $a$ is the distance from the center to a vertex, and $b$ is the distance from the center to a co-vertex.
5. How does the eccentricity affect the latus rectum of a hyperbola?
Ans. The eccentricity of a hyperbola does not directly affect the length of the latus rectum. The length of the latus rectum is solely determined by the distances $a$ and $b$. However, the eccentricity indirectly influences the shape of the hyperbola, which in turn affects the position and orientation of the latus rectum.
Introduction of Hyperbola : Eccentricity Standard Equations Latus Rectum in English is available as part of our Mathematics (Maths) Class 11 for Commerce & Hyperbola : Eccentricity Standard Equations Latus Rectum in
Hindi for Mathematics (Maths) Class 11 course. Download more important topics related with notes, lectures and mock test series for
Commerce Exam by signing up for free.
Students of Commerce can study Hyperbola : Eccentricity Standard Equations Latus Rectum alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Hyperbola : Eccentricity Standard Equations Hyperbola : Eccentricity Standard Equations Latus Rectum is prepared as per the latest Commerce syllabus. | 677.169 | 1 |
Hint: Here, in the question, we have been given position vectors of three points, which are collinear. And we are asked to find the relation between the variables present in their vectors. We will first understand the position vectors, meaning of three collinear points and then solve to find the desired relation.
Note: We had to find out the relation between two variables, that's why we used this \[\overrightarrow {AB} = \lambda \overrightarrow {BC} \]. In case we are given three position vectors and we have to find one common variable between them, we can use the fact that the scalar triple product of all three vectors is zero if the three points given are collinear points. | 677.169 | 1 |
What is the Meaning of "Locus"?
A recent question asked about an interesting locus, which led me to realize we haven't talked about that topic in general. Here we'll look at what a locus is, using three simple examples, and then dig into a question about the wording.
What it is
We'll start with this question from 2003:
The Meaning of Locus
What is the locus of points equidistant from two parallel lines 8 meters apart?
I have trouble understanding the meaning of locus.
Since it was assumed in the answer, I'll point out that "equidistant" means "the same distance from both lines".
I answered:
Hi, Marina.
The idea of locus is very simple, but a little subtle. You can almost leave the word out of many problems; this one asks you simply to describe all points that are equidistant from the two lines. The point of the word "locus" is merely that you are to think of all the points that fit that description as a single entity, which might be a curve or line or a set of discrete points.
We'll look further into the origin of the concept at the end. You could consider it an old word for "set"; in Latin the word locus means "place". Our problem, then, is simply "Describe the set consisting of all points equidistant from two parallel lines 8 meters apart?" or "Where are all the points equidistant from two parallel lines 8 meters apart?"
Here are our two parallel lines:
Now we can answer the question:
So picture it one point at a time. What sort of point is the same distance from both lines? Imagine a point somewhere; how do you measure the distance from each line? You draw a perpendicular from the point to each line, and their lengths are the two distances. Now, what does it mean if they are the same? You will find that the points you are looking for are exactly between the two lines.
Here is one such point:
Once you get that idea, think about how you can describe all such points. Imagine that you were forced somehow to walk only in places that are the same distance from the two lines, perhaps because there is a rope from you to each of the lines and some mechanism keeps them the same length. Where will the grass be trampled down? That is the locus.
We'll see this idea of trampled grass again later! Here is where all those points are (that is, the locus):
Notice that we can't just think about some points that fit; we have to find them all, and to be sure no other points do. Once we see that, we need to describe what we see: the parallel line halfway between the two given lines.
Although the question doesn't require it, we could prove that all points on this new line are equidistant from the given lines, and that every point equidistant from the two lines is on the new line. The first step, though, is to see it as we have done, so we can make a conjecture, and then prove that.
Perpendicular bisector as locus
For another example, consider this question from 1999:
Locus
What do you have to do to get the perpendicular bisector of all the loci of a triangle? What does it mean when it asks you about equidistance from certain points?
The question this time is about the meaning of "equidistant"; but it's not stated quite right, so we'll have to rephrase the question as it was presumably given, explaining the terms that made it easy to misstate.
But one thing Jo got right: The plural of "locus" (which, as I said, is a Latin word), is "loci" (pronounced "low-sigh", or, if you learned Latin as I did, "low-key").
I answered:
Hi, Jo. Thanks for writing.
Your first question doesn't make a lot of sense, because there is no "locus of a triangle." But I think I can see what you're asking about.
A "locus" is the set of all points that satisfy some rule or description, and a perpendicular bisector is one of the simplest examples of a locus. If I asked you "what is the locus of all points equidistant from two given points?", the answer would be "the perpendicular bisector of the segment determined by the two points."
Here's what the question means:
What is the locus of all points ...
(What geometrical object consists of all points X ...)
that are equidistant from A and B?
(... for which the distances AX and BX are equal?)
That is, "equidistant from two points" means "the same distance from both points."
Here is a point that is equidistant from points A and B:
Here's what the answer means:
The perpendicular bisector ...
(the line through the midpoint of the segment,
perpendicular to the segment)
of the segment determined by A and B.
(you make the segment by connecting
A and B with a straight line)
That is, this line bisects segment AB (cuts it exactly in half) and is perpendicular to it (at right angles):
You can prove (and probably your text did this) that any point on the perpendicular bisector of a segment AB is the same distance from both endpoints; you can see this by drawing in the segments and seeing that you have an isosceles triangle, so that AX = BX:
|
+ X
/|\
/ | \
/ | \
/ | \
/ | \
/ | \
/ | \
/ | \
/ | \
/ | \
/ | \
/ | \
/ |_ \
/ | | \
+--------------+--------------+
A | B
|
|
This tells us that any point on the perpendicular bisector is equidistant from A and B, so it is part of the locus we are looking for.
This is only half of the requirement: Every point on the locus must meet the requirements, and also every point that meets the requirements must be part of the locus:
Conversely, you can show that if any point is equidistant from A and B, it must be on the perpendicular bisector, so there are no other points in the locus: the perpendicular bisector IS the entire locus we're looking for.
That's what a locus is: the set of ALL points that fit some description (in this case, being equidistant from A and B). In our case, the perpendicular bisector is "the locus, the whole locus, and nothing but the locus" of equidistant points.
The word "all" is essential!
There are many other examples of a locus. For example, the locus of points 1 inch from point A is the circle of radius 1 centered at A. Every point of the circle is 1 inch from A, and every point one inch from A is part of the circle.
I hope that helps a little to clarify the idea of a locus, and the meaning of equidistant. If you have problems you still have trouble with, maybe you can write back with a specific problem so I can see the wording of it.
Here is a point 1 unit away from point A:
Here is that circle:
We'll see more examples next time.
Locus of "a point" or "points"?
I want to close for now with this question, from Navneet in 2016:
Locus Locution
Which of these is the correct definition of a circle?
1. A circle is the locus of *a point* which moves in a plane in such a way that its distance from a given fixed point is always constant.
OR
2. A circle is the locus of *points* which moves in a plane in such a way that its distance from a given fixed point is always constant.
Is one of these incorrect? Or are they both acceptable?
I answered, yet again (yes, this is a favorite topic of mine):
Hi, Navneet.
The proper (older) terminology is "locus of a point." We define a constraint that applies to a point, and imagine following ONE POINT as it moves around subject to that constraint.
We can also describe this as "the locus of all points that ...," thinking not of a single point moving, but just of the set of points satisfying the condition. This is the more modern usage that emerged with the development of the set concept. Your second version uses the idea of moving, which is not appropriate when talking about a set of points, so it is not good wording.
The first definition, "A circle is the locus of a point which moves in a plane in such a way that its distance from a given fixed point is always constant," envisions the process of drawing a circle, in which a point, namely the point of a compass, is moving around, keeping a constant distance from the center. The locus is the circle it draws, which is the path it followed. We can call this the dynamic definition.
The second definition, which would have been better written as, "A circle is the locus of points in a plane in whose distance from a given fixed point is a given constant," looks at the result of drawing the circle and sees that it consists of all points a given distance from the center. It doesn't consider the motion, just the resulting figure. We can call this the static definition.
See this Wikipedia article, which explains the two usages (key phrases capitalized by me for emphasis):
"Until the beginning of the 20th century, a geometrical shape (for example, a curve) was not considered as an infinite set of points; rather, it was considered as an entity on which a point may be located or on which it moves. Thus a circle in the Euclidean plane was defined as THE LOCUS OF A POINT that is at a given distance of a fixed point, the center of the circle. In modern mathematics, similar concepts are more frequently reformulated by describing shapes as sets; for instance, one says that the circle is THE SET OF POINTS that are at a given distance of the center."
The first perspective sees a locus as "where a point lives", as a river is the "locus" of fish; the second, as the collection of such points themselves.
Thus, the common wording today appears in MathWorld:
"For example, THE LOCUS OF POINTS in the plane equidistant from a given point is a circle, and THE SET OF POINTS in three-space equidistant from a given point is a sphere."
You can see that the word "locus" can be replaced by the word "set", as we see also here:
See also the Math Open Reference:
"We can say "THE LOCUS OF ALL POINTS on a plane at distance R from a center point is a circle of radius R." In other words, we tend to use the word locus to mean the shape formed by a set of points. An odd thing is that you can often just drop the word locus, and it still makes sense: "The set of all coplanar points distance R from a central point forms a circle."
"... Sometimes the idea of locus has a slightly different explanation. If you think of a point moving along some path, we sometimes say that the path is the locus of the point. So for example a point that moves a fixed distance from another point draws out a circle. So we could say
"THE LOCUS OF A POINT moving at a fixed distance from a center point is a circle.
One problem is that a locus is not necessarily a path; it can be a plane in space, or a region such as the interior of a circle; these probably didn't occur in the original uses of the concept.
Your reference books may well use the old form ("locus of a point") with or without explicit mention of motion. Other sources you find, especially online, are likely to use the new form ("locus of points") with no reference to motion at all.
Does that help?
More on the locus of a moving point
Navneet wasn't quite sure about the "old form":
What should be the appropriate and precise definition of the word "locus" in mathematics?
We know that the definition of locus is "the *set of all points* (usually forming a curve or surface) satisfying some condition." But how can we use it for one point, as in "A circle is the locus of *a point* which...."?
I answered:
You aren't really asking about the DEFINITION, but about how the older WORDING fits that definition. I'll try to explain that in more detail.
In the old wording, "the locus of a point MOVING under some condition" refers to the set of all places that that point will reach as it moves. Think of the point as a pen point moving around, and the locus as the ink mark it leaves behind. The pen is a MOVING point, not a single location; the locus (drawn curve) consists of all the points (fixed locations) where it ever traveled. In particular, a circle is the locus of the pencil point of a compass as it turns. I imagine that may be exactly what was in the mind of whoever first used the word "locus" in this way.
Again, the locus can be thought of as the path traced by the pen, or as the set of ink molecules on the paper. They both represent the same static entity.
Incidentally, we can describe a curve as a function of time (the path of a point), or we can eliminate time from the equation(s) and be left with just an equation in x and y (the equation of a curve). These are two views of the same locus. But the time factor is ignored in thinking about the locus.
Another example I've given is this: suppose you tether a dog to a ten-foot rope that you stake in the middle of your yard. What part of your yard will it trample?
The dog can move around, but is constrained by the rope to stay within ten feet of the stake; so all the places that the dog could trample will lie in a ten-foot disk with that center. This is the "locus of the dog": the set of all points where the dog will ever travel. (A disk is the locus of points whose distance from the center is less than or equal to a radius.)
Or, if the dog is like some I have seen, the actual part of the lawn it tramples may be not the interior of the circle, but the circle itself, since it will spend all its time at the end of its rope trying to get away. And in fact this is how we define the circle as a locus: the set of all points a fixed distance from the center. But we are describing it as the locus of "a point" (the dog) as it moves.
Long ago I had a neighbor whose dog behaved this way. Today, another neighbor sometimes attaches his dog to a "zip line", producing a locus consisting of a pair of parallel lines if the dog is on a similar mood, or a rectangular region otherwise.
In the more modern wording, we don't think of a single point moving, but go directly to the idea of a set of points. Perhaps this change in common usage was made in part to prevent the confusion you are expressing due to the use of the word "point" in two different ways (moving and fixed). The meaning is the same whether we speak of many points or a single one.
Next time we'll have more examples to illustrate these concepts, and will also move on to algebraic descriptions of loci | 677.169 | 1 |
Degree Symbol | Copy symbol + Degree sign uses
Copy the degree symbol and use the below query to learn more about Degree Sign
Section 1: What is Degree Symbol (°)
The degree symbol (°) is a small circle used in various fields to denote degrees. It is commonly associated with measuring temperature (Celsius and Fahrenheit), angles, and geographic coordinates.
Definition and Visual Representation
Definition: The degree symbol (°) is a typographical mark that signifies degrees of arc, degrees of temperature, and other measurements that employ degrees as a unit. It is a key symbol in many scientific, mathematical, and everyday contexts.
Visual Representation: The degree symbol looks like a small, raised circle (°) and is typically placed immediately after the number it modifies, with no space in between. For example, 90° (ninety degrees).
Section 2: History and Origin of Degree Symbol
Historical Background
The use of the degree symbol (°) dates back to ancient civilizations that developed early methods for measuring angles and temperature. The Babylonians, around 3000 BCE, are credited with introducing the sexagesimal (base-60) system, which greatly influenced how we measure angles and time today. They divided the circle into 360 degrees, a practice that has persisted through millennia.
Origin of the Degree Symbol
The degree symbol itself likely originated from a small raised zero, used to denote the concept of a "complete" unit in the sexagesimal system. This small circle was convenient for scribes and scholars to write and has been adopted in various manuscripts and scholarly works over centuries. The symbol (°) was used by Greek astronomers and mathematicians such as Hipparchus and Ptolemy, who further solidified its use in their astronomical observations and mathematical treatises.
Evolution of its Usage Over Time
Ancient Greece and Rome: The degree symbol was primarily used in astronomical calculations and geometry.
Middle Ages: The symbol's use expanded into trigonometry and navigation, critical for explorers and scientists during this period.
Modern Era: The degree symbol has become ubiquitous in various fields, including physics, engineering, and computer science, to denote angles, temperature, and even in programming for defining rotations and directions.
How the Degree Symbol Became Standardized in Different Fields
The degree symbol's standardization can be traced through its adoption in various scientific and technical standards:
Temperature Measurement: The degree symbol became standardized in temperature scales like Celsius (°C) and Fahrenheit (°F) in the 18th century, thanks to the works of scientists like Anders Celsius and Daniel Gabriel Fahrenheit.
Geodesy and Cartography: In the field of geography and map-making, the degree symbol is used to denote geographic coordinates (latitude and longitude), helping in the accurate depiction of locations on Earth.
Mathematics and Engineering: The degree symbol is universally recognized in mathematics to represent angular measurements. Engineering standards and technical drawings also used extensively to ensure precision and uniformity.
Digital and Computing Standards: With the advent of computing, the degree symbol was included in character sets like ASCII and Unicode, ensuring its availability in digital formats and across different platforms and devices.
Section 3: How to Type Degree Symbol
Operating Systems
On Windows
Typing the degree symbol on a Windows computer is straightforward and can be done using the following methods:
Using the Keyboard (Alt Code):
Hold down the Alt key.
Type 0176 on the numeric keypad.
Release the Alt key, and the degree symbol (°) will appear.
Using Character Map:
Open the Character Map application by typing "Character Map" in the Start menu search bar and selecting it from the results.
In the Character Map, find the degree symbol (°), select it, and click "Copy."
Paste the symbol where you need it.
On macOS
Mac users can easily insert the degree symbol using the following methods:
Using the Keyboard Shortcut:
Press Option + Shift + 8.
The degree symbol (°) will appear where the cursor is located.
Using the Emoji & Symbols Menu:
Press Control + Command + Space to open the Emoji & Symbols menu.
In the search bar, type "degree" to find the degree symbol.
Double-click the symbol to insert it at the cursor's location.
On Linux
Linux users can type the degree symbol using various methods depending on the distribution and keyboard layout:
Using Unicode:
Press Ctrl + Shift + U to enter Unicode input mode.
Type 00B0 and press Enter to insert the degree symbol (°).
Using the Compose Key:
If the Compose key is enabled, press Compose followed by Shift + o, o.
The degree symbol (°) will be inserted.
On Mobile Devices
iOS
Typing the degree symbol on an iPhone or iPad is simpleAndroid
On Android devices, you can type the degree symbol using the following stepsThese methods ensure you can easily and quickly insert the degree symbol on any device or operating system, facilitating accurate and efficient communication.
Section 4: Degree Symbol in Temperature Measurements
Scientific and Everyday Use
The degree symbol (°) is widely used in both scientific contexts and everyday life to represent temperature measurements. It is essential for denoting units of temperature, such as Celsius (°C) and Fahrenheit (°F), which are commonly used in weather forecasts, cooking, and scientific research.
Examples of Correct Usage
Celsius (°C):
Celsius is a metric unit of temperature used worldwide, especially in scientific contexts and by most countries outside the United States.
Example: The average room temperature is 25°C.
Example: Water freezes at 0°C and boils at 100°C under standard atmospheric pressure.
Fahrenheit (°F):
Fahrenheit is primarily used in the United States for everyday temperature measurements.
Example: A typical summer day might have a temperature of 77°F.
Example: Water freezes at 32°F and boils at 212°F under standard atmospheric pressure.
Additional Examples
Weather Forecasts:
Example (Celsius): The weather forecast predicts a high of 30°C and a low of 20°C tomorrow.
Example (Fahrenheit): The weather forecast predicts a high of 86°F and a low of 68°F tomorrow.
Cooking:
Example (Celsius): Preheat the oven to 180°C for baking.
Example (Fahrenheit): Preheat the oven to 350°F for baking.
Scientific Research:
Example (Celsius): The experiment was conducted at a controlled temperature of 37°C.
Example (Fahrenheit): The experiment was conducted at a controlled temperature of 98.6°F.
The degree symbol is placed immediately after the number, with no space in between, followed by the letter indicating the unit of temperature (C for Celsius and F for Fahrenheit). This convention ensures clarity and consistency in temperature representation across different contexts and regions.
Section 5: Degree Symbol in Geometry
Mathematical Notation
In geometry and trigonometry, the degree symbol (°) is used to denote measurements of angles. An angle is formed by two rays emanating from a common endpoint, and its measure is crucial in various mathematical calculations and applications. The degree symbol helps to differentiate angle measurements from other numerical values.
Examples in Trigonometry and Geometry
Trigonometry
Right Angles:
A right angle is exactly 90°.
Example: In a right-angled triangle, one of the angles is 90°.
Acute Angles:
An acute angle is less than 90°.
Example: An angle of 45° is an acute angle often found in trigonometric problems involving sine, cosine, and tangent functions.
Obtuse Angles:
An obtuse angle is greater than 90° but less than 180°.
Example: An angle of 120° is an obtuse angle, commonly used in problems involving supplementary angles.
Geometry
Triangle Angles:
The sum of the interior angles of a triangle is always 180°.
Example: In an equilateral triangle, each interior angle is 60°.
Example: In a right-angled triangle, if one angle is 90°, the other two angles must add up to 90°.
Quadrilateral Angles:
The sum of the interior angles of a quadrilateral is always 360°.
Example: In a rectangle, each interior angle is 90°, summing to 360°.
Circle:
A full circle is 360°, and this is fundamental in defining and working with circles and circular sectors.
Example: A semicircle has an angle of 180°, while a quarter circle has an angle of 90°.
Polygon Angles:
The sum of the interior angles of a polygon can be calculated using the formula: (n−2)×180°, where n is the number of sides.
Example: For a pentagon (5-sided polygon), the sum of the interior angles is (5−2)×180°=540°.
Practical Applications
Construction and Engineering:
Precise angle measurements are crucial for constructing buildings, bridges, and other structures.
Example: Ensuring that corners are exactly 90° in a square room.
Navigation:
Angles are used in navigation to determine direction and course.
Example: A ship might change its course by 30° to the east.
Art and Design:
Angles play a vital role in creating perspective and accurate proportions in art and design.
Example: Using a 45° angle to create isometric drawings.
The degree symbol is essential in geometry and trigonometry for representing angles accurately, making it indispensable for solving mathematical problems, designing structures, and understanding the physical world.
Section 6: Degree Symbol in Geographic Coordinates
Latitude and Longitude Representation
In geography, the degree symbol (°) is used to represent geographic coordinates, which specify the precise location of a point on the Earth's surface. Geographic coordinates are expressed in terms of latitude and longitude:
Latitude: Measures how far north or south a point is from the Equator, ranging from 0° at the Equator to 90° at the poles.
Longitude: Measures how far east or west a point is from the Prime Meridian, ranging from 0° at the Prime Meridian to 180° at the International Date Line.
Latitude and longitude are typically written in degrees, minutes, and seconds (DMS) or in decimal degrees.
The degree symbol (°) in geographic coordinates is fundamental for accurately representing locations on Earth. It plays a critical role in mapping, navigation, and various practical applications, ensuring precise communication and data recording in numerous fields.
Section 7: Conclusion
The degree symbol (°) is a versatile and essential character used across multiple fields and disciplines. From its historical origins and evolution to its standardized use in modern contexts, the degree symbol remains integral to scientific, mathematical, geographic, and everyday applications.
Understanding the degree symbol's diverse uses and proper application is essential for clear and accurate communication. Whether you are a student, scientist, engineer, navigator, or someone engaging in everyday tasks, the degree symbol helps convey precise measurements and information across different domains.
By mastering the use of the degree symbol, you enhance your ability to interact with a wide range of data and information, ensuring clarity and accuracy in your work and daily life. The degree symbol's rich history and continued relevance underscore its significance as a universal tool in both professional and personal contexts.
Explore our website to learn more about the degree symbol, its uses, and how to copy it easily. Whether you need it for scientific research, mathematical calculations, or everyday applications, understanding the degree symbol is a valuable skill in today's information-rich world.
Section 9: Frequently Asked Questions: Degree Symbol
How to type degree symbol?
To type the degree symbol (°), you can use various methods depending on your device and operating system. Here are some common methods:
How to type the degree symbol?
On Windows, hold down the Alt key and type 0176 on the numeric keypad. On macOS, press Option + Shift + 8.
How to make degree symbol?
You can make the degree symbol by using the keyboard shortcuts specific to your operating system. For example, on Windows, use Alt + 0176, and on macOS, use Option + Shift + 8.
How to make a degree symbol?
To make a degree symbol, press and hold the appropriate keys on your keyboard based on your device. For Windows, it's Alt + 0176, and for macOS, it's Option + Shift + 8.
How to get degree symbol on iPhone?
To get the degree symbol on an iPhone, press and hold the 0 (zero) key until a pop-up menu appears, then select the degree symbol (°).
How to insert degree symbol?
You can insert the degree symbol using keyboard shortcuts or the character map. On Windows, press Alt + 0176, and on macOS, press Option + Shift + 8.
How to use degree symbol?
To use the degree symbol, type it with the appropriate keyboard shortcut and place it after the number to represent degrees. For example, 90° for angles or 25°C for temperature | 677.169 | 1 |
Degrees vs. Radians: What's the Difference?
Degrees and radians are two units for measuring angles; degrees divide a circle into 360 parts, while radians use the radius to measure around a circle, equating to about 57.3 degrees.
Key Differences
Degrees are a measure of angles where a full circle is divided into 360 equal parts. Each part is one degree. In contrast, radians offer a different approach, defining angles based on the radius of the circle. One radian is the angle created when the arc length equals the radius.
The concept of degrees is historically older and more familiar in everyday contexts. It is widely used in various fields like navigation and geometry. Radians, however, are more common in higher mathematics and physics as they provide a direct relationship between the angle and the arc length.
When measuring in degrees, simple fractions of a circle, like a quarter or half, are easily represented (90° and 180°, respectively). Radians, on the other hand, express these fractions in terms of π, with π radians being equivalent to 180 degrees, making them more intuitive in mathematical analysis.
In practical applications, degrees are often preferred for their simplicity and ease of understanding. Radians, however, are essential in calculus and trigonometry, as they simplify the integration and differentiation of trigonometric functions.
To convert between degrees and radians, one can use the relation 180 degrees equals π radians. This conversion is crucial in various scientific and engineering fields where both units are used interchangeably.
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Comparison Chart
Unit Definition
1/360 of a circle's circumference
Angle subtended by an arc equal to the radius
Historical Origin
Ancient civilizations' use of 360-day calendars
Conceptualized in 18th century mathematics
Use in Mathematics
Common in basic geometry and navigation
Essential in advanced mathematics, like calculus
Relationship to π
No direct relationship
Directly related (π radians = 180 degrees)
Conversion Factor
1 degree = π/180 radians
1 radian = 180/π degrees
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Degrees and Radians Definitions
Degrees
A level or step in a scale, often used to denote intensity or severity.
The illness escalated to a dangerous degree.
Radians
A dimensionless unit, derived from the ratio of two lengths.
Radians simplify the formulas in circular motion.
Degrees
A stage in a process or series.
The project is in its final degrees of completion.
Radians
A unit of angular measure where the angle subtended at the center of a circle is equal to the radius.
In trigonometry, π/2 radians represent a right angle.
Degrees
A measure of temperature.
The temperature dropped to 30 degrees overnight.
Radians
A measure used in mathematics to relate the length of an arc to its radius.
A full circle is 2π radians.
Degrees
A unit for measuring angles, one degree is 1/360 of a circle's circumference.
The corner of a square measures 90 degrees.
Radians
A standard unit in calculus for differentiating and integrating trigonometric functions.
The derivative of sin(x) is cos(x) when x is measured in radians.
Degrees
An academic award conferred by a university or college.
She earned her degree in biology.
Radians
A unit used in physics to describe rotational motion.
Angular velocity is often expressed in radians per second.
Degrees
One of a series of steps in a process, course, or progression; a stage
Proceeded to the next degree of difficulty.
Radians
A unit of angular measure equal to the angle subtended at the center of a circle by an arc equal in length to the radius of the circle, approximately 57°17′44.6".
Degrees
A step in a direct hereditary line of descent or ascent
First cousins are two degrees from their common ancestor.
Radians
Plural of radian
FAQs
Can degrees be converted to radians?
Yes, degrees can be converted to radians using the formula 1 degree = π/180 radians.
What is a degree?
A degree is a unit of measurement for angles, representing 1/360 of a circle.
Are radians more complex than degrees?
Radians can seem more complex due to their mathematical nature, but they simplify many calculations in advanced mathematics.
Are radians used in physics?
Yes, radians are often used in physics, especially in describing rotational motion and wave phenomena.
Why are degrees commonly used in everyday life?
Degrees are simpler and more intuitive for everyday use, like in navigation and basic geometry.
Why are radians preferred in calculus?
Radians provide a natural approach to dealing with arc lengths and simplify the differentiation and integration of trigonometric functions.
How is a radian defined?
A radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
How are angles measured in astronomy?
Angles in astronomy are typically measured in degrees, arcminutes, and arcseconds.
Do all calculators support both degrees and radians?
Most scientific calculators allow switching between degrees and radians.
Do engineers use degrees or radians?
Engineers use both, depending on the specific requirements of their work.
Is it necessary to learn both systems?
Understanding both systems is beneficial, especially in fields involving mathematics, physics, and engineering.
Is there a simple way to remember how to convert between degrees and radians?
Yes, remember that 180 degrees equal π radians.
Is one unit better than the other?
Neither unit is inherently better; the choice depends on the context and application.
What is the significance of 360 degrees in a circle?
The use of 360 degrees is based on historical and practical reasons, likely derived from ancient astronomical observations.
Are degrees and radians interchangeable in formulas | 677.169 | 1 |
Learning Target: Students identify specific angle pairs by name and know all the properties of angles formed by parallel lines. Essential Question(s): When you have two parallel lines cut by a transversal, which angle pairs are congruent? which are supplementary? | 677.169 | 1 |
Short Answer
Step by step solution
Collinearity Condition
Three points \( A(x_1,y_1), B(x_2,y_2) \) and \( C(x_3,y_3) \) are collinear if the determinant formed by their coordinates is zero, according to the formula \(x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) = 0\).
02
Apply Collinearity Condition to Statement (i)
Apply the collinearity condition to the coordinates of the points given in the first statement. After substituting the values and simplifying, we get the equation \( -2 \cdot \cos 2\theta - 4(1 - \sin 2\theta) = 0 \). Solving it gives \(\theta = n \pi\), where n is any integer.
03
Apply Collinearity Condition to Statement (ii)
Use the same collinearity condition for the second statement. After substituting values and simplifying, we get the equation \( -4(1 - \tan^2\theta) + 2(1 + \tan^2\theta) \cdot \tan\theta = 0 \). Upon solving, we get \(\theta = n \pi, (2n+1) \frac{\pi}{4}\), where n is an integer.
04
Compare the Solutions of the Statements
Compare the solutions of the two statements. It is evident that the second statement \( (ii) \) has more solutions for \( \theta \) compared to statement \( (i) \). Therefore, the answer is option (c | 677.169 | 1 |
The Cartesian equation of a plane which passes through the points $$\mathrm{A}(2,2,2)$$ and making equal nonzero intercepts on the co-ordinate axes is
A
$$x+y+z=6$$
B
$$x-2 y+z=0$$
C
$$2 x+y+z=7$$
D
$$x-y+z=$$
2
MHT CET 2021 23rd September Evening Shift
MCQ (Single Correct Answer)
+2
-0
The co-ordinates of the foot of the perpendicular drawn from the point $$2 \hat{i}-\hat{j}+5 \hat{k}$$ to the line $$\vec{r}=(11 \hat{i}-2 \hat{j}-8 \hat{k})+\lambda(10 \hat{i}-4 \hat{j}-11 \hat{k})$$ are
A
$$(1,-2,3)$$
B
$$(1,2,-3)$$
C
$$(-1,2,3)$$
D
$$(1,2,3)$$
3
MHT CET 2021 23th September Morning Shift
MCQ (Single Correct Answer)
+2
-0
If A(3, 2, $$-$$1), B($$-$$2, 2, $$-$$3) and D($$-$$2, 5, $$-$$4) are the vertices of a parallelogram, then the area of the parallelogram is
A
286 sq. units
B
$$\sqrt{286}$$ sq. units
C
300 sq. units
D
$$\sqrt{300}$$ sq. units
4
MHT CET 2021 23th September Morning Shift
MCQ (Single Correct Answer)
+2
-0
The distance between the parallel lines $$\frac{x-2}{3}=\frac{y-4}{5}=\frac{z-1}{2}$$ and $$\frac{x-1}{3}=\frac{y+2}{5}=\frac{z+3}{2}$$ is | 677.169 | 1 |
Calculation of Earth's model using results of Struve Geodetic Arc
Calculation of Earth's model using results of Struve Geodetic Arc
How meridian measurements alow to determine the size of the Earth
Since the time of Eratosthenes, attempts to estimate the size and shape of the Earth have led to the realisation that the shape of the Earth essentially follows a mathematical model - an ellipsoid or spheroid, made up of a major semi-axis a, and a minor semi-axis b (Figure 1).
Figure 1. Elipsoide of the Earth
Eratosthenes calculated that if we could measure the Earth's ellipsoid as an irregular circle of 360 degrees, and if we were able to measure the length of one degree in the circle, we would have the entire circumference of the Earth. However, the problem is not so simple because, as can be seen from Figure 1, the radius of curvature of the Earth's ellipsoid is variable, i.e. the Earth varies unequally at the equator and on the North or South axes. It is therefore necessary to determine the angle of curvature of the ellipsoid, α, as close as possible to the actual surface of the Earth by ground and astronomical measurements, and once this has been determined, we can obtain the flattening of the Earth, which is the difference between the two axes divided by the angle of curvature
Also the term eccentricity , a positive real number expressing the characteristics of a tendency:
The meridian radius of curvature can be shown to be equal to
The arc length of an section S, of the meridian from the equator to latitude φ is dm = M(φ) dφ is cakculated φ as following.
Expression for a given meridian segment:
It should be considered that the shape of the Earth causes critical differences in flattening on the Earth's axes and at the equator. If we try to measure the distance along the meridian at the same angle of change on the axes of the Earth's ellipsoid and at the equator (Figure 2) , the distance S1 would be greater than the distance S2. S1 būtų didesnis nei atstumas S2.
Figure 2. Difference in flatenning of the Earth.
It is clear from the above example that if we could determine the distance on the Earth's meridian and find out how many degrees of latitude this distance covers on the Earth's ellipsoid, we would subtract 1 degree on the meridian, and multiply it by 360 to get the entire circumference of the Earth. The longer meridian length S is, the closer to the real flattening of the Earth we could apply to the curvature (Figure 3). Once the meridian length S is determined to 1 degree, the circumference of the Earth is easy to calculate as following:, also the radius of the Earth
Figure 3. Length of meridian on elipsoide
Determining the meridian by a triangulation arc chain
The triangulation method, introduced in the 17th century, made it possible to measure long distances more accurately by eliminating elevation issues on the ground. The Struve geodetic arc measurements were based on a modified triangulation method, introduced by the Dutch scientist Willebrord Snel van Royen. Triangular segments were projected in the geodetic chain so, that the sides (baseline) of one triangle in both sides of the segment was surveyed in high precision on the ground, as shown by a red double line (Figure 4). Vilebordas Snel van Rojenas (Willebrord Snel van Royen) trianguliacijos metodas. Struvės geodezinio lanko grandinėje buvo suprojektuoti trikampių segmentai, kurių vieno trikampio kraštinė (bazė) buvo labai tiksliai pamatuota vietovėje segmento pradžioje ir pabaigoje, kaip parodyta raudona dviguba linija 4 pav.
Figure 4. Triangulation chain along a meridian.
The triangulation measurements were made by surveying on the ground all the angles of the triangles of the chain, and the distances of each side were calculated applying the trigonometric formula of sinus theorem (Figure 5). To ensure control of the calculations, precise measurements of the distance of one triangle side was again surveyed in the field at the end of each segment (Figure 4).
Figure 5. Applies of principles of sinus trigonometric theory.
In total, 10 sides (baselines) have been accurately surveyed in the Struve geodetic arc, and astronomical observations have been made at 13 points to get the positioning of the points in space and to calculate the earth's radius of curvature by calculating astronomical latitudes and the astronomical azimuths, i.e. the directions to the neighbouring points. The astronomical observations were carried out using high-precision telescopes. The results have been achieved based on positioning of stars at certain timing. The astronomical parameters were adjusted using astronomical reference books.
Triangulation measurements were not necessarily taken exactly on the meridian, as the terrain and vegetation often made this impossible. Once the results of the calculations had been processed and the astronomical latitudes of each point on the chain had been calculated, they were mathematically projected onto the meridian line, thus fixing the longitude of the meridian. The Struve geodetic arc is 2822 km long, with Fuglenes (Norway) as the northernmost point and Staro Nekrasovka as the southernmost. Fuglenes (Norway) as the northernmost point Staro Nekrasovka. The points of the Arc in prof. F.G.W. Struve's report were projected onto the so-called Tartu meridian - 26 degrees and 43 minutes East of Greenwich.
The first practical application of the data of Struve geodetic arc were utilized by Friedrich Wilhelm Bessel who calculated the parameters of the so-called Bessel ellipsoid: whom the radius from the center of the Earth to the equator a = 6 337 397 m, Earth's flattening, α = 1 / 299,15. The Bessel ellipsoid has served for more than 100 years as a reliable basis for the Earth model in a number of applied geodetic coordinate systems.
Figure 6. Section of meridian length on an ellipsoid.
When considering the spherical transformation of the latitudes and longitudes of the Struve geodetic arc, several aspects of the mathematical analysis need to be highlighted. Consider two points on a spherical surface, A, with latitude φ1 and longitude λ1 ir B, with latitude φ2 and longitude λ2 (Figure 6). The connecting spherical segment (from A to B) is AB with the length s12 and azimuths α1 and α2 at both ends .1 .2Once we have determined A, α1, and s12 we can solve the geodesic problem by determining the parameters of the next point B and its azimuth in the north direction α. .2.
Using the given example, by solving a spherical trigonometrical issue where the meridian's north direction is known, we can calculate the triangle NAB by applying the mathematical trigonometric (triangle) formula., Where a sum of angles of the triangle always is
Figure. 7. Element of a meridian projection
For the change in curvature ρ on the surface of an ellipsoid F.W. Bessel derives a relationship between the change in azimuth α, the change in longitude ds, and the change in latitude dφ (Figure 7.).
This information was prepared during the implementation of the Interreg V-A Latvia-Lithuania Cross Border Cooperation Programme 2014-2020 project No. LLI-477 "Creation of the International Tourist Route "Struve Geodetic Arc" / STRUVE, the aim of which is to strengthen the development of educational tourism, increase the number of visitors and extend the duration of their visit in the regions by providing a variety of tourism opportunities.
This information was developed with the financial support of the European Union. Total financing of the project: EUR 850,5 thousand (including ERDF financing – EUR 723 thousand).
The contents of this information are the sole responsibility of Directorate of Aukštaitija Protected Areas and can under no circumstances be regarded as reflecting the position of the European Union. | 677.169 | 1 |
phase angle formula sine wave
{\displaystyle t} By measuring the rate of motion of the test signal the offset between frequencies can be determined. As a sword and board Eldritch Knight do I need to put away my sword on my turn if I want to use Shield as a reaction? t If you know the frequency, it is probably more accurate to fit a sine wave to each of the two vectors than to use the FFT. Let sin (This claim assumes that the starting time In this device, the voltage is ahead of the current by 90 degrees. Understanding the relationship between the sine curve and the unit circle is a basic trigonometric concept which you need to understand for this class. However, there is no need to use degrees or any other particular unit inside the micro otherwise. {\displaystyle t} In these types of waveforms, the retardation of wavelengths is the whole number like 0, 1, 2, 3…etc. {\displaystyle t} I two sinewave signals with same frequency. This lets you see how the right triangle is formed by the angle of the radius moving counterclockwise around the unit circle. t ) with a shifted and possibly scaled version from between the phases of two periodic signals π Instantaneous phase (ϕ) represents an angular shift between two sine waves and is measured in radians (or degrees).A sine wave and a cosine wave are 90 ° (π /2 radians) out of phase with each other. It also makes it easy to index into a table, like to get computing sine or cosine for example. Can it be justified that an economic contraction of 11.3% is "the largest fall for more than 300 years"? Mathematical curve that describes a smooth repetitive oscillation; continuous wave, "Sinusoid" redirects here. {\displaystyle F} which tells us that the vertical side is 1/√2 = .707. = In this equipment, the current leads the voltage by 90 degrees. Ans. Similarly, if ɸ < 0, then the wave has a negative phase of the phase angle. Hence, it is a relative property of more than one waveform. 2 {\displaystyle \textstyle {\frac {T}{4}}} Choose a web site to get translated content where available and see local events and offers. relative to ϕ At other positions of the sine wave (B, D, F, H) the EMF will be as per the formula, e = Vmax*sinθ. {\displaystyle t} {\displaystyle \phi (t_{1})=\phi (t_{2})} Resistor: The voltage and current in the same phase in a resistor. π You don't need to have a full install of Max to run this. In the adjacent image, the top sine signal is the test frequency, and the bottom sine signal represents a signal from the reference. The timer capture hardware in micros can usually be set up for that. τ In this case the phase difference is increasing, indicating that the test signal is lower in frequency than the reference.[2]. {\displaystyle \phi (t)} {\displaystyle F} as Since you have sine waves least squares curve fitting will give you the best result. . The difference between the neg to pos zero crossings (or pos to neg) gives the period. is the length seen at the same time at a longitude 30° west of that point, then the phase difference between the two signals will be 30° (assuming that, in each signal, each period starts when the shadow is shortest). {\displaystyle t} . The periodic changes from reinforcement and opposition cause a phenomenon called beating. If the frequencies are different, the phase difference Thanks for contributing an answer to Electrical Engineering Stack Exchange! Thus at 45º, the length of the sin is .707. There are several contributions on the file exchange for fitting sine waves (search "harmonic fit"), and my own function is attached, or you could write your own. It can be possible if both the waves have the same frequency and same phase. F Phase Angle Formula and its Relation with Phase Difference, The equation of the phase difference of a sine wave using maximum amplitude and voltage is. When the phase difference Convey 'is raised' in mathematical context, SFMC Rest api to create/delete Data Extensions. 1 ( There are two conditions in these types of waveforms, namely leading phase and lagging phase. @verendra Without being able to measure the instantaneous value of the sine wave to find the zero crossings, I can't think of a way to find the phase shift. t {\displaystyle \pi } This wave pattern occurs often in nature, including wind waves, sound waves, and light waves. I have implemented timer functionality from the datasheet of ATmega32-A.I have updated my code.But i am thinking about Reset_timer function. (At 45º, the length of the horizontal and vertical sides must be equal. The Pythagorean theorem tells us that f F . In the clock analogy, this situation corresponds to the two hands turning at the same speed, so that the angle between them is constant. {\displaystyle G} ) {\displaystyle T} Vedantu academic counsellor will be calling you shortly for your Online Counselling session. MathWorks is the leading developer of mathematical computing software for engineers and scientists. Often we will have two sinusoidal or other periodic waveforms having the same frequency, but is phase shifted. Can a late passport renewal affect getting visas? {\displaystyle F} = Capacitor: The current and voltage in a capacitor are not in the same phase with each other. ( F There is no phase of a sine wave. ] π I am using ATmega32-A micro controller and external ADC AD7798 to read the voltage of both signal. calculating phase shift is not related to ADC reading? It follows that, for two sinusoidal signals The leading phase represents that the wave is ahead of another one having the same frequency. In any waveform, the complete phase is 360 degrees or 2π radians. x Any suggestions please. | 677.169 | 1 |
Tangent rule
The tangent rule, also known as the tangent-secant theorem, states that the tangent to a circle is perpendicular to the radius at the point of contact. Additionally, it asserts that the product of the lengths of the entire secant segment and its external segment is equal for any two secants intersecting outside a circle. This geometric principle is instrumental in solving various problems involving circles, tangents, and secants.
Definition of Tangent Rule
The tangent rule is an essential concept in trigonometry that helps in finding the tangent of the sum or difference of two angles. Whether you are solving complex equations or simple problems, this rule is highly valuable.
Definition of the Tangent Rule
The tangent rule, also known as the tangent-sum formula, is expressed as:For the sum of two angles, \[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \] For the difference of two angles, \ Explained with Examples
Understanding the tangent rule is an integral part of enhancing your mathematical skills. This trigonometric identity is especially useful in solving problems related to the sum or difference of angles.
Definition of the Tangent Rule
The tangent rule, also known as the tangent-sum formula, is expressed as in Trigonometry Techniques
The tangent rule is a powerful tool in trigonometry that allows you to calculate the tangent of the sum or difference of angles. Knowing how to apply this rule can simplify many trigonometric problems. Let's explore the tangent rule in detail.
Definition of the Tangent Rule
The \textbf{tangent rule}, or tangent-sum formula, is given byEnsure that your angle measurements are consistent. The tangent rule applies to both radians and degrees.
Learn with 12 Tangent rule flashcards in the free Vaia app
Frequently Asked Questions about Tangent rule
What is the tangent rule in trigonometry?
The tangent rule in trigonometry relates the tangent of angles in a triangle to the sides of the triangle. It states that for any triangle, the ratio of the length of a side to the tangent of the opposite angle is constant: \\( \\frac{a}{\\tan A} = \\frac{b}{\\tan B} = \\frac{c}{\\tan C} \\).
How is the tangent rule applied to solve triangles?
The tangent rule, also known as the tangent formula, is used in trigonometry to find unknown sides or angles in non-right-angled triangles. It states that (a-b)/(a+b) = tan[(A-B)/2]/tan[(A+B)/2], where a and b are sides opposite angles A and B, respectively. This helps solve triangles by relating the sides and angles.
What is an intuitive explanation of the tangent rule?
The tangent rule relates the angles and sides in a non-right-angled triangle, stating that the ratio of the tangent of half the sum of two angles to the tangent of half their difference is equal to the ratio of the lengths of the opposite sides. This helps in solving triangles when certain angle relationships are known.
What are some real-world applications of the tangent rule?
The tangent rule is applied in surveying for determining distances and angles, in navigation for plotting courses on maps, in architecture for designing angles and slopes, and in physics for analysing forces and motion in mechanics.
How does the tangent rule relate to other trigonometric rules?
The tangent rule relates to other trigonometric rules by expressing the tangent of an angle in a triangle in terms of the sides, akin to the sine and cosine rules. It complements the sine rule and cosine rule by solving for different angles or sides using tangent relationships.
A. Identify the angles involved in the problem.
B. Apply the cosine addition formula to find the tangent.
C. Simplify the expression to arrive at the solution.
D. Convert all angles to radians for consistency.
Learn with 12 Tangent rule | 677.169 | 1 |
Unit angles and triangles homework 2 answer key
2 DO NOT EDIT--Changes must be made through File info 6 enceCommon Core Math StandardsThe student is expected to:COMMON COREG-SRT5Use congruence ometric figures. Here is the answer key for Homework 2 on Similar Triangles in Unit 6. If two angles of a triangle are acute, they must be equal, as they all share a side. Geometry Unit 2: Lines, Angles, and Triangles vertical angles. Since the sum of the interior angles in a triangle is always 180 ∘ , we can use an equation to find the measure of a missing angle.
Did you know?
Homework answer key unit 8 right triangles and trigonometry : unit 8Unit 8 right triangles and trigonometry key / unit 3 right triangle Trigonometry triangles mcalpin theresaTriangles trigonometry. The Twelve Triangles quilt block looks good from any angle. When finding missing angles, there are a few key concepts to keep in mind: Understanding basic angle relationships, such as vertical angles. e Argumentative, Popular Problem Solving Editor Sites, Resume Objective For Bank, Yurt Business Plan, Women Of War And Gender Based Violence Essay In this video solutions to all the homework problems from Homework 2 (Unit 4 - Congruent Triangles, Angles of Triangles) are shown with the exceptions of num. | 677.169 | 1 |
10 Daily Life Examples of Triangular Prism
There are many examples of triangular prisms we come across regularly in common objects around us. For example, the classic shape of a pencil is a triangular prism, its long, slim body is made of wood shaped into a three-sided form. Pieces of chocolate broken off a Toblerone bar also display triangular prisms, which come from imprinting its unique mold.
Other everyday examples of Triangular Prism include triangular road signs that are highly visible to drivers, separation of garlic into tiny three-sided pyramids, and of course, slices of cake or pizza cut into those familiar triangular wedges we grab eagerly.
The triangle design minimizes how often the pencil rolls off desks and tables.
Experiment: Roll several pencils on a flat table. Notice how the triangular shape makes them roll unevenly compared to objects with circular or square cross-sections. They move jerkily and settle into a corner of the table.
6. Cut birthday cake
Some claim mathematician Pythagoras invented the wedge cake cut to celebrate geometrical progress with his students.
Experiment: Cut a cake or brownie into wedges rather than squares. Compare how much crust people get with each slice. Wedges guarantee crunchy corners in every portion.
7. Garlic cloves
The individual segments of a garlic bulb separate into small triangular prisms. Their tapered tips help push up through the soil as they grow.
Do You Know?
Digging up wild garlic reveals an underground system of stringy triangular roots spreading to sprout more growth.
Experiment: Try planting garlic cloves pointy-side up vs sideways to observe which grows more easily upward. The orientation should make the shoots emerge more readily.
8. Pool shark fins
The dorsal fins on some pool sharks protrude in roughly triangular shapes. The hydrodynamic form may aid fast swimming motions.
Do You Know?
Many such sharks can swim up to 40 mph assisted by these specialized fins evolved over millions of years.
Experiment: Test model shark fins with square vs triangular shapes to see which design cuts most smoothly through water in simulated swimming motions. The triangle should have noticeably less drag.
9. Pine tree air fresheners
Car air fresheners infused with pine tree scent often have a triangular prism form that clips onto vents. The design maximizes flow of fragrance into circulating air.
Do You Know?
Research shows the iconic pine tree shape triggers consumers to psychologically associate the product with freshness and cleanliness.
Experiment: Hang experimental paper flower, square, and triangular shapes near a fan blowing lightly. See which flutters enough to release scent quickest. The stiffness of the triangle should make it waiver productively. | 677.169 | 1 |
deed description
I don't know if this is common anywhere, but it's the first time that I've come across this.
"…..200′ to a pipe, deflecting thence from said line 93-36 to the right, a distance of 412.2′ to a stake, deflecting thence 86-46 to the right a distance of 264.8'…."
When I first ran the deed in it didn't close. Then I ran it in going the opposite direction, and still not good. Then I used the angles as the exterior angle, but ran it's interior angle that distance, and the lot closed just over a tenth.
Weird, since instead they could've just said, "turning left an angle of 86-24….."
They're definitely angles. But as said, they're using some sort of odd deflection method, like they can only describe angles right, but they're running the lot clockwise, so they should be stating the left angle.
Instead of listing the interior angle, they're calling out the supplementary angle, and calling it a deflection.
The lot closes that way, and makes sense with the monuments and abutters.
Now I'm confused. The next line should run at the stated angle relative to a line continuing from the prior line either to the left or right. So, instead of continuing in the same direction you deflect from that direction by the angle stated.
If you pick up surveying books from the black transit days it's very common.
Deflections are what the name implies; a deviation from the direction you are currently traveling. If you are heading east and deflect 45 left, you will be heading northeast. If you are heading east and deflect 45 right, you will be heading southeast. Pretty simple stuff…
One advantage of deflection angles is when the traverse lines are to be connected by curves, the deflection angle is equal to the delta, or central angle, of the curve. That solves some of the problems where surveyors make the most mistakes: adding and subtracting.
I went to school with the son of a paving contractor. He was all about using a T-16. Apparently it was a very useful instrument for turning deflection angles right or left that were common in NCDOT plan sets back in the 1980ƒ??s. -tying this thread back to the ƒ??40 years agoƒ? thread.
You are right, no deflection angles in the description, but there are also no bearings. Using angle rights or deflection angles were a way to make the surveyors job easier. No need to establish and carry through a basis of bearing.
Surveying was approached by many as a kind of sub-engineer. The more dumbed down the better. Unfortunately some new surveyors still have that attitude. | 677.169 | 1 |
MCQ Three Dimensional Geometry CBSE Maths 12 Science Answers in English to enable students to get Answers in a
narrative video format for the specific question.
Expert Teacher provides MCQ Three Dimensional Geometry 1 : Find the distance of the point (-1, -5, -10) from the point of intersection of the line and the plane (View Answer Video)
Question 2 : Find the distance of the point (-1, -5, -10) from the point of intersection of the line and plane (View Answer Video)
Question 3 4 : Find the vector and cartesian equations of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes x + 2y +3z = 5 and
3x + 3y + z = 0. (View Answer Video)
Question 5 : If a line has direction ratios 2, -1, -2, then what are its direction cosines? (View Answer Video)
Questions from Other Chapters of CBSE, 12th Science, Maths
Continuity and Differentiability
Question 1 : Find the value of k, if the area of the triangle is 4 sq unit and vertices are (-2, 0), (0, 4), (0, k). (View Answer Video) | 677.169 | 1 |
To find the center of gravity of a particular quadrilateral
In summary, the center of gravity of a uniform triangular lamina is the same as that of three equal particles placed at the vertices of the lamina.
Apr 28, 2021
#1
gnits
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Homework Statement
To find the center of gravity of a quadrilateral
Relevant Equations
Moments
Could I please ask for help with the following question. Part 2 is my problem. I have no idea how to begin, any hints would be much appreciated:
1) Prove that the center of gravity of a uniform triangular lamina is the same as that of three equal particles placed at the vertices of the lamina
I'm ok here, done that.
2) A uniform lamina of weight W is in the shape of a quadrilateral ABCD. The diagonals AC, BC meet at P, where AP < PC, BP < PD and Q, R are points on AC, BD respectively such that QC = AP, RD = BP. By replacing the triangles ABD, BCD by equivalent systems of particles, or otherwise, prove that the center of gravity of the lamina is the same as that of a particle of weight W/3 at Q and a particle of weight 2W/3 at the midpoint of BD.
Here is a diagram:
(The blue and orange strokes are meant to show the equality of the lengths of the segments that they are on)
I don't mind which way it is proved, by equivalent particle systems or "otherwise".
I started by setting up cartesian coordinates at A, but I don't think that helps.Apr 28, 2021
#3
onatirec
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Lnewqban said:
It seems that the problem does not give specific location for point Q
QC = AP
LikesLnewqban
Apr 28, 2021
#4
gnits
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Lnewqban said:Yes, this is what I first started to do but then I couldn't see how to assign the weights. Let's take triangle ABD. I want to put three equal weights at each vertex, each weight should be 1/3 of the weight of triangle ABD. But what is the weight of triangle ABD in terms of W ? I only know that the weight of triangle ABD + weight of triangle BCD = W.
If you can calculate the location of the centroid of each triangle; then, the summation of the triangle areas x centroid distances about the total CG should be equal to zero.
Apr 28, 2021
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gnits
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So I could proceed be setting up Cartesian axes at A(0,0), B(Bx, By), C(Cx, Cy) and D(Dx, Dy). then the centroid of triangle ABD would be ( (Bx + Dx) / 3 , (By + Dy) / 3 ) and that of BCD would be ( (Bx + Cx + Dx) / 3 , (By + Cy + Dy) / 3 ). Triangle areas could be calculated in terms of these coordinates and then multiplied by distances of these centroids from centroid of quadrilateral and equated to zero. Am I missing a more elegant way?LikesLnewqban and BvU
Apr 29, 2021
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gnits
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haruspex said:Thanks you very much indeed. I wouldn't have seen this way of approaching it. I've definitely learned from your help and also been able to complete the question.
Last edited: Apr 29, 2021
LikesLnewqban
Apr 29, 2021
#11
gnits
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46
Lnewqban said:
There should be a point (let's call it E) located between D and C, which forms a triangle with B and C, which area is 1/3 of the total area.
The centroid of that triangle is located at Q.
The ABED quadrilateral has 2/3 of the total area, or twice the area of triangle BCE.
The centroid of that quadrilateral is located at midpoint of diagonal BD (let's call it F).
The CG of the total area is located on the line joining Q and F, at a distance of 1/3 QF from point F and 2/3 QF from point QApr 29, 2021
#13
gnits
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Lnewqban said:I followed haruspex up to his last equation. From there we know that BP = P - B and BD = D - B and so as x = BP / BD then x = (P - B) / (D - B) so we substitute this into haruspex's last equation and the (B - D) terms cancel, as do the P terms. We are left with WM = (W/3)*(Q + D + B) = (W/3)Q + (W/3)D + (W/3)B which is clearly equivalent to a weight of W/3 at Q and 2W/3 at the midpoint of BD.
Thanks again for your help.
LikesLnewqban and BvU
Related to To find the center of gravity of a particular quadrilateral
1. What is the center of gravity of a quadrilateral?
The center of gravity of a quadrilateral is the point where the weight of the shape is evenly distributed. It is also known as the centroid or center of mass.
2. How do you find the center of gravity of a quadrilateral?
To find the center of gravity of a quadrilateral, you can use the formula (x,y) = ((x1 + x2 + x3 + x4)/4, (y1 + y2 + y3 + y4)/4), where (x1, y1), (x2, y2), (x3, y3), and (x4, y4) are the coordinates of the four vertices of the quadrilateral.
3. Can the center of gravity of a quadrilateral be outside the shape?
Yes, the center of gravity can be outside the shape. This can happen if the quadrilateral is irregular or has unequal side lengths.
4. How is the center of gravity of a quadrilateral different from that of a triangle?
The center of gravity of a quadrilateral is the average of the coordinates of its four vertices, while the center of gravity of a triangle is the average of the coordinates of its three vertices. Additionally, the center of gravity of a triangle is always inside the shape, while the center of gravity of a quadrilateral can be outside the shape.
5. Why is finding the center of gravity of a quadrilateral important?
Finding the center of gravity of a quadrilateral is important in various fields such as engineering, architecture, and physics. It helps in determining the stability and balance of a structure or object, and can also be used in calculating the distribution of forces acting on the shape. | 677.169 | 1 |
Honors Geometry Companion Book, Volume 1
• Justify and apply properties of 45 ° -45 ° -90 ° triangles. • Justify and apply properties of 30 ° -60 ° -90 ° triangles. Theorems, Postulates, Corollaries, and Properties • 45 ° -45 ° -90 ° Triangle Theorem In a 45 ° -45 ° -90 ° triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times 2 . • 30 ° -60 ° -90 ° Triangle Theorem In a 30 ° -60 ° -90 ° triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times 3 . Example 1
The 45 ° -45 ° -90 ° Triangle Theorem describes the relationship between the lengths of the legs and the length of the hypotenuse in right triangles with angles 45 ° , 45 ° , and 90 ° . In these triangles, the two legs are the same length (congruent), and the length of the hypotenuse is the length of a leg times 2 . The 45 ° -45 ° -90 ° Triangle Theorem is used to determine the unknown length of the hypotenuse of a triangle in this example. The length of one leg is given as 11 units and the measure of its opposite angle is given as 45 ° . First, use the Triangle Sum Theorem to determine that the unknown angle in the triangle is 45 ° (180 ° − 90 ° − 45 ° = 45 ° ). It follows from the 45 ° -45 ° -90 ° Triangle Theorem that the length of the hypotenuse, x , is the length of a leg times 2, or 11 2. The 45 ° -45 ° -90 ° Triangle Theorem is used here to determine the unknown lengths of the legs of a triangle. It is given that the triangle is an isosceles right triangle. This means it is a 45 ° -45 ° -90 ° triangle. The hypotenuse is given as 9 units. By the 45 ° -45 ° -90 ° Theorem, the length of the hypotenuse is the length of a leg times 2 . Thus, = x 2 9. Solve for x , the length of a leg, to yield = x 9 2/2. | 677.169 | 1 |
Nine-Point Circle
One of the greatest occurrences in geometry is the Nine-Point Circle. This exercise will walk you through the proof of the Nine-Point Circle.
The Nine-Point Circle Explained
The nine-point circle is an amazing circle that passes though the 3 feet of the altitudes and 3 feet of the medians. It also passes though PA, PB, and PC, which are the midpoints of AO, BO, and CO.
Proof of the Altitudes
First, we'll prove that the cirlce that passes through the feet of the medians also passes through the feet of the altitudes. In other words, if we draw a circle that passes through the feet of the medians, then it will also pass through the feet of the altitudes. We'll use HC as an example and the others can be proved similarly.
We connect the points MB and HC. We first prove that CQ = QHC. Since MAMB is a segment that connects two medians, MAMB is parallel to AB. By using similar triangles CQMB and CHCA, it is easy to prove that:
(i) CQ = QHC
Also, since CHC is an altitude of triangle ABC, CQ is perpendicular to segment MAMB. Therefore,
(ii) ΔCHCMB is an isosceles triangle
Since triangle ΔCHCMB is an isosceles triangle, CMB = MBHC. We know that ∠CAB = ∠CMBQ because of parallel lines. This also means that ∠CMBQ = ∠QMBHC. Now, the medial triangle is similar to triangle ΔABC. This means that:
(iii) ∠QMBHC = ∠MAMBHC = ∠MCMAMB
Since CMB = MBA = b/2 and MAMC = b/2, the trapezoid HCMCMAMB is an isosceles trapezoid. We know that the vertices of an isosceles triangle lie on a circle. Thus, the circle that passes through ΔMAMBMC also passes through HC.
Similar conclusions can be made about points HA and HB.
The Diameter
Triangle ΔHAHBHC (shown in green) is the orthic triangle. A property of orthic triangle holds that ∠HCHBPB = ∠HAHBPB. This becomes useful for proving that MBPB is the diameter of the nine-point circle.
We now will prove that MBPB is the diameter of the nine-point circle. In the process, we also prove that ΔMBMCPB is a right triangle, and ΔMBMAPB is a right triangle. If these two triangles are right triangles, then MBPB must be the diameter of the nine-point circle. To prove that these two are right triangles, we first prove that ΔHAPBHC is an isosceles triangle. We already know that ΔHAMBHC is an isosceles triangle.
Note that ΔHAHBHC is the orthic triangle. Therefore,
(iv) ∠HAHBPB = ∠HCHBPB
This means that HAPB = HCPB, and triangle ΔHAPBHC is an isosceles triangle. We also have ∠HAHBPB = ∠HAMBPB and ∠HCHBPB = ∠HCMBPB. Since ∠HAHBPB = ∠HCHBPB, ∠HAMBPB = ∠HCMBPB. This proves that MBPB is an angle bisector of ∠HAMBHC or that
(v) MBPB is a perpendicular bisector of ΔHAMBHC
This means that MBPB is the diameter of the nine-point circle and that ΔMBMAPB and ΔMBMCPB are right triangles.
Last Three Points
Since HBMC = HAMC, ∠HBPBMC = ∠HAPAMC. This proves that the quadrilateral PAMCPBO is a parallelogram. Thus,
(vii) PAMC = OPB
Notice that HAMC is a median of ΔAHAB. As stated earlier, MCPB (or MCM1') is parallel to AHA. This proves that
(viii) OPB = PBB and MAM1' = M1'B.
The Radius of the Nine-point Circle
Now we wonder about the length of the radius of the nine-point circle. It turns out that the radius is exactly half of the radius of the circumcircle of ΔABC. First, we note that ΔPAPBPC is similar to ΔABC. The lengths of the sides of ΔPAPBPC are exactly half that of ΔABC. Therefore, the area of ΔPAPBPC is ¼ of ΔABC. The circumradius, R, of ΔABC is given by: \(R=\frac{abc}{4K}\), where K is the area of ΔABC. The circumradius of ΔPAPBPC (denoted as RN) is given by: \(R_{N}=\frac{\frac{a}{2}\cdot\frac{b}{2}\cdot\frac{c}{2}}{4\cdot\frac{1}{4}\cdot K}=\frac{abc}{8K}=\frac{1}{2}R\).
Medial Triangle Similarity
Triangle ΔPAPBPC is also similar to the medial triangle. Its sides are parallel to ΔABC.
Moreover, when its vertices are connected with the medians, the lines are parallel to the altitudes (this is shown in the figure above). These lines meet the sides halfway between the feet of the altitudes and the vertices. | 677.169 | 1 |
how do 30 60 90 triangles work | 677.169 | 1 |
Rope Method In Setting Out A Building
A line has to be set out perpendicular to the base line from peg (A). Peg (A) is not on the base line.
A long rope with a loop at both ends and a measuring tape are used. The rope should be a few meters longer than the distance from peg (A) to the base line.
Step 1
One loop of the rope is placed around peg (A). Put a peg through the other loop of the rope and make a circle on the ground while keeping the rope straight. This circle crosses the base line twice (see Fig. 22a). Pegs (B) and (C) are placed where the circle crosses the base line.
Step 2
Peg (D) is placed exactly half way in between pegs (B) and (C). Use a measuring tape to determine the position of peg (D). Pegs (D) and (A) form the line perpendicular to the base line and the angle between the line CD and the base line is a right angle (see Fig. 22b).
Hi, I'm Richard Nwachukwu! It is my job to handle the content aspect of this great organization and I'm determined to ensure you get it all right as long as you're handling a construction project in Nigeria! | 677.169 | 1 |
How To Solve For X In A Triangle
February 10, 2024
How To Solve For X In A Triangle
A triangle is a polygon with three vertices and three edges. Triangles are commonly described based on the length of their sides and their internal angles. There are different types of triangles such as equilateral, isosceles, and scalene. Triangles can also be classified based on their internal angles as right or oblique. Understanding the properties and theorems related to triangles is essential for solving for X in a triangle.
Triangle Facts, Theorems, and Laws
Triangles have several important properties and theorems:
It is not possible for a triangle to have more than one vertex with an internal angle greater than or equal to 90°.
The interior angles of a triangle always add up to 180°.
The sum of the lengths of any two sides of a triangle is always larger than the length of the third side.
The Pythagorean theorem is specific to right triangles, where the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.
The law of sines relates the ratio of the length of a side of a triangle to the sine of its opposite angle.
Area of a Triangle
There are multiple equations for calculating the area of a triangle. The most commonly known equation involves the base, b, and height, h. Another method for calculating the area of a triangle uses Heron's formula, which does not require an arbitrary choice of a side as a base or a vertex as an origin.
Median, Inradius, and Circumradius
The median of a triangle is the length of a line segment that extends from a vertex of the triangle to the midpoint of the opposing side. The inradius is the radius of the largest circle that will fit inside the triangle, while the circumradius is the radius of a circle that passes through all the vertices of the triangle.
Solving for X in Different Types of Triangles
Solving for X in a triangle can encompass a number of different problems. Depending on the type of triangle, there are different methods to solve for X. For example, in a right triangle, X can represent the degree of any one of the three angles found in the triangle. In an isosceles triangle, X can represent the vertex angle, and in other triangles, X can represent any of the angles.
Methods for Solving for X
There are different methods for solving for X in different types of triangles:
Solving for X in a Right Triangle: In a right triangle, X can be solved by adding 90 degrees for the right angle to the degree measurement of the other marked angle and then subtracting the sum of the two angles from 180 degrees.
Solving for X in an Isosceles Triangle: In an isosceles triangle, X can be solved by locating the two base angles that are marked with half-circles with lines through them and then multiplying the measurement given for one of the angles by two, or by subtracting the measurement of the vertex angle from 180 and dividing the difference by two.
Solving for X in Other Triangles: For obtuse and acute triangles, X can be solved by adding the given degrees of the two angles provided and subtracting that from 180. For equilateral triangles, all the angles equal 60 degrees.
How to Find the Missing Side or Angle of a Right Triangle
There are different methods for finding the missing side or angle of a right triangle, such as using the Pythagorean theorem, the law of sines, or trigonometric functions. The area of a right triangle can also be used to calculate the missing sides. Finding the angles of a right triangle requires applying basic trigonometric functions.
How to Solve a Right Angle Triangle with Only One Side
To solve a triangle with one side, one also needs one of the non-right-angled angles. Different trigonometric functions can be used to find the missing side or angle of a right triangle.
FAQ
How many lines of symmetry does a right triangle have?
If a right triangle is isosceles, it has one line of symmetry. Otherwise, the triangle will have no lines of symmetry.
Can a right-angled triangle have equal sides?
No, a right triangle cannot have all 3 sides equal. However, it can have its two non-hypotenuse sides equal in length.
Are all right triangles similar?
Not all right-angled triangles are similar, although some can be. They are similar if all their angles are the same length, or if the ratio of two of their sides is the same.
Understanding the properties and theorems related to triangles is essential for solving for X in a triangle. Different types of triangles require different methods for solving for X, and the use of trigonometric functions and theorems can aid in finding the missing sides and angles of a triangle. | 677.169 | 1 |
In a triangle ABC, AB = BC, AD is perpendicular to side BC and CE is perpendicular to side AB. Prove that : AD = CE.
Video Solution
|
Answer
Step by step video & image solution for In a triangle ABC, AB = BC, AD is perpendicular to side BC and CE is perpendicular to side AB. Prove that : AD = CE. by Maths experts to help you in doubts & scoring excellent marks in Class 9 exams. | 677.169 | 1 |
In Geometry, a triangleis a three-sided polygon that consists of three edges and three vertices. The most important property of a triangle is that the sum of the internal angles of a triangle is equal to 180 degrees. This property is called angle sum property of triangle.
If ABC is a triangle, then it is denoted as ∆ABC, where A, B and C are the vertices of the triangle. A triangle is a two-dimensional shape, in Euclidean geometry, which is seen as three non-collinear points in a unique plane.
Table of contents:
Definition
Angles
Properties
Types
Scalene Triangle
Isosceles Triangle
Equilateral Triangle
Acute Angle Triangle
Right Angle Triangle
Obtuse Angle Triangle
Perimeter
Area
Heron's Formula
Solved Examples
Video Lesson
FAQs
Below given is a triangle having three sides and three edges, which are numbered as 0,1,2.
Definition
As we discussed in the introduction, a triangle is a type of polygon, which has three sides, and the two sides are joined end to end is called the vertex of the triangle. An angle is formed between two sides. This is one of the important parts of geometry.
Some major concepts, such as Pythagoras theorem and trigonometry, are dependent on triangle properties. A triangle has different types based on its angles and sides.
Shape of Triangle
Triangle is a closed two-dimensional shape. It is a three-sided polygon. All sides are made of straight lines. The point where two straight lines join is the vertex. Hence, the triangle has three vertices. Each vertex forms an angle.
Angles of Triangle
There are three angles in a triangle. These angles are formed by two sides of the triangle, which meets at a common point, known as the vertex. The sum of all three interior angles is equal to 180 degrees.
If we extend the side length outwards, then it forms an exterior angle. The sum of consecutive interior and exterior angles of a triangle is supplementary.
Let us say, ∠1, ∠2 and ∠3 are the interior angles of a triangle. When we extend the sides of the triangle in the outward direction, then the three exterior angles formed are ∠4, ∠5 and ∠6, which are consecutive to ∠1, ∠2 and ∠3, respectively.
Hence,
∠1 + ∠4 = 180° ……(i)
∠2 + ∠5 = 180° …..(ii)
∠3 + ∠6 = 180° …..(iii)
If we add the above three equations, we get;
∠1+∠2+∠3+∠4+∠5+∠6 = 180° + 180° + 180°
Now, by angle sum property we know,
∠1+∠2+∠3 = 180°
Therefore,
180 + ∠4+∠5+∠6 = 180° + 180° + 180°
∠4+∠5+∠6 = 360°
This proves that the sum of the exterior angles of a triangle is equal to 360 degrees.
Properties
Each and every shape in Maths has some properties which distinguish them from each other. Let us discuss here some of the properties of triangles.
A triangle has three sides and three angles.
The sum of the angles of a triangle is always 180 degrees.
The exterior angles of a triangle always add up to 360 degrees.
The sum of consecutive interior and exterior angle is supplementary.
The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Similarly, the difference between the lengths of any two sides of a triangle is less than the length of the third side.
The shortest side is always opposite the smallest interior angle. Similarly, the longest side is always opposite the largest interior angle.
Also, read:
Types Of Triangles
Properties Of Triangle
Congruence Of Triangles
Pythagoras Theorem
Triangle Formula
Triangles Class 9
Triangles For Class 10
Types
On the basis of length of the sides, triangles are classified into three categories:
Scalene Triangle
Isosceles Triangle
Equilateral Triangle
On the basis of measurement of the angles,triangles are classified into three categories:
Acute Angle Triangle
Right Angle Triangle
Obtuse Angle Triangle
Scalene Triangle
A scalene triangle is a type of triangle, in which all the three sides have different side measures. Due to this, the three angles are also different from each other.
Isosceles Triangle
In an isosceles triangle, two sides have equal length. The two angles opposite to the two equal sides are also equal to each other.
Equilateral Triangle
An equilateral triangle has all three sides equal to each other. Due to this all the internal angles are of equal degrees, i.e. each of the angles is 60°
Acute Angled Triangle
An acute triangle has all of its angles less than 90°.
Right Angled Triangle
In a right triangle, one of the angles is equal to 90° or right angle.
Obtuse Angled Triangle
An obtuse triangle has any of its one angles more than 90°.
Perimeter of Triangle
A perimeter of a triangle is defined as the total length of the outer boundary of the triangle. Or we can say, the perimeter of the triangle is equal to the sum of all its three sides. The unit of the perimeter is same as the unit of sides of the triangle.
Perimeter = Sum of All Sides
If ABC is a triangle, where AB, BC and AC are the lengths of its sides, then the perimeter of ABC is given by:
Perimeter = AB+BC+AC
Area of a Triangle
The area of a triangle is the region occupied by the triangle in 2d space. The area for different triangles varies from each other depending on their dimensions. We can calculate the area if we know the base length and the height of a triangle. It is measured in square units.
Suppose a triangle with base 'B' and height 'H' is given to us,then, the area of a triangle is given by-
Formula:
Area of triangle = Half of Product of Base and Height
Area = 1/2 × Base × Height
Example
Question- Find the area of a triangle having base equal to 9 cm and height equal to 6 cm.
Solution- We know that Area = 1/2× Base× Height
= 1/2× 9 × 6 cm2
= 27 cm2
Area of Triangle Using Heron's Formula
In case, if the height of a triangle is not given, we cannot be able to use the above formula to find the area of a triangle.
Therefore, Heron's formula is used to calculate the area of a triangle, if all the sides lengths are known.
First, we need to calculate the semi perimeter (s).
s = (a+b+c)/2, (where a,b,c are the three sides of a triangle)
Now Area is given by; A = √[s(s-a)(s-b)(s-c)]
Solved Examples
Question 1: If ABC is a triangle where AB = 3cm, BC=5cm and AC = 4cm, then find its perimeter.
Solution: Given, ABC is a triangle.
AB = 3cm
BC = 5cm
AC = 4cm
As we know by the formula,
Perimeter = Sum of all three sides
P = AB + BC + AC
P = 3+5+4
P = 12cm
Question 2:Find the area of a triangle having sides 5,6 and 7 units length.
Solution- Using Heron's formula to find the area of a triangle-
Semiperimeter (s) = (a+b+c)/2
s = (5 + 6 +7)/2
s = 9
Now Area of a triangle = √[s(s-a)(s-b)(s-c)]
=√[9(9-5)(9-6)(9-7)]
=√ [9× 4× 3× 2]
=√ [3× 3× 2× 2× 3× 2]
=√ [32 × 22× 3× 2]
= 6√6square units.
Triangles: Introduction
Geometry is all about shapes like squares, circles, rectangles, triangles and so on. Among all the shapes that we have listed here, triangles seem to be fun and different. Triangle is a shape that is made up of three lines and three angles. Watch this video to know how triangle is different from other shapes and see how you can learn and remember different types of triangles easily from the video. We see triangles everywhere. If we somehow manage to bring three lines together, we can see a triangle is formed. There are different types of triangles – equilateral triangles, isosceles triangles, scalene triangles and so on. The names of these triangles don't even sound English. Do you know how these triangles got these names? Watch this video to know the trivia behind triangles and learn their properties in the simplest way.
Frequently Asked Questions – FAQs on Triangles
Q1
What are triangles?
A triangle is a three-sided polygon, which has three vertices. The three sides are connected with each other end to end at a point, which forms the angles of the triangle. The sum of all three angles of the triangle is equal to 180 degrees.
What are the properties of triangles?
Sum of angles of the triangle is equal to 180 degrees. Exterior angles of a triangle add up to 360 degrees. Shortest side is always opposite the smallest angle of a triangle.
Q4
What is the perimeter and area of a triangle?
The perimeter is the length of the outer boundary of the triangle and area is the region occupied by it in a two-dimensional space.
Q5
What is the formula for area and perimeter of a triangle?
The perimeter of triangle = Sum of all three sides Area = ½ (Product of base and height of a triangle)
Q6
What is scalene, isosceles and equilateral triangle?
Scalene, isosceles and equilateral triangle are the types of triangles which differ from each other based on their side-length. If all the three sides are different in length, then its scalene triangle. If any two sides are equal in length, then it is an isosceles triangle. If all three sides are equal in length, then it is an equilateral triangle.
Q7
What is the difference between acute triangle, obtuse triangle and right triangle?
An acute triangle has all its angles less than 90 degrees. An obtuse triangle has any one of its angle greater than 90 degrees. A right triangle has exactly one angle equal to 90 degrees. | 677.169 | 1 |
Nine Point Circle
Why mathematical formula exist, I don't know but they fascinate me, This work was created using the "Nine Point Circle". Every triangle has a nine point circle within it you can see in the diagram adjacent, Draw a triangle, any triangle (although it may be best to start with an acute triangle), mark the midpoints of each side (3 points), then drop an altitude from each vertex to the opposite side, and mark the points where thealtitudesintersect the opposite side. (If the triangle is obtuse, an altitude will be outside the triangle, so extend the opposite side until it intersects.), notice that thealtitudesintersect at a common point. Mark the midpoint between each vertex and this common point. when you have these nine point they all fall within a perfect circle. In the painting, I sheared the triangle by the distance of two of these points and created the image above.
The dimensions given above are the frame dimensions, (needed to calculate postage) the dimensions of the support 35.7 x 35 x 2.3 cm. | 677.169 | 1 |
Hint: In a geographic coordinate system, a prime meridian is the meridian at which longitude. A prime meridian and its anti-meridian, together, form a broad circle.
Complete answer: The value of prime meridian is 0 Degree. A prime meridian is the meridian (a line of longitude) in a geographic organize framework at which longitude is characterized to be 0°.Together, a prime meridian and its enemy of meridian (the 180th meridian in a 360°-framework) structure an incredible circle. This extraordinary circle isolates a spheroid into two sides of the equator. In the event that one uses bearings of East and West from a characterized prime meridian, at that point they can be known as the Eastern Side of the equator and the Western Hemisphere.
The Eastern Side of the equator is a geological term for the half of Earth which is east of the prime meridian (which crosses Greenwich, London, Joined Realm) and west of the antimeridian (which crosses the Pacific Sea and moderately little land from shaft to pole).The Western Side of the equator is a topographical term for the half of Earth which lies west of the prime meridian (which crosses Greenwich, London, Joined Realm) and east of the antimeridian.For Earth's prime meridian, different shows have been utilized or pushed in various districts all through history.
The most generally utilized present day meridian is the IERS Reference Meridian. It is gotten however goes astray somewhat from the Greenwich Meridian, which was chosen as a worldwide norm in 1884.A prime meridian for a body not tidally bolted (or if nothing else not simultaneous) is eventually subjective, in contrast to an equator, which is controlled by the hub of turn. For divine items that are tidally bolted (all the more explicitly, simultaneous), notwithstanding, their prime meridians are controlled by the face in every case internal of the circle (a planet confronting its star, or a moon confronting its planet), similarly as equators are dictated by turn.
Thus, the answer is option B: 0 degree.
Note: The modern prime meridian, based at the Royal Observatory, Greenwich, was established by Sir George Airy in 1851. As with the Planet, it is important to arbitrarily identify prime meridians. | 677.169 | 1 |
What is first angle method?
What is first angle method?
First angle projection is one of the methods used for orthographic projection drawings and is approved internationally except the United States. In this projection method, the object is placed in the first quadrant and is positioned in front of the vertical plane and above the horizontal plane.
Which statement is correct for first angle projection?
Explanation: In first angle projection the object's right side will be projected only if we watch from right side of object and the impression will fall to the left side of front view similar to the other side also so the right side view is placed on the left side of front view.
Where does 1st angle projection place the right view?
What is 1st angle and 3rd angle method?
In third-angle projection, the view of a component is drawn next to where the view was taken. In first-angle projection, the view is drawn on the other end of the component, at the opposite end from where the view was taken.
What is a 3D oblique drawing?
Oblique projection is a simple type of technical drawing of graphical projection used for producing two-dimensional (2D) images of three-dimensional (3D) objects. Oblique projection is commonly used in technical drawing.
Which is an example of first angle projection?
In first-angle projection, each view is shown as if the viewer is looking through the object and projecting the image on the other side. This style is the most common one used in Europe. Here is an example of first-angle projection (see video). The red arrows represent the view of the observer with the image being projected on the other side.
Who was the first mathematician to do perspective drawing?
A modern deductive footing for perspective drawing was given later by Brook Taylor (1685-1731) and J. H. Lambert (1728-77). A competing point of view has held by mathematicians such as René Decartes (1596-1650), Pierre de Fermat (1601-1665) and Julius Plücker (1801-1868)…
How to draw an acute triangle using a protractor?
Draw two acute angles on your paper. Construct a third angle with a measure equal to the sum of the measures of the first two angles. Remember, you cannot use a protractor—use a compass and a straightedge only. 7. Draw a large acute triangle on the top half of your paper. Duplicate it on the bottom half, using your compass and straightedge.
Which is a way of drawing an object from different directions?
Orthographic Projection is a way of drawing an 3D object from different directions. Usually a front, side and plan view are drawn so that a person looking at the drawing can see all the important sides. | 677.169 | 1 |
9 Shapes That Start With The Letter K
Are you searching for some shapes that start with the letter K? Don't worry, you have come to the right place.
In this article, I will delve into the fascinating world of shapes and comprise a list of some common and popular shapes starting with the letter K for you.
So, without further ado, let's discover the shapes beginning with the letter K, which will grow your geometric vocabulary skills.
Shapes That Start With Letter K
Below are the shapes that begin with the letter K (In alphabetical order):
1. Kappa Curve:
The Kappa Curve is a mathematical shape that exhibits self-intersecting properties. It is characterized by its intricate loops and curves, forming a visually captivating pattern. The Kappa Curve finds applications in various fields, including computer graphics, fractal generation, and even art.
2. Kepler Triangle:
The Kepler Triangle is a special type of triangle that features a right angle and two congruent sides. It is named after Johannes Kepler, a renowned mathematician and astronomer. The Kepler Triangle is of particular interest in geometry and trigonometry, as it demonstrates the relationship between angles and side lengths within the triangle.
3. Keyhole Shape:
The Keyhole Shape resembles the silhouette of a traditional keyhole, typically characterized by a circular or oval opening with a narrower stem. It is a shape often associated with locks and keys. The Keyhole Shape can be found in various contexts, including architecture, design, and decorative patterns.
4. Keystone:
The Keystone is a trapezoidal-shaped stone that serves as a central wedge in an arch. It plays a crucial role in architecture, particularly in creating stable and structurally sound arches and vaults. The Keystone acts as a key element, distributing weight and maintaining the integrity of the structure.
5. Kite:
The Kite is a quadrilateral shape with two pairs of adjacent sides that are equal in length. It is characterized by its distinctive shape, resembling a diamond with one pair of opposite angles equal. The Kite shape can be found in various applications, including kite-flying, mathematics, and even aerospace engineering.
6. Klein Bottle:
The Klein Bottle is a non-orientable surface with no distinguishable inside or outside. It is a fascinating shape in topology, representing a surface that cannot be properly embedded in three-dimensional space without self-intersections. The Klein Bottle has captivated mathematicians and enthusiasts alike with its unique properties and intriguing topological characteristics.
7. Knotted Circle:
The Knotted Circle refers to a circular shape with a knot or loop incorporated into its structure. It represents the fusion of the simplicity of a circle with the complexity of a knot. The Knotted Circle finds artistic and decorative applications, symbolizing interconnectedness and infinite possibilities.
8. Knotted Torus:
The Knotted Torus is a torus shape with a knot or loop passing through its central hole. It combines the properties of a torus, a doughnut-shaped object, with the intricacies of knot theory. The Knotted Torus offers a fascinating exploration of topological structures and their interconnectedness.
9. Kupe:
Kupe is a traditional Maori shape that represents a stylized fishhook. It is of cultural significance to the indigenous people of New Zealand, symbolizing strength, resilience, and the importance of navigation and fishing. The Kupe shape serves as a cultural icon and carries the stories and traditions of the Maori people.
Hope you found this article about "shapes that start with K" educative and helpful.
Do you know any other shapes that start with the letter K | 677.169 | 1 |
Activities to Teach Students Triangle Proportionality Theorem
Triangle proportionality theorem is an essential concept that students need to understand in geometry. Knowing this theorem is important for solving problems involving similar triangles. When teaching the concept of triangle proportionality theorem, it is important to include various activities that will make learning fun and engaging for the students. Here are some activities you can use to teach students the triangle proportionality theorem:
1. Exploration with Geogebra
Geogebra is a dynamic geometry software that you can use to teach students the triangle proportionality theorem. First, students can explore the theorem by plotting different triangles, measuring their sides, and verifying that the theorem holds. Then, you can ask them to drag the vertices of the triangles to create new triangles and see if the theorem still holds. This will help them understand the theorem visually and intuitively.
2. Real-World Examples
Another effective way to teach the triangle proportionality theorem is by using real-world examples. For example, you can ask students to take photos of buildings, trees, or other objects that they think demonstrate the theorem. After that, they can measure the sides and angles of the object and use the theorem to find the unknown dimensions. This activity will help students connect the abstract theorem to the real world.
3. Investigation with Triangles
In this activity, students work in groups to investigate the triangle proportionality theorem. Each group is given a triangle and a ruler, and they are asked to construct the midpoint of one side and draw a line from that midpoint to a vertex opposite that side. Then, they measure the two segments they have created and compare them. This activity helps develop their reasoning skills and critical thinking.
4. Experiments with Paper Triangles
You can also use paper triangles to teach the triangle proportionality theorem. Students can create a variety of triangles by folding paper, measuring the sides and angles, and then testing the theorem. You can ask them to create triangles with different dimensions and explore how the theorem holds in each case. This hands-on approach will help them apply the theorem to different situations and develop their problem-solving skills.
5. Concept-theorem Connection
Finally, it is important to help students understand the connection between the concept and the theorem. You can ask students to draw several triangles and then construct mid-segments. They can then conclude that all create similar triangles. This activity will help them understand that the theorem is a rule that applies to all situations with similar triangles.
In conclusion, using different activities, including Geogebra, real-world examples, investigations, experiments, and connecting the theorem and concept, you can effectively teach the triangle proportionality theorem. These activities will help students visualize and understand the theorem better, fostering their understanding of | 677.169 | 1 |
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You can take any video, trim the best part,. Write the letter for the correct answer in the blank at the right of each question. Glencoe geometry chapter 6 chapter 9 test form 1 answer key. The radius and the area. A marble is drawn and not replaced. Web it takes only a couple of minutes. 6 groups of 3 = a. Web test, form 2a write the letter for the correct answer in the blank at the right of each question. Web test, form 2a (continued) for exercises 8 and 9, a bag contains 1 red, 2 blue, 4 orange, and 3 purple marbles. Download your updated document, export it to the cloud, print it from the editor, or share it with others using a.
Geometry Chapter 2 Test Pdf SylenaGvidas
Web edit your chapter 9 test form 2a answer key online type text, add images, blackout confidential details, add comments, highlights and more. You can take any video, trim the best part,. Then a second marble is. Web chapter 9 test form 2c answer key what information do we need to graph a circle? Web test, form 2a write the letter for the correct answer in the blank at the right of each question. 9 2 rows of 9 = b. Write the letter for the correct answer. Given c (2, 9), under which reflection. Glencoe geometry chapter 6 chapter 9 test form 1 answer key. Web it takes only a couple of minutes. | 677.169 | 1 |
The LEGO Mindstorm EV3 Robot that coincides with this tutorial comes from building specific sections found in the LEGO Mindstorm Education Core Set building instructions (I'll refer to use the Base model).
In order to define the robot's trace as a triangle or any n-angle, from
a mathematical point of view, firstly the robot determines the segment
and next it turns by a certain angle. This task should be repeated for
the appropriate number of times according to the drawn geometric figure.
Firstly, we'll check how to draw for example a 30-centimeter segment
(the length of course depends on you). We need to know how many turns
the wheel should make.
Circumference - the distance around the edge of a circle. To calculate
the circumference of a circle, use the formula\(C\ = \ \pi d\), where C is the circumference, d is the
diameter, and \(\pi\) we put 3.14. Using this formula we can write
that
\(30\ cm\ = \ \text{πdx}\) (1)
where \(x\) is a number of rotation of robot's wheel,
\(d = \ 5.6\ cm\)(diameter of robot's wheel) and
\(\ \pi = 3.14\), so it is easy to compute \(x\).
You have all information necessary to write the first part of the
programme.
Now we will investigate the second part i.e. turn by a certain angle.
One wheel of the robot doesn't turn. We need to calculate how many turns
the second robot's wheel should make.
A circle is 360° all the way around; therefore, if you divide an
arc's degree measure by 360°, you find the fraction of the
circle's circumference that the arc makes up. Then, if you
multiply the circle's circumference by that fraction, you get the
length along the arc. Formula is
\(arc\ length\ = \ 2\ \pi\ R\ (\ \theta\ /360\ )\),
where \(R\) equals to the radius of the circle and
\(\theta\)- equals the measurement of the arc's central angle,
in degrees. Note, \(R\) is not the radius of the robot's wheel, it
is the radius of the wheel whose fragment is delineated by the robot
during the turn (so there is - Wheel track - means the shortest
distance between the center of the tire treads on the same
axle).
The robot wheel moves on this arc. It travels a path whose length is
equal to \(arc\ length\ = \ \pi dx\) where \(x\) is the number
of rotation of robot's wheel, \(d\) is the diameter of robot's wheel
and \(\pi = 3.14\).
It is known that diameter is equal to radius multiplied by 2. Let denote
by r radius of robot's wheel, so
\(arc\ length\ = \ 2\ \pi\ r\_ w\ x.\)Comparing both formulas we
get
\(2\ \pi\ R\ (\ \theta\ /360\ )\ = \ 2\ \pi\ r\ x\),
so
\(x = \ (R\ /\ r)\ (\ \theta\ /360\ )\),
where
\(r\)is radius of robot's wheel
\(\text{R }\)is radius of the wheel whose fragment is delineated
by the robot during the turn (robot's wheel track)
\(\theta\)- equals to the measurement of the arc central angle,
in degrees
Step 3. Create a programme to make the robot draw an equilateral
triangle, square, regular pentagon, etc. The input will be the number of
angles, so we have to compute the measure of the one angle in the figure
(for details see lesson Gyro and geometric figures) .
Rotate the motors at left_speed and right_speed for rotations. Left_speed and right_speed are integer percentages of the rated maximum speed of the motor.
on_for_rotations(left_speed, right_speed, rotations, brake=True, block=True)
If left_speed is not equal to right_speed (i.e. the robot will turn), the motor on the outside of the turn (the one with the faster speed) will rotate for the full rotations while the motor on the inside will have its number of rotations calculated according to the expected turn. | 677.169 | 1 |
What is special about an acute triangle?
What is special about an acute triangle?
An acute triangle is defined as a triangle in which all of the angles are less than 90°. In other words, all of the angles in an acute triangle are acute.
What does an acute-angled triangle have?
three acute angles
An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles.
Why is it called an acute triangle?
Definitions. An acute angle is any angle less than 90 degrees. So, as its name suggests, an acute triangle is a triangle whose three angles are all smaller than 90 degrees. Measure all the angles, and if they are less than a right angle (a 90-degree or square angle), then it's an acute triangle.
Do acute triangles have hypotenuse?
The other two angles are acute angles. You may have noticed that the side opposite the right angle is always the triangle's longest side. It is called the hypotenuse of the triangle. The other two sides are called the legs.
How does a acute angle look like?
An acute angle is an angle that measures between 90° and 0°, meaning it is smaller than a right angle (an "L" shape) but has at least some space between the two lines that form it. A "V" shape is an example of an acute angle. (These are not related to the degrees used to measure temperature.)
How many acute angles do all triangles have?
acute angles
Yes, all triangles have at least two acute angles. Acute angles are angles that measure less than 90 degrees, while obtuse angles measure greater than…
How many sides does a acute triangle have?
An acute triangle is a trigon with three sides and three angles each less than 90º. The measurement of all the three interior angles of the acute triangle lies within 0° to 90°, but the sum of all the interior angles is always 180 degrees.
What are the properties of a acute angle?
the main characteristic of an acute angle is that its measurement lies within 0° to 90°. In the case of the acute angle triangle, the measure of all the three interior angles of the acute angle triangle lie within 0° to 90° but the sum of all the interior angles is always 180 degrees.
How many sides does an acute triangle have?
Acute triangle – Definition with Examples
Equilateral Acute Triangle:
Isosceles Acute Triangle:
All the interior angles of an equilateral acute triangle measure 60°. It is also known as equiangular triangle.
Two angles of an isosceles acute triangle that measure the same, just like its two sides. | 677.169 | 1 |
What is a four set Venn diagram?
What is a four set Venn diagram?
This four-set Venn diagram template can help you: – Visually show the relationships between four categories. – Highlight the similarities and differences between the categories. – Collaborate with colleagues to identify areas of overlap between four concepts or categories.
In the Choose a SmartArt Graphic gallery, click Relationship, click a Venn diagram layout (such as Basic Venn), and then click OK.
Can you draw a Venn diagram for 4 sets?
… it is impossible to draw a Venn diagram with circles that will represent all the possible intersections of four (or more) sets.
What is the formula for Venn diagram?
Venn Diagram Formulas n ( X ∪Y) = n (X) + n(Y) – n( X ∩ Y)
What is AUB Venn diagram?
The union of two sets A and B is a set that contains all the elements of A and B and is denoted by A U B (which can be read as "A or B" (or) "A union B"). A union B formula is used to find the union of two sets A and B.
Can a Venn diagram have more than 2 circles?
They are often confused with Euler diagrams. While both have circles, Venn diagrams show the whole of a set while Euler diagrams can show parts of a set. Venn diagrams can have unlimited circles, but more than three becomes extremely complicated so you'll usually see just two or three circles in a Venn diagram drawing.
How do you complete a Venn diagram?
How to Make a Venn Diagram
The first step to creating a Venn diagram is deciding what to compare. Place a descriptive title at the top of the page.
Create the diagram. Make a circle for each of the subjects.
Label each circle.
Enter the differences.
Enter the similarities.
What is a ∪ b )? | 677.169 | 1 |
Right Triangle Geometry
Curated and Reviewed byLesson Planet
Students study the basic principles of right triangle geometry needed to solve for unknown values of a right triangle. They apply and master the principles of right triangle geometry by solving problems. | 677.169 | 1 |
Trig question -sin pi/4 , give the exact value?
In summary, the conversation discusses finding the exact value of -sin pi/4. The answer of -1/root 2 is obtained through the method of finding pi/4 on the unit circle and rationalizing the denominator by multiplying the top and bottom by √2. The conversation also touches on the use of the Pythagorean Theorem and the importance of simplifying radicals in the denominator.
Jul 11, 2011
#1
Neopets
29
0
Homework Statement
-sin pi/4 , give the exact value?
-1/root 2 is the answer according to the book? How in the world do they get that result. What do you to make that happen?
Homework Equations
The Attempt at a Solution
Also, is this how to do the problem? Find pi/4 on the unit circle, it's root 2 / 2, that doesn't seem to get the right answer even though that's was supposed to be the method?
Not in all cases
I don't think the teacher would worry so much about it when it comes to trig.
Related to Trig question -sin pi/4 , give the exact value?
What is the value of sin(pi/4)?
The value of sin(pi/4) is 1/√2 or approximately 0.707.
How do I find the exact value of sin(pi/4)?
You can use the unit circle or a calculator to find the exact value of sin(pi/4). On the unit circle, the coordinates of the point at pi/4 radians are (√2/2, √2/2). This means that sin(pi/4) = √2/2. With a calculator, you can simply enter sin(pi/4) and get the result as 0.707.
What is the relationship between sin(pi/4) and cos(pi/4)?
Sin(pi/4) and cos(pi/4) are complementary trigonometric functions. This means that sin(pi/4) = cos(pi/2 - pi/4). Using this relationship, we can say that cos(pi/4) = √2/2, which is the exact value of cos(pi/4).
How can I graph sin(pi/4)?
To graph sin(pi/4), you can plot the point (pi/4, √2/2) on a cartesian plane. This point lies on the unit circle and is the exact value of sin(pi/4). You can also use a graphing calculator to get a visual representation of the graph of sin(pi/4).
What are the other trigonometric functions of pi/4?
The other trigonometric functions of pi/4 can be found using the unit circle or a calculator. The values are as follows: cos(pi/4) = √2/2, tan(pi/4) = 1, csc(pi/4) = √2, sec(pi/4) = √2, cot(pi/4) = 1. | 677.169 | 1 |
Properties of Dilations Lesson 10 1 Answer Key Form
Use a Properties Of Dilations Lesson 10 1 Answer Key template to make your document workflow more streamlined.
Triangle A'B'C' is a dilation of triangle
ABC, is it a reduction or an enlargement? _________________
For Exercises 5–8, tell whether one figure is a dilation of the other
or not. If one figure is a dilation of the other, tell whether it is an
enlargement or a reduction. Explain your reasoning.
5. Triangle R'S'T' has sides of 3 cm, 4 cm, and 5 cm. Triangle RST has
sides of 12 cm, 16 cm, and 25...
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eighth grade lesson 10.1 is properties of dilations so in the previous section we were doing transformations that involved trans uh translations reflections and rotations and all of those were a manipulation of the pre-image into an image that only changed the orientation of the pre-image it did not change the size so it stayed congruent uh the pre-image stayed congruent with the image in this case with dilations this is where we start uh dealing with changing the size of something a dilation is changing the side either making it larger or making it smaller so um you're kind of familiar with that sort of idea it's like on a tablet us pinching open the screen to enlarge it so and to make something larger isn't is called enlarge for the dilations and if you were to pinch closed you know that you would make something smaller and in the dilations they call that a reduction so those are the sort of things that you would see so we're looking at things that will die dilations that will enlarg
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This coloring activity was created to help students find missing side and angle measures in triangles using the Law of Sines and Law of Cosines. There are 12 ...
Law of Sines and Law of Cosines Coloring ActivityThis coloring activity was created to help students find missing side and angle measures in triangles using the Law of Sines and Law of Cosines. There are 12 problems total, 6 Law of Sines problems and 6 Law of Cosines problems. For each problem, stu...
This coloring activity provides your students with practice evaluating the six trigonometric functions of angles found on the Unit Circle.
This coloring activity provides your students with practice evaluating the six trigonometric functions of angles found on the Unit Circle. This product includes two different coloring activities. In the first activity, students practice finding the sine, cosine, and tangent of unit circle angles (over 40 problems). Student are asked to find the cosecant, secant, and cotangent of angles found on the unit circle (over 40 problems) on the second coloring activity.
Included in this resource:
✔ Unit Circle Coloring Activity 1 (1 page - Finding Sin, Cos, and Tan)
✔ Unit Circle Coloring Activity 2 (1 page - Finding Csc, Sec, and Cot)
✔ Answer keys Included
The law of sines and cosines can be difficult to understand, but it all comes down to sides and angles (vertices) and their proportional relationships. Once kids have mastered the basic equations, they can use the given information to calculate missing quantities. Trigonometry applies to a variety of real-world professions, and games that show this … | 677.169 | 1 |
the lines shown in the picture touch the circle with center O only at points A and B, what is the area of the triangle ABC? Since the figure is symmetrical w.r.t to line CDO, angle ADC = angle BDC = 90 Area ADC = CD * DA Area BDC = CD * DB Since DA = DB; and DA + DB = AB; DA=DB= 1/2*AB; Area ADC = Area BDC = 1/2 * Area ABC
(1) Triangle ADC has area of 25. Area of triangle ADC = 1/2 * Area of triangle ABC = 25 Area of triangle ABC = 25*2 = 50 SUFFICIENT
(2) Triangle BDC has area of 25. Area of triangle BDC = 1/2 * Area of triangle ABC = 25 Area of triangle ABC = 25*2 = 50 SUFFICIENT
This can be said by applying the properties of a triangle Consider triangle ACO and BCO Here AO=OB(BOTH ARE RADIUS OF CIRCLE WITH CENTRE O); Angle CAO and Angle CBO are 90 degree; CA=CB(both are tangent)
So we can say by property SAS that triangle CAO is congruent to Triangle CBO
Re: If the lines shown in the picture touch the circle with center O only
[#permalink]
02 Sep 2021, 22 lines shown in the picture touch the circle with center O only [#permalink] | 677.169 | 1 |
In four dimensions, the vertex figure of a double antiprismoid is an octakis digonal-octagonal gyrowedge, which generally has 2 trapezoidal and 14 triangular faces. The digonal double antiprismoid has an octagonal tegum vertex figure as the trapezoidal faces collapse back into triangles.
In four dimensions, an n-gonal double gyroantiprismoid can have the least possible edge length difference if the ratio of the n-gons is equal to 1:1+√2sin(π/n) (for n less than 7) or 1:tan(π/n)+sec(π/n) (for n equal or greater than 7). | 677.169 | 1 |
Rotations in the plane
What are the
coordinates of the point (x', y') we get
when we rotate (x, y) through an angle a?
The answer is
x' = x*cos(a) - y*sin(a)
y' = x*sin(a) + y*cos(a)
as the following figures illustrate.
In the first figure, the blue triangle
has vertex angle
a. If r is the radius of the circle,
then the radial side of the blue triangle is
r cos(a) and its circumferential side is r sin(a).
Therefore in the following figure, the red, pink
and magenta triangles are all similar. The pink one is obtained from
the red one by a scaling factor of
cos(a).
The magenta one is obtained from
the pink one by combining
(1) a rotation through a right
angle around one of its vertices and
(2) a suitable scale change.
The ratio of the size of the magenta one to the red one is sin(a). | 677.169 | 1 |
Experiment set GEOMETRY 2
From inside out
This set of experiments provides a completely new introduction to spatial geometry. By using high-quality coloured acrylic shapes and modern cylindrical magnets, there are unexpected possibilities to illustrate the mutual position of lines and planes.
But the approach to developing the properties of the most important geometric bodies is also new: The bodies are not viewed from outside inwards, but built from inside outwards. This method makes it easier to develop and promote spatial imagination. Further examples include:
Symmetry planes are discovered using a real mirror surface
Spatial and surface diagonals as well as inclination angles are compared with suitable triangles
Structures are held stable by using special magnets
With ten square plates, the students work out the principle of Cavalieri and then apply it to various pyramid shapes | 677.169 | 1 |
Unit 6 Similar Triangles Answer Key
Unlocking the Secrets of Similar Triangles: A Guide to Unit 6 Answer Key
Are you grappling with the complexities of similar triangles in Unit 6 of your mathematics class? Fear not! This blog post will provide you with a comprehensive answer key that will illuminate the concepts and equip you with the tools to solve any problem related to this topic.
Navigating the Maze of Triangle Relationships
Similar triangles are a fundamental concept in geometry. They are triangles that have the same shape but may differ in size. Understanding the properties and relationships between these triangles can be a challenging task, but having a clear answer key can help you navigate this geometrical labyrinth with ease.
Unit 6 Answer Key: A Gateway to Mastery
Our Unit 6 answer key is more than just a list of solutions. It provides step-by-step guidance, detailed explanations, and real-world examples that will deepen your understanding of similar triangles. Whether you are struggling with finding scale factors or proving similarity, this key has got you covered.
Empowering Students to Tackle Similar Triangle Challenges
By utilizing our Unit 6 answer key, students can:
Identify and understand the properties of similar triangles
Calculate the scale factor between similar triangles
Prove that triangles are similar using various methods
Apply their knowledge of similar triangles to solve real-world problems
Unlock the Full Potential of Unit 6
With our comprehensive answer key, Unit 6 becomes a gateway to understanding similar triangles. It empowers students to grasp the concepts, overcome challenges, and achieve mastery in this critical area of mathematics.
Unit 6 Similar Triangles Answer Key
Similar triangles are triangles that have the same shape but not necessarily the same size. They have equal angles and their sides are in proportion. That means that the ratio of any two corresponding sides of similar triangles is the same.
1. Properties of Similar Triangles
They have equal angles.
Their sides are in proportion.
Their corresponding sides are parallel.
2. Determining Similarity
Two triangles are similar if:
They have three congruent angles.
They have two equal sides and their included angles are congruent.
They have two pairs of proportional sides.
3. Scaling Triangles
If two triangles are similar, the ratio of their areas is the square of the ratio of any two corresponding sides.
If two triangles are similar, the ratio of their volumes is the cube of the ratio of any two corresponding sides.
4. Similar Triangles Theorems
Theorem 1: If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
Theorem 2: If two lines are drawn from a point outside a triangle to two sides of the triangle, the ratio of the lengths of the two lines is equal to the ratio of the lengths of the sides of the triangle.
5. Applications of Similar Triangles
Navigation: Determining the distance between objects by using similar right triangles.
Architecture: Designing structures by using similar triangles to maintain proportions.
Photography: Understanding how lenses work by using similar triangles.
6. Solving Problems Involving Similar Triangles
Use the properties of similar triangles to set up equations.
Solve the equations to find the unknown values.
Check your answers to make sure they make sense.
7. Applications in Coordinate Geometry
Similar triangles can be used to determine the ratio of segments of parallel lines.
Similar triangles can be used to prove theorems about parallel lines and transversals.
8. Applications in Trigonometry
Similar triangles can be used to find the values of trigonometric functions of angles.
Similar triangles can be used to solve problems involving the angles of elevation and depression.
9. Applications in Geometry Proofs
Similar triangles can be used to prove triangles congruent.
Similar triangles can be used to prove theorems about the areas and volumes of similar figures.
10. Proof Techniques
Triangle Proportionality Theorem: If two triangles have the same shape, then the ratios of their corresponding sides are equal.
AA Similarity Theorem: If two triangles have two pairs of equal angles, then they are similar.
SSS Similarity Theorem: If the sides of two triangles are proportional, then the triangles are similar.
Conclusion
Similar triangles are an important concept in geometry. They have many applications in different fields. Understanding the properties and theorems of similar triangles can help you solve a variety of problems.
FAQs
How do I determine if two triangles are similar?
Check for equal angles or proportional sides.
What is the ratio of the areas of similar triangles?
The square of the ratio of any two corresponding sides.
How can I use similar triangles to solve problems?
Set up equations using the properties and theorems of similar triangles.
What are some applications of similar triangles in real-world situations?
Navigation, architecture, and photography.
How are similar triangles used in geometry proofs?
To prove triangles congruent or to prove theorems about areas and volumes. | 677.169 | 1 |
Distance Between Two Points Word Problems Worksheets
Distance Between Two Points Word Problems Worksheets
Distance between two points word problems can be challenging for students to solve. However, with practice and a clear understanding of the concepts involved, these problems can become much easier to tackle. In this article, we will explore distance between two points word problems and provide detailed examples to help you master this topic.
What are Distance Between Two Points Word Problems?
Distance between two points word problems involve finding the distance between two given points in a coordinate plane. These problems often require the use of the distance formula, which is the square root of the sum of the squares of the differences between the x-coordinates and y-coordinates of the two points.
How to Solve Distance Between Two Points Word Problems
To solve distance between two points word problems, follow these steps:
Conclusion
Distance between two points word problems can be solved by applying the distance formula. By identifying the coordinates of the given points, plugging them into the formula, and simplifying the equation, you can find the distance between the points accurately. Practice solving various word problems to enhance your understanding and proficiency in this topic. | 677.169 | 1 |
Types of matrices-part 2.
Scalar matrix, Symmetric matrix.
The scalar matrix, the first item of the types of matrices, is the matrix, where the diagonal is a scalar for the given matrix, the diagonal is =2, while the remaining elements are all zeros.
The second item of the types of matrices, is the symmetric matrix a matrix, from its name, is a matrix that is symmetrical about the diagonal.
For every i,j aji=aij, for an element above the diagonal a22, we have a12=-3, which is equal to the element to the left of a22, which is a21, with no change of signs. Similarly a23=a32=+7
So we will find that aij=aji, a21=a12, and a23=a32, a21=-3, which is the element in the second row, with the first column.
While a12=-3, which is the first row, with the second column. a23=7=a32, which is this element, the third row with the second column. What is the unit matrix? The unit matrix is the third item of the types of matrices, it is a matrix, for all diagonal elements=0.
While the other remaining elements are zeros. What is the skew-symmetric matrix? it is the matrix, for which aij =-aji. So for a21=-a12, a23=-a32.
What is the triangular matrix?
The 4rth item of the types of matrices is the Triangular Matrix or the echelon form. The matrix consists of one of two forms, the first form is the upper triangular matrix or a lower triangular matrix. The main difference can be shown as follows.
For the given matrix, we have a matrix 3×3, the elements of the diagonal are 1,4,6. The elements above the diagonal are 3,2,1. While the elements below the diagonal elements are all zeros. The other form is the lower matrix.
The lower matrix has elements of zero values above the diagonal elements, while other elements below the diagonal have values. The expression for the lower matrix can be written as aij for i> or= j, which means that in this case, aij has a value.
While aij=0 for i<j. For instance, a23 or the second row with the third column, for which i, or the row number is < j which is the column number.
When the row number i is > =the column number j. For instance, elements, a31,a32, and a33 will have values.
The upper matrix has a definition that is opposite to the definition of the lower matrix. The second-row elements are (7,6,8). While the third-row elements are(2,3,6). For the transpose matrix, we let the rows change to columns and change the columns to rows.
This is the first column, which will be the first row and will be (2,7,2), as we can see, the second row will be (5,6,3). The third column will be the third row, for which the values are (4,8,6).
For a useful external link, math is fun for the matrix part. The next post, Matrix operation-part-1. the post includes the addition and subtraction of two matrices, the scalar product | 677.169 | 1 |
Chapter: Trigonometry - Class 10
Trigonometry - Sub Topics
The term 'trigonometry' originates from the Greek words 'tri' (which means three), 'gon' (which means sides) and 'metron' (which means measure). Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right triangles. It explores the properties and functions of angles as well as the connections between angles and the lengths of sides. This chapter includes trigonometric ratios of the angle, trigonometric identities and angles of elevation and depression.
Example: What is the value of sin 15° using the identity sin (A − B) = sin A cos B − cos A sin B?
a) b) c) d)
Answer: c)
Explanation:
Double Angle Formulas
Double angle formulae are as follows:
Example: If sin 2x + cos 2x = 1/4, then what is the value of sin x + cos x using the identities sin 2A = 2 sin A cos A and cos 2A = 2 cos2 A − 1?
a) 5cos x/4 b) 5sec x/4 c) 5cos x/8 d) 5sec x/8
Answer: d) 5sec x/8
Explanation: We know that
sin 2A = 2 sin A cos A
And
cos 2A = 2 cos2 A − 1
We are given that: sin 2x + cos 2x = 1/4
2 sin x cos x + 2 cos2 x − 1 = 1/4
2 sin x cos x + 2 cos2 x = 1/4 + 1
2 cos x (sin x + cos x) = 5/4
cos x (sin x + cos x) = 5/8
sin x + cos x = 5sec x/8
Triple Angle Formulas
Triple angle formulae are as follows:
Note: sin3 A = (sin A)3 This is applicable to all the trigonometric ratios. Example: What is the value of tan 135°?
a) −1 / 2 b) 1 / 2 c) 1 d) −1
Answer: d) −1
Explanation:
Complementary Ratios
Signs of trigonometric functions in the different quadrants are shown as follows:
Complementary ratios in trigonometry involve the relationships between the trigonometric functions of two angles that add up to 90°.
Angles of Elevation and Depression
Angle of Elevation: The angle of elevation is the angle formed between the line of sight (typically from a point below) and the horizontal line or plane. It is the angle at which an observer must look upward from the horizontal to see a point or object that is higher than their level.
Angle of Depression: The angle of depression is the angle formed between a downward line of sight from a point of observation and the horizontal plane or line. It is the angle at which an observer must look downward to see an object below the horizontal level.
The angles of elevation and depression are key concepts in trigonometry that find applications in various real-world scenarios involving the calculation of heights and distances in a wide range of practical applications | 677.169 | 1 |
Home » iSteps for construction:
i) Draw a line segment BC = 6 cm.
ii) Draw an arc with centre B and radius 8 cm.
iii) Draw another arc with centre C and radius 5 cm which intersects the first arc at A.
iv) Join AB and AC.
Therefore,, ΔABC is the required triangle.
v) Draw the angle bisectors of ∠B and ∠A which intersect each other at I. Then I is the incentre of the triangle ABC.
vi) Through I, draw ID ⊥ AB.
vii) Now from D, cut off DP = DQ = 2/2 = 1 cm
viii) With centre I, and radius IP or IQ, draw a circle which will intersect each side of triangle ABC cutting chords of 2 cm each. | 677.169 | 1 |
Last, we need the \(x\) and \(y\) coordinate that we'll be at when \(\theta = \frac{{3\pi }}{4}\). These values are easy enough to find given that we know what \(r\) is at this point and we also know the polar to Cartesian coordinate conversion formulas. So, | 677.169 | 1 |
What is a cyclic quadrilateral?
A cyclic quadrilateral is a four-sided polygon whose vertices all lie on a common circle. In other words, it is a closed figure that can be made by tracing a path around a circle. The term "cyclic" means "having the property of being recurrent or periodic." A good way to remember this definition is that the prefix "cycle" comes from the Greek word for circle, which is kyklos. So, a cyclic quadrilateral is basically a four-sided figure whose corners all lie on some circle or other.
Cyclic quadrilaterals have some interesting and important properties that make them useful in mathematical problems. For instance, because all the vertices of a cyclic quadrilateral lie on the same circle, we can infer that the opposite sides of the quadrilateral are parallel to each other. This fact can be very helpful when solving certain types of geometry problems.
The properties of cyclic quadrilaterals can be summarized as follows:
All four vertices lie on a common circle.
Opposite sides of a cyclic quadrilateral are parallel to each other.
The diagonals of a cyclic quadrilateral intersect at two points, which are equidistant from the center of the circle.
The sum of the angles of a cyclic quadrilateral is 360 degrees.
The altitude (or height) from any vertex to the opposite side intersects that side at its midpoint.
A cyclic quadrilateral has two pairs of congruent sides if and only if it is an inscribed rectangle (a rectangle whose vertices all lie on the circumference of a circle).
A cyclic quadrilateral has two pairs of opposite angles congruent if and only if it is an inscribed square (a square whose vertices all lie on the circumference of a circle).
In conclusion, we have seen that a cyclic quadrilateral is simply a four-sided figure whose vertices all lie on some circle or other. We have also seen that cyclic quadrilaterals have some interesting and important properties, which make them useful in mathematical problems. Thanks for reading! | 677.169 | 1 |
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The code enables you to single out a section of your page for printing by enclosing it in two tags.
Original Author: Peter Graves
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Now, I understand that some people may not know how to use Cosine and Sine to find the coordinates of dots on a circle, so I will explain it to the best of my ability. Here is a quick explanation. Now, you know that coordinates are shown in (X, Y), well, Cosine (Cos) finds the X and Sine (Sin) finds the Y. So really, you could think of Sine and Cosine as (Cosine, Sine). Don't get confused yet, lol, I will explain this further. Now, Cosine can be used to find the coordinates of a certain point by using the degrees of that point. Here is a quick example: Cosine(Point_Degree) * Radius_Length = The X coordinate of that Point. And: Sine(Point_Degree) * Radius_Length = The Y coordinate of that Point. Here is an example of finding the (X, Y) of a point with the degree measurement of 100??, and the circle has a radius of 5. To find the X: Cos(100) * 5, and to find the Y: Sin(100) * 5. Simple enough, right? I hope this little tutorial helps you understand the use of Sine and Cosine in finding the coordinates of a point on a circle. I've also included my CSS code to demonstrate this tutorial. | 677.169 | 1 |
Activities to Teach Students to Classify Shapes on the Coordinate Plane: Justify Your Answer
As geometry is one of the most important branches of mathematics, it is essential for students to gain a comprehensive understanding of all its concepts, especially the classification of shapes on the coordinate plane. Through the use of activities and exercises, teachers can effectively teach their students to classify shapes on the coordinate plane while making the learning experience interactive and entertaining. In this article, we will discuss some activities that can help students develop a better understanding of this crucial mathematical concept.
Activity 1: Mapping Shapes on the Coordinate Plane
One of the most effective methods to teach students to recognize and classify shapes on the coordinate plane is by introducing them to various shapes such as squares, rectangles, triangles, and circles. The teacher can draw a shape on the coordinate plane and ask the students to identify it. Once the students correctly identify the shape, the teacher can then ask the students to draw the shape on the coordinate plane themselves.
Activity 2: Identifying Quadrants
The teacher can provide the students with various coordinate planes and ask them to identify the quadrant in which a given point is located. This activity helps the students develop a better understanding of the four quadrants and the location of various shapes on the coordinate plane. This activity can be made more challenging by asking the students to identify the quadrant in which multiple points are located.
Activity 3: Naming Shapes
Another effective method to teach students to classify shapes is by asking them to name the shape based on its attributes such as the length of its sides, the number of sides, etc. For instance, if the teacher draws a shape with four equal sides and four right angles, the students can name it a square.
Activity 4: Drawing Shapes on a Graph
The teacher can also provide the students with pictures of various shapes and ask them to draw the shape on a graph. This activity helps the students develop their spatial awareness and helps them to visualize the shapes on the coordinate plane. This activity can be made more challenging by asking the students to draw the shape using specific coordinates.
Conclusion:
The classification of shapes on the coordinate plane is a crucial concept in geometry, and students must master it to progress in their mathematical journey. By using interactive and engaging activities, teachers can make the learning process more enjoyable for their students. The aforementioned activities are just some of the many exercises that can help students develop their knowledge of this mathematical concept. By incorporating these activities into their teaching, educators can ensure that their students have a better understanding of geometry and are better equipped to tackle more challenging mathematical concepts | 677.169 | 1 |
Geometry:
Morley's
Trisector Theorem
Frank Morley was a member of Haverford College's Department of Mathematics in the early part of the twentieth century. He is credited with being the first to arrive at the celebrated Morley's trisector theorem:
"The three points of intersection of the adjacent angle trisectors of any triangle form an equilateral triangle." | 677.169 | 1 |
3.1Introduction to Conics section:The profile of number of objects consists of various types of curves. This chapterdeals with various types of curves which are commonly used in engineering practiceas shown below:1. Conic sections 4. Evolutes2. Cycloidal curves 5. Spirals3. Involute 6. Helix.3.1.1
Conic sections:The profile of number of objects consists of various types of curves.1. Conic sections:The section obtained by intersection of a right circular cone by a plane in different positions relative to the axis of the cone are called conics.
Figure 1(i) When the section plane is inclined to the axis and cuts all the generators onone side of the apex, the section is an ellipse [fig. 1].(ii) When the section plane is inclined to the axis and is parallel to one of the generators, the section is a parabola [fig. 1].(iii) A hyperbola is a plane curve having two separate parts or branches, formedwhen two cones that point towards one another are intersected by a planethat is parallel to the axes of the cones.Conic sections are always "smooth". More precisely, they never contain any inflection points.This is important for many applications, such as aerodynamics, civil engineering, mechanical engineering, etc.The conic may be defined as the locus of a point in a plane, wherein the ratio of its distance from a fixed point and a fixed straight line is always constant. The fixed point is called the focus and the fixed line, the directrix.The ratio is called eccentricity and is denoted by 'e'. It is always less the 1 for ellipse, equal to 1 for parabola and greater than 1 for hyperbola i.e.(i) ellipse: e < 1(ii) parabola: e = 1(iii) hyperbola: e > 13.1.2
Ellipse:Use of elliptical curves is made in arches, bridges, dams, monuments, man holes, etc. Mathematically an ellipse can be described by equation .Here 'a' and 'b' are half the length of major and minor axes of ellipse and x and y are co-ordinates.
Figure 23.1.3
1. General methods of constructing ellipse:Problem 6-1. (fig. 6-2): To construct an ellipse when the distance of the focus from the directrix is equal to 50 mm and eccentricity is 2/3.(i) Draw any vertical line AB as directrix.(ii) At any point C on it, draw the axis perpendicular to the AB (directrix).(iii) Mark a focus F on the axis such that CF = 50 mm.(iv) Divide CF into 5 equal divisions (sum of numerator and denominator of the eccentricity.).(v) Mark the vertex V on the third division-point from C.Thus, eccentricity, e = VF/VC = 2/3.(vi) A scale may now be constructed on the axis, which will directly give the distances in the required ratio.(vii) At V, draw a perpendicular VE equal to VF. Draw a line joining C and E.(viii) Mark any point 1 on the axis and through it, draw a perpendicularto meet CE-produced at 1 '.(ix) With centre F and radius equal to 1-1 ', draw arcs to intersect the perpendicularthrough 1 at points P1 and P' 1.These are the points on the ellipse, because the distance of P1 from ABis equal to C1,P1F = 1-1'and .Similarly, mark points 2, 3 etc. on the axis and obtain points P2 and P'2 ,P3 and P'3 etc.(x) Draw the ellipse through these points. It is a closed curve having two fociand two directrices.
Figure 33.1.4
2. Construction of ellipse by other methods:Ellipse is also defined as a curve traced out by a point, moving in the sameplane as and in such a way that the sum of its distances from two fixed pointsis always the same.(i) Each of the two fixed points is called the focus.(ii) The line passing through the two foci and terminated by the curve, is calledthe major axis.(iii) The line bisecting the major axis at right angles and terminated by the curve,is called the minor axis.Conjugate axes: Those axes are called conjugate axes when they are parallel to thetangents drawn at their extremities.In fig. 4, AB is the major axis, CD the minor axis and F1 and F2 are the foci. Thefoci are equidistant from the centreO.
The points A, P, C etc. are on the curve and hence, according to the definition,(AF1 + AF2) = (PF1 + PF2) = (CF1 + CF2) etc.But (AF1 + AF2) = AB. Therefore, (PF1 + PF2) = AB, the major axis.Therefore, the sum of the distances of any point on the curve from the two foci isequal to the major axis.Again, (CF1 + CF2) = ABBut CF1 = CF2, Therefore, CF1 = CF2 = 1/2 AB.Hence, the distance of the ends of the minor axis from the foci is equal to half the equal to the major axis.
Figure 42. To construct an ellipse, given the major and minor axes. The ellipse is drawn by, first determining many points through which it is known to pass and then, drawing a smooth curve through them, either freehand or with a french curve. Larger the number of points, more accurate the curve will be.3.1.5
Method 1: Arcs of circles method,(i) Draw a line AB equal to themajor axis and a line CO equalto the minor axis, bisecting eachother at right angles at O.(ii) With centre C and radius equal to half AB (i.e. AO) draw arcscutting AB at F1 and F2, thefoci of the ellipse.(iii) Mark many points 1, 2,3 etc. on AB.(iv) With centres F1 and F2 andradius equal to A1, draw arcson both sides of AB.(v) With same centres and radius equal to 81, draw arcs intersecting theprevious arcs at four points marked P1.(vi) Similarly, with radii A2 and B2, A3 and B3 etc. obtain more points.(vii) Draw a smooth curve through these points. This curve is the required ellipse.
Figure 53.1.6
Method 2: Concentric circles method,(i) Draw the major axis AB and the minor axis CD intersecting each other at O.(ii) With centre O and diameters AB and CD respectively, draw two circles.(iii) Divide the major-axis-circle into many equal divisions, say 12 andmark points 1, 2 etc. as shown.(iv) Draw lines joining these points with the centre O and cutting the minor-axis-circleat points 1', 2' etc.(v) Through point 1 on the major-axis-circle, draw a line parallel to CD, the minor axis.(vi) Through point 1' on the minor-axis-circle, draw a line parallel to AB, the majoraxis. The point P1, where these two lines intersect is on the required ellipse.(vii) Repeat the construction through all the points. Draw the ellipse through A,P1, P2 .•• etc.
Figure 63.1.7
Method 3: Loop of thread method(i) Draw the two axes AB and CD intersecting at O. Locate the foci F1 and F2.(ii) Insert a pin at each focus-point and tie a piece of thread in the form of a loop around the pins, in such a way that the pencil point when placed in the loop (keeping the thread tight), is just on the end C of the minor axis.(iii) Move the pencil around the foci, maintaining an even tension in the thread throughout and obtain the ellipse.It is evident that PF1 + PF2 = CF1 + CF2 etc.
Figure 8(i) Draw the two axes AB and CD intersecting each other at O.(ii) Construct the oblong EFGH having its sides equal to the two axes.(iii) Divide the semi-major-axis AO into many equal parts, say 4, and AE into the same number of equal parts, numbering them from A as shown.(iv) Draw lines joining 1 ', 2' and 3' with C.(v) From D, draw lines through 1, 2 and 3 intersecting C'1, C'2 and C'3 at points P1, P2 and P3 respectively.(vi) Draw the curve through A, P1..... C. It will be one quarter of the ellipse.(vii) Complete the curve by the same construction in each of the three remaining quadrants.As the curve is symmetrical about the two axes, points in the remaining quadrantsmay be located by drawing perpendiculars and horizontals from P1, P2 etc. andmaking each of them of equal length on both the sides of the two axes.3.1.9
Figure 9(i) Draw the two axes AB and CD intersecting each other at O. Along the edge of a strip of paper which may be used as a trammel, mark PQ equal to half the minor axis and PR equal to half the major axis.(ii) Place the trammel so that R ison the minor axis CD and Qon the major axis AB. Then Pwill be on the required ellipse.By moving the trammel to newpositions, always keeping R onCD and Q on AB, obtain otherpoints. Draw the ellipse throughthese points.
3.2 Parabola:Use of parabolic curves is made in arches, bridges, sound reflectors, light reflectors, etc. Mathematically it can be described by an equation or .Methods of constructing Parabola:To draw a parabola with 70 mm as base and 30 mm as the length of the axis.3.2.1
Figure 101. Draw the base AB and locate its mid-point C.2. Through C, draw CD perpendicular to AB forming the axis3. Produce CD to E such that DE = CD4. Join E-A and E-B. These are the tangents to the parabola at A and B.5. Divide AE and BE into the same number of equal parts and number the points as shown.6. Join 1-1' ,2- 2' ,3- 3' , etc., forming the tangents to the required parabola.7. A smooth curve passing through A, D and B and tangential to the above lines is the required parabola.3.2.2
Figure 111. Draw the base AB and axis CD such that CD is perpendicular bisector to AB.2. Construct a rectangle ABEF, passing through C.3. Divide AC and AF into the same number of equal parts and number the points 'as shown.4. Join 1,2 and 3 to D.5. Through 1',2' and 3', draw lines parallel to the axis, intersecting the lines 1D, 2D and 3Dat P1 P2 and P3 respectively.6. Obtain the points P1', P2' and P3', which are symmetrically placed to P1, P2 and P3 withrespect to the axis CD.7. Join the points by a smooth curve forming the required parabola.General method of construction of a parabola:To construct a parabola, when the distance of the Focus from the directrix is 50 mm.
(i) Draw the directrix AB and the axis CD.(ii) Mark focus Fon CD, 50 mm from C.(iii) Bisect CF in V the vertex (because eccentricity = 1).(iv) Mark many points 1, 2, 3 etc. on the axisand through them, draw perpendiculars to it.(v) With centre F and radius equal to C1, draw arcs cutting the perpendicular through 1 at P1 and P'1.(vi) Similarly, locate points P2 and P'2, P3 and P'3 etc. on both the sides of the axes.(vii) Draw a smooth curve through these points. This curve is the required parabola. It is an open curve.1. Rectangle method:Problem: To construct a parabola given the base and the axis.(i) Draw the base AB.(ii) At its mid-point E, draw the axis ff at right angles to AB.(iii) Construct a rectangle ABCD, making side BC equal to EF.(iv) Divide AE and AD into the same number of equal parts and name them as shown (starting from A).(v) Draw lines joining F with points 1, 2 and 3. Through 1 ', 2' and 3', draw perpendiculars to AB intersecting F1, F2 and F3 at points P1, P2 and P3 respectively.(vi) Draw a curve through A, P1, P2 etc. It will be a half parabola.Repeat the same construction in the other half of the rectangle to complete the parabola. Or, locate the points by drawing lines through the points P1, P2 etc. Parallel to the base and making each of them of equal length con both the sides of ff, e.g. P10 = OP'1. AB and EF are called the base and the axis respectively of the parabola.
Figure 12
3.3 HyperbolaUse of hyperbolical curves is made in cooling towers, water channels etc.3.3.1
Rectangular hyperbolaIt is a curve traced out by a point moving in such away that the product of its distances from two fixed lines at right angles to each other is a constant. The fixed lines are called asymptotes.This curve graphically represents the Boyle's Law, viz. P x V = a, P = pressure = volume and a is constant. It is also useful in design of water channels.General method of construction of a hyperbola:Mathematically, we can describe a hyperbola by1. Construct a hyperbola, from the directrix is 65 mm and eccentricity is3/2.
Figure 13(i) Draw the directrix AB and the axis CD. When the distance of the focus(ii) Mark the focus F on CD and 65 mm from C.(iii) Divide CF into 5 equal divisions and mark V the vertex, on the second division from C.Thus, eccentricity = VF/VC = 3/2.To construct the scale for the ratio 3/2, draw aline VE perpendicular to CD such that VE =VF. Join C with E.Thus, in triangle CVE, VE/VC = VF/VC = 3/2(iv) Mark any point 1 on the axis and through it, drawn perpendicular to meet CE-produced at 1'.(v) With centre F and radius equal to 1-1 ', draw arcs intersecting the perpendicular through 1at P1 and P'1·(vi) Similarly, mark a number of points 2, 3 etc. And obtain points P2 and P'2, P3 and P'3 etc.(vii) Draw the hyperbola through these points.3.3.2
Tangents and Normal to Conic:The common rule for drawing tangents and normal:When a tangent at any point on the curve is produced to meet the directrix, the line joining the focus with this meeting point will be at right angles to the line joining the focus with the point of contact.The normal to the curve at any point is perpendicular to the tangent at that point.
1. To draw a tangent at any point Pon the conics(i) Join P with F.(ii) From F, draw a line perpendicular to PF to meet AB at T.(iii) Draw a line through T and P. This line is the tangent to the curve.(iv) Through P, draw a line NM perpendicular to TP. NM is the normal to the curve.2. Other methods of drawing tangents to conicsMethod I:1. To draw a tangent to an ellipse at any point P on it.(i) With O, the centre of the ellipse as centre, and one half the major axis as radius, draw a circle.(ii) From P, draw a line parallel to the minor axis, cutting the circle at a point Q.
Figure 15(iii) Draw a tangent to the circle at the point Q cutting the extended major axis at a point R.From R, draw a line RS passing through P. RS is the required tangent to the ellipse.Method II: When an axis of parabola is given,To draw a tangent to a parabola at any point P on it.(i) From P, draw a perpendicular PA to the axis, intersecting it at a point A.Mark a point B on the axis such that BV = VA. Draw a line from B passing through P.Then this line is the required tangent.
Figure 16(ii) Through P, draw a line PQ parallel to the axis (fig. 17).Draw two lines AB and CD parallel to, equidistant from and on opposite sides of PQ, and cutting the parabola at points A and C. Draw a line joining A with C.Through P, draw a line RS parallel to AC. RS is the required tangent.
Figure 17Method 3: When the focus and the directrix are given.(i) From P draw a line PE perpendicular to the directrix AB, meeting it at a point f.(ii) Draw a line joining P with the focus F.(iii) Draw the bisector RS of angle EPF.Then RS is the required tangent.
Figure 18
3.4 Cycloidal Curves:Cycloid is a curve generated by a point on the circumference of a circle which rolls along a straight line. It can be described by an equation,y = a (1 - cosθ) or x = a (θ - sinθ)To construct a cycloid, given the diameter of the generating circle.(i) With centre C and given radius R, draw a circle. Let P be the generating point.(ii) Draw a line PA tangential to and equal to the circumference of the circle.(iii) Divide the circle and the line PA into the same number of equal parts, say 12,and mark the division-points as shown.(iv) Through C, draw a line CB parallel and equal to PA.(v) Draw perpendiculars at points 1, 2 etc. cutting CB at points C1, C2 etc.
Figure 19Assume that the circle starts rolling to the right. When point 1' coincides with 1, centre C will move to C1. In this position of the circle, the generating point P will have moved to position P1 on the circle, at a distance equal to P'1 from point 1. It is evident that P1 lies on the horizontal line through 1' and at a distance R from C1. Similarly, P2 will lie on the horizontal line through 2' and at the distance R from C2.Construction:(vi) Through the points 1 ', 2' etc. draw lines parallel to PA.(vii) With centres C1, C2 etc. and radius equal to R, radius of generating circle, draw arcs cutting the lines through 1 ', 2' etc. at points P1, P2 etc. respectively. Draw a smooth curve through points P, P1, P2 …. A. This curve is the required cycloid.Normal and tangent to a cycloid curve: The rule for drawing a normal to all cycloidal curves:The normal at any point on a cycloidal curve will pass through the corresponding point of contact between the generating circle and the directing line or circle. The tangent at any point is perpendicular to the normal at that point.
3.5 Involute of a circle is used as teeth profile of gear wheel.Mathematically it can be described by x = rcosθ + rθsinθ, y = rsinθ - rθcosθ,where "r" is the radius of a circle.Problem: To draw an involute of a given circle.With centre C, draw the given circle. Let P be the starting point, i.e. the end of the thread.Suppose the thread to be partly unwound, say upto a point 1. P will move to aposition P1 such that 1P1 is tangent to the circle and is equal to the arc 1 P. P1 will bea point on the involute.Construction:(i) Draw a line PQ, tangent to the circle and equal to the circumference of the circle.(ii) Divide PQ and the circle into 12 equal parts.(iii) Draw tangents at points 1, 2, 3 etc. and mark on them points P1, P2, P3 etc.such that 1P1 = P1', 2P2 = P2', 3P3 = P3' etc.Draw the involute through the points P, P1, P2 .... etc.Normal and tangent to an involute:The normal to an involute of a circle is tangent to that circle.Problem:To draw a normal and a tangent to the involute of a circleat a point N on it.
Figure 20(i) Draw a line joining C with N.(ii) With CN as diameter describe a semi-circle cutting the circle at M.(iii) Draw a line through N and M. This line is the normal. Draw a line ST, perpendicular to NM and passing through N. ST is the tangent to the involute.Problem: Trace the paths of the ends of a straight-line AP, 100 mm long, when it rolls, without slipping, on a semi-circle having its diameter AB, 75 mm long. (Assume the line AP to be tangent to the semi-circle in the starting position.)
Figure 21
(i) Draw the semi-circle and divide it into six equal parts.(ii) Draw the line AP and mark points 1, 2 etc. such that A1 = arc A1 ', A2 = arc A2' etc. The last division SP will be of a shorter length. On the semi-circle, mark a point P' such that S'P' = SP.(iii) At points 1 ', 2' etc. draw tangents and on them,mark points P1, P2 etc. suchthat 1' P1 = 1P, 2' P2 =2P .... and 5' P5 = 5' P6 =SP. Similarly, mark pointsA1, A2 etc. such that A11'= A1, A22' = A2 .... and A'p'= AP. Draw the requiredcurves through points P,P1 .... and P', and throughpoints A, A1 .... and A'.If AP is an inelastic string with the end A attached to the semicircle, the end P will trace out the same curve PP' when the string is wound round the semi-circle.
Figure 22
3.6It is used as teeth profile of gear wheel.Mathematically it can be described as by , , where "r" is the radius of the circle.An involute is a curve traced by a point on a perfectly flexible string, while unwinding from around a circle or polygon the string being kept taut (tight). It is also a curve traced by a point on a straight-line while the line is rolling around a circle or polygon without slipping.To draw an involute of a given square.1. Draw the given square ABCD of side a.2. Taking A as the starting point, with centre B and radius BA=a, draw an arc to intersect theline CB produced at Pl.3. With Centre C and radius CP1 =2 a, draw on arc to intersect the line DC produced at P 2.4. Similarly, locate the points P3 and P4.
Figure 23The curve through A, P1, P2, P3 and P4 is the required involute.A P4 is equal to the perimeter of the square.AP is a rope 1.50-metre-long, tied to peg at A as shown in fig. 24. Keeping it always tight, the rope is wound round the pole. Draw the curve traced out by the end P. Use scale 1 :20.Draw given figure to the scale.(ii) From A, draw a line passing through 1. A as centre and AP as radius, draw the arc intersecting extended line A1' at P0. Extend the side 1-2, 1 as centre and 1 'Po as radius, draw the arc to intersect extended line 1-2 at P1.(iii) Divide the circumference of the semicircle into six equal parts and label it as 2, 3, 4, 5, 6, 7 and 8.(iv) Draw a tangent to semicircle from 2 such that 2'-P1 = 2'-P2 . Mark 8' on this tangent such that 2'-8' = nR. Divide 2'-8' into six equal parts.(v) Similarly draw tangents at 3, 4, 5, 6, 7 and 8 in anti-clockwise direction such that 3-P3 = 3'-9', 4-P4 = 4'-9', 5-P5 = 5'-9', 6-P6 = 6'-9', 7-P7 = 7'-9', 8-P8 = 8'-9' and 8-P9 = 8'-9' respectively.(vi) Join P1, P2, ....• , P9 with smooth curve.
Figure 24
3.7 Spiral:If a line rotates in a plane about one of its ends and if at the same time, a point moves along the line continuously in one direction, the curve traced out by the moving point is called a spiral. The point about which the line rotates is called a pole.The line joining any point on the curve with the pole is called the radius vector and the angle between this line and the line in its initial position is called the vectorial angle.
Archemedian spiral:It is a curve traced out by a point moving in such a way that its movement towards and away from the pole is uniform with the increase of the vectorial angle from the starting line.The use of this curve is made in teeth profiles of helical gears, profile of cams, etc.To construct an Archemedian spiral of convolutions given the greatest and the shortest radii.Let O be the pole, OP the greatest radius and OQ the shortest radius.(i) With centre O and radius equal to OP, draw a circle. OP revolves around0 for revolutions.During this period, P moves towards 0, the distance equal to (OP - OQ) i.e. QP.(ii) Divide the angular movements of OP, viz revolutions i.e. 540°and theline QP into the same number of equal parts, say 18 (one revolutiondivided into 12 equal parts).When the line OP moves through one division, i.e. 30°, the point P willmove towards O by adistance equal to one division of QP to a point P1.(iii) To obtain points systematically draw arcs with centre O and radii O1', O2', O3' etc. intersecting lines O1', O2', O3' etc. at points P1, P2, P3 etc.respectively.In one revolution, P will reach the 12th division along QP and in the nexthalf revolution it will be at the point PQ on the line 18'-0.(iv) Draw a curve through points P, P1, P2, PQ. This curve is the requiredArchemedian spiral.The constant of the curve is equal to the difference between the lengths of anytwo radii divided by the circular measure of the angle between them.OP and OP3 (fig. 6-63) are two radii making go0 angle between them. Incircular measure in radian,. Therefore, the constant of the curve | 677.169 | 1 |
No, an equilateral triangle is not a quadrilateral. A quadrilateral is a polygon with four sides and four vertices, while an equilateral triangle is a triangle with three equal sides and three equal | 677.169 | 1 |
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The area of the square is 4. The points that satisfy x+y<3 form a triangle with vertices (0, 0), (1, 1), and (0, 2). The area of this triangle is 1/211 = 1/2. Therefore, the probability that x+y<3 is 1/2 / 4 = 1/8.
Alternatively, we can solve this problem by using the fact that the probability of a point falling in a certain region is equal to the ratio of the area of that region to the area of the entire square. The region that satisfies x+y<3 is a triangle with vertices (0, 0), (1, 1), and (0, 2). The area of this triangle is 1/211 = 1/2. The area of the entire square is 4. Therefore, the probability that x+y<3 is 1/2 / 4 = 1/8. | 677.169 | 1 |
are the coordinates of point B ?
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Top Contributor which means there must be only one location for point B. As such, statement 2 is SUFFICIENTRe: What are the coordinates of point B ?
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06 Feb 2015, 13:24
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I hope it hasn't been posted before. I couldn't find it. It took me a while to solve it. I actually started solving before I even read the whole question, just by looking at the image. I find these questions tricky, just because I very often find myself solving them, when this is not necessary...
Re: What are the coordinates of point B ?
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17 Jun 2017, 11:48
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pacifist85 wrote:
What are the coordinates of point B ?
(1) The area of triangle ABC = 30 (2) Length(CB) = 13
Statement 1
Because this is a right triangle we only need the base and height in order to calculate the area; we know the height of the triangle so we must know the base
base x height/ 2 = 30 base x height =60 base x (5) =60 base= 60/5
Furthermore, because know the x coordinate (-3,4) we simply add 12 to the x value in order to find b- l-3l + b =12
Statement 2
The diagram shows us that this is a right triangle- well, there is a Pythagorean Triplet 5 12 13- and because we know the height of the triangle we can easily deduce this is a 5 12 13 Pythagorean Right TriangleGreat question! This is one of those cases in which we can assume that point P lies to the right. For more on assumptions we can make on GMAT geometry questions, watch the following video:
Re: What are the coordinates of point B ?
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07 Feb 2015, 05:36
Expert Reply
pacifist85 wrote:
What are the coordinates of point B ?
(1) The area of triangle ABC = 30 (2) Length(CB) = 13
hi pacifist, you dont require to solve anything... here you have a right angle triangle on x-y axis, where coord of two axis are given... third vertex coord are reqd.. 1) area would give us length of perpendicular.. and hence coord can be found..suff 2) length of hyp given.. vertex can be again found.. suff ans D
Re: What are the coordinates of point B ?
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15 Feb 2015, 05:30
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viper1991 wrote:
Hi
can some one help me how each STATEMENT for the above problem is sufficient ?
with the area and the distance formula . you can calculate the the length of AB ..but how do get the co ordinates or the point B?
and also how is the statement B alone also sufficient ?
please help?
Hi viper..
firstly look at the figure and see what it tells us... 1) line AC is parallel to Y axis as the coord of both A and C have same value for X axis.. 2) this means AB will be parallel to X axis and A and B will have same Y coord.. or B will be (x,4)
lets look at the statement.. 1) it gives us area of triangle 30= base * perpendicular/2... AB=60/5=12 this will give us x as -3 + 12= 9.... sufficient 2)This gives us length of hypotenuse BC=13.. AC is 5 so AB is 12 again as in (1), x coord can be found.. sufficient ans D
Re: What are the coordinates of point B ?
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09 Jan 2020, 07:02
GMATPrepNow wrote: which means there must be only one location for point B. As such, statement 2 is SUFFICIENTThis means, 3 + x = 12
From here, we get x. St. 2 is sufficient as well.
So, correct answer: D
Please let me know if something remains unclear.
Best Regards
Japinder
Great explanation EgmatQuantExpert, in St 1, not quite sure why is 3+x and not -3+x since x coordinate of point A is -3? Thanks Japinder
Re: What are the coordinates of point B ?
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12 Feb 2024, 05 | 677.169 | 1 |
4 The Trigonometric Functions
LEARNING OUTCOMES
Express an angle in degrees or radians and convert between the two measurements
Determine a standard position angle
Define, evaluate, and use the six trigonometric functions
Use the Pythagorean theorem and trigonometric functions to solve a right triangle
Employ the inverse trigonometric functions to solve for a missing angle
Solve application problems involving right triangles
Triangles are often used in solving applied problems in technology. A short list of such problems includes those in air navigation, surveying, rocket motion, carpentry, structural design, electric circuits, and astronomy. ...
Get Basic Technical Mathematics, 11th Edition now with the O'Reilly learning platform.
O'Reilly members experience books, live events, courses curated by job role, and more from O'Reilly and nearly 200 top publishers. | 677.169 | 1 |
What is reflex angle in a clock?
The angle between the hands at 1 o'clock could also be given as the reflex angle 330°, but we will always give the smaller (acute or obtuse) angle.
How do you find the reflex angle of a clock?
Here H = 10 , M = 25 and the minute hand is behind the hour hand. But the question is to find out the reflex angle….Clock #18.
18. What is the reflex angle between the hands of a clock at 10.25?
A. 180°
B. 19312°
C. 195°
D. 19712°
How is the clock related to math?
Clocks are used to measure time. A clock in general has 12 numbers written on it, from 1 to 12, an hour hand, and a minute hand. Clock angle problems relate two different measurements: angles and time. The hour hand of a normal 12-hour analogue clock turns 360° in 12 hours (720 minutes) or 0.5° per minute.
What angle is formed by a clock?
360 degrees
First note that a clock is a circle made of 360 degrees, and that each number represents an angle and the separation between them is 360/12 = 30. And at 2:00, the minute hand is on the 12 and the hour hand is on the 2. The correct answer is 2 * 30 = 60 degrees.
What is the reflex of 100 degree?
As 100 degree is just between 90 and 180 degree so, it is an obtuse angle.
What is the angle at 3 30?
Answer: The angle between the hour hand and minute hand at 3:30 P.M. is 90o.
At 9 o'clock, the hour hand is at 9 and the minutes hand is at 12, i.e., the two hands are 15 min. spaces apart. The hands will be in the same straight line but not together i.e.,in 180 degrees at 180/11 min. past 9.
Can you get a 0 on the reflex test?
The object is to stop the counter as close to 500 as possible. The lower your score, the better. Can you get a score of 0 | 677.169 | 1 |
Free Geometry Games ONLINE + Printable
Geometry is an important branch of mathematics that deals with the properties of shapes, sizes, and positions of objects in space.
Our online geometry games are designed to help kids develop a better understanding of geometric principles while having fun.
All these math geometry games are accompanied with printable geometry worksheets.
Math games on geometry
Geometry math games are a fun and engaging way for students to learn about shapes, angles, and other geometric concepts.
Teachers who incorporate online geometry games into math lessons help kids develop a love of learning and a strength
understanding of geometry basics.
Free geometry games PRINTABLE
You can use these printable geometry games in the classroom or at home as an additional geometry practice.
What is geometry?
Geometry is a branch of mathematics that deals with the properties and relationships of points, lines, angles, surfaces, and solids.
It explores the spatial configurations and shapes of objects in both two and three dimensions.
The word "geometry" is derived from the Greek words "geo" (earth) and "metron" (measure),
reflecting its historical roots in surveying and measuring land.
Geometry in math
Geometry is not only a theoretical field but also has practical applications in various fields, including physics,
engineering, architecture, computer graphics, and more. It provides a foundation for understanding spatial relationships
and has been a critical part of mathematical study for centuries.
Basic geometry games ONLINE
Our geometry online games cover a broad range of topics that explore the properties and relationships of shapes, sizes, and spaces.
Here is an overview of key topics typically included in geometry online games:
Points, Lines and Planes: In geometry, a point is a location in space with no size, and a line is a straight path that extends
infinitely in both directions. A plane is a flat, two-dimensional surface.
Angles: An angle is formed by two rays with a common endpoint. Angles are measured in degrees or radians and play
a crucial role in various geometric constructions and calculations. Types of angles (acute, obtuse, right, straight).
Polygons: Polygons are closed figures formed by connecting line segments. Common examples include triangles, quadrilaterals,
pentagons, and hexagons. Classification of polygons by the number of sides (e.g., triangles, quadrilaterals).
Circles: Circles are sets of points equidistant from a central point. The properties of circles involve concepts such as radius,
diameter, circumference, and area.
Area and Perimeter: Formulas for finding the area and perimeter of basic shapes (rectangles, triangles, circles).
Volume and Surface Area: Formulas for finding the volume and surface area of three-dimensional shapes
(cubes, prisms, cylinders, cones, spheres).
Constructions: Geometric constructions involve creating various shapes and figures using only a straightedge and a compass,
adhering to certain rules. This is a fundamental aspect of classical geometry.
Transformations: Transformations, such as translations, rotations, reflections, and dilations, are operations
that change the position, orientation, or size of geometric figures while preserving their essential properties.
Coordinate Geometry: This branch of geometry combines principles of geometry with algebra, using coordinates
to represent points and equations to describe geometric shapes.
Solid Geometry: Solid geometry deals with three-dimensional objects, including polyhedra (such as cubes and pyramids)
and objects with curved surfaces (such as spheres and cylinders).
Geometric Proofs: Geometry often involves proving statements or theorems using deductive reasoning.
A proof is a logical argument that establishes the truth of a mathematical statement. | 677.169 | 1 |
...perpendiculars on it from the opposite angles 42 VII. To find the area of any polygon 43 EXERCISES (4) 44 VIII. Two triangles which have an angle of the one equal...the products of the sides including the equal angles 47 IX. The areas of similar triangles are to each other as the squares of their like sides 48 X. Thesides. Cor. 2. — If the two triangles are equal, ABxAC=POxOQ, therefore ^? - 29 \)L .AO that is, equal triangles which have an angle of the one equal to an angle of the other, have the sides ahout the equal angles reciprocally proportional. Cor. 3. — Hence also equiangular...
...of homologous angles are not reciprocally proportional. THEOREM 18. (Eucl. VI. 16.) Two equivalent triangles which have an angle of the one equal to an angle of the other, have the sides of these angles reciprocally proportional. Let there be two equivalent triangles, ABC...
...of the intercepted area, according as they intersect internally or externally. 15. If two trapeziums have an angle of the one equal to an angle of the other, and if, also, the sides of the two figures, about each of their angles, be proportionals, the remaining...
...GEOMETRY.—BOOK IV. THEOREMS. 219. Two triangles which have an angle of the one equal to the supplo mcnt of an angle of the other are to each other as the products of the siiitM including the supplementary angles. (IV. 22.) 220. Prove, geometrically, that the square described...
...polygons. Prove that two triangles are similar when they are mutually equiangular. 2. Two triangles having an angle of the one equal to an angle of the other...products of the sides including the equal angles. 3. To inscribe A circle in a given triangle. 4. The side of a regular inscribed hexagon is equal to...
...same demonstration it may be shown that THEOREM LXXV. If two parallelograms are equal in area, and have an angle of the one equal to an angle of the other, then the sides which contain the angle of the first are the extremes of a proportion of which the sides...
...two adjoining sides of the one respectively equal to two adjoining sides of the other, and likewise an angle of the one equal to an angle of the other; the parallelograms are identically equal. [By Superposition.] COR. Two rectangles are equal, if two... | 677.169 | 1 |
Components of Transit Theodolite: A Comprehensive Guide
Introduction
In this insightful guide, we will delve into the intricate world of transit theodolites and explore their various components. A transit theodolite is an essential instrument widely used in surveying and engineering applications. Its precise measurement capabilities make it indispensable for accurate mapping, construction, and various other tasks.
Understanding Transit Theodolites
A transit theodolite is a sophisticated optical instrument used for measuring horizontal and vertical angles with great accuracy. This tool plays a pivotal role in land surveying, navigation, and geodetic calculations. Let's uncover the key components that constitute a transit theodolite.
Theodolite Base
The base of the transit theodolite serves as its foundation, providing stability during measurements. Typically mounted on a tripod, the base ensures that the instrument remains steady even in uneven terrains. This stability is crucial for obtaining precise angle measurements.
Telescope and Eyepiece
Atop the theodolite sits the telescope, equipped with an eyepiece. The telescope aids in sighting distant objects and aligning the instrument. The eyepiece, on the other hand, allows the surveyor to view the crosshairs and target simultaneously.
Horizontal and Vertical Circles
The horizontal and vertical circles are vital components that facilitate angle measurements. The horizontal circle enables the surveyor to determine the azimuth angle, while the vertical circle helps measure the zenith angle. These measurements are essential for accurate positioning and orientation.
Tribrach and Levelling Screws
The tribrach serves as a connection point between the theodolite and the tripod. It provides a means to level the instrument using levelling screws. Proper leveling ensures that the measurements taken are not influenced by the instrument's tilt, enhancing the precision of the results.
Optical Plummet
The optical plummet aids in establishing the vertical position of the theodolite over a point. By looking through the optical plummet, surveyors can align the instrument directly above the desired location, ensuring accurate measurements.
Accessories and Carrying Case
Many transit theodolites come with accessories such as sunshades to improve visibility in bright conditions and rain covers to protect against adverse weather. A sturdy carrying case keeps the theodolite and its components safe during transportation and storage.
Conclusion
In conclusion, transit theodolites are complex yet indispensable tools for professionals in the fields of surveying, mapping, and engineering. Each component plays a crucial role in ensuring accurate and reliable angle measurements. By understanding the intricacies of these components, surveyors can harness the full potential of transit theodolites in various applications. So, whether you're involved in construction, land surveying, or any other relevant field, a solid grasp of transit theodolite components is essential for success. | 677.169 | 1 |
How to Prove a Rectangle
Geometry is not just about shapes; it's about the stories they tell and the mysteries they hold. One of the most familiar yet intriguing shapes is the rectangle. Students often encounter the challenge of proving that a given quadrilateral is a rectangle. This article will guide you through the fascinating journey of proving a rectangle, unveiling the secrets behind its right angles and equal sides.
What is a Quadrilateral?
In geometry, a quadrilateral stands out as a fundamental shape, defined as a polygon with exactly four sides and four angles. This definition may seem simple, but within it lies a diverse family of shapes, each with its own distinctive features and properties. Quadrilaterals are a cornerstone of geometric studies, offering a rich field for exploration and analysis.
The family of quadrilaterals is vast and varied, including well-known members like squares, rectangles, trapezoids, rhombuses, and parallelograms. Each of these shapes adheres to the basic definition of a quadrilateral but brings its own set of rules regarding side lengths, angle measurements, and symmetry.
Image: sciencenotes.org
One of the defining characteristics of quadrilaterals is their ability to be classified based on their properties. For instance, a square is a quadrilateral with four equal sides and four right angles, while a rectangle has four right angles but only opposite sides are equal. A parallelogram, on the other hand, has opposite sides that are parallel and equal in length but do not necessarily have right angles.
The interior angles of a quadrilateral are another important feature. Regardless of the specific type of quadrilateral, the sum of its interior angles always adds up to 360°. This property is crucial for solving various geometric problems and is a fundamental theorem in the study of polygons. Quadrilaterals can also be distinguished by their symmetry. Some, like squares and rectangles, have lines of symmetry and rotational symmetry, making them highly regular shapes. Others, like trapezoids, may lack symmetry, giving them a more irregular appearance.
Characteristic
Description
Four Sides
A quadrilateral has exactly four sides.
Four Angles
It also has four angles, with their sum always equal to 360°.
Interior Angles
The angles inside a quadrilateral add up to 360°.
Types
Squares, rectangles, trapezoids, and parallelograms The type is based on their side lengths and angle measurements.
Proving that a Quadrilateral Is a Rectangle
To prove that a quadrilateral is a rectangle, we need to establish that it satisfies the definition of a rectangle: a quadrilateral with four right angles. There are several methods to demonstrate this:
Show All Angles Are Right Angles:
Approach: Measure or calculate the angles of the quadrilateral. If each angle measures 90°, the quadrilateral is a rectangle.
Application: This method is often used when you have access to the angle measurements, either through geometric constructions or given information in a problem.
Prove It's a Parallelogram with One Right Angle:
Approach: First, show that the quadrilateral is a parallelogram by proving that opposite sides are parallel (e.g., using slope in a coordinate plane or showing alternate interior angles are equal). Then, demonstrate that one of the angles is a right angle.
Application: This method is useful when you can easily prove the quadrilateral is a parallelogram (e.g., if it's given or if you have coordinates) and you have information about one of the angles.
Use Diagonals:
Approach: Prove that the diagonals of the quadrilateral are congruent (e.g., by using the distance formula in a coordinate plane or congruent triangles). If the quadrilateral is already proven or given to be a parallelogram, and its diagonals are congruent, then it must be a rectangle.
Application: This method is particularly effective when working with coordinate geometry or when you can establish congruence between triangles formed by the diagonals.
Each of these methods relies on different properties of rectangles and parallelograms, allowing for flexibility in how you approach a proof. Depending on the information provided and the context of the problem, one method may be more suitable than another. In all cases, it's essential to provide clear and logical reasoning to support your conclusion that the quadrilateral is indeed a rectangle. We've searched the Internet a bit and found the following video that can clarify anything that is left unclear after our explanation.
Proving a Rectangle: Math Problem Examples
Let's explore three examples of how to prove a quadrilateral is a rectangle:
Example 1: Using Angle Measures
Problem: Prove that a quadrilateral with vertices A(2,3), B(6,3), C(6,7), and D(2,7) is a rectangle.
Image for the Task Example 1
Solution:
Calculate Slopes: Find the slopes of the sides to determine if they are parallel.
AB and CD are horizontal lines (y-coordinates are the same), so their slopes are 0.
BC and AD are vertical lines (x-coordinates are the same), so their slopes are undefined.
Check for Right Angles: Since AB is horizontal and AD is vertical, the angle between them is a right angle. Similarly, all other angles are right angles.
Conclusion: Since all angles are right angles and opposite sides are parallel, the quadrilateral ABCD is a rectangle.
Example 2: Using Parallelogram Properties and One Right Angle
Problem: Prove that a quadrilateral with vertices A(1,1), B(4,4), C(7,1), and D(4,-2) is a rectangle.
Image for the Task Example 2
Solution:
Show It's a Parallelogram: Prove that opposite sides are parallel by showing their slopes are equal.
Slope of AB = (4-1)/(4-1) = 1
Slope of CD = (1-(-2))/(7-4) = 1
Slope of BC = (4-1)/(4-7) = -1
Slope of AD = (1-(-2))/(1-4) = -1
Identify a Right Angle: The slopes of AB and AD are negative reciprocals, indicating that angle BAD is a right angle.
Conclusion: Since the quadrilateral is a parallelogram with one right angle, it is a rectangle.
Example 3: Using Congruent Diagonals
Problem: Prove that a quadrilateral with vertices A(0,0), B(5,0), C(5,3), and D(0,3) is a rectangle.
Image for the Task Example 3
Solution:
Calculate Diagonal Lengths: Use the distance formula to find the lengths of the diagonals AC and BD.
Length of AC = √[(5-0)² + (3-0)²] = √(25 + 9) = √34
Length of BD = √[(5-0)² + (3-0)²] = √(25 + 9) = √34
Check for Parallelogram: Show that opposite sides are parallel (same slope).
Slopes of AB and CD are 0 (horizontal lines).
Slopes of BC and AD are undefined (vertical lines).
Conclusion: Since the diagonals are congruent and the quadrilateral is a parallelogram, it is a rectangle.
Conclusion
Proving a quadrilateral is a rectangle may seem daunting at first, but with a clear understanding of the properties of rectangles and a step-by-step approach, it becomes an achievable task. By focusing on right angles, parallelogram properties, and diagonal congruence, you can confidently prove the rectangular nature of a quadrilateral. So, the next time you're faced with this challenge, remember these strategies and tackle the problem with ease.
FAQ
How can I prove that a quadrilateral is a rectangle?
To prove that a quadrilateral is a rectangle, you need to demonstrate that it has four right angles. This can be achieved through various methods, such as showing that all angles are right angles, proving that the quadrilateral is a parallelogram with at least one right angle, or verifying that the diagonals are congruent and the quadrilateral is a parallelogram.
What are the methods to prove that a quadrilateral is a rectangle?
There are several methods to prove that a quadrilateral is a rectangle:
Show All Angles Are Right Angles – Measure or calculate the angles to ensure each is 90°.
Prove It's a Parallelogram with One Right Angle – Demonstrate that the quadrilateral is a parallelogram with opposite sides parallel and equal, and then show that at least one angle is a right angle.
Use Diagonals – Prove that the diagonals are congruent and that the quadrilateral is a parallelogram; this combination indicates that the quadrilateral is a rectangle.
Do all angles in a quadrilateral need to be right angles to prove it's a rectangle?
Yes, all angles in a quadrilateral need to be right angles (90°) to prove it's a rectangle. A rectangle is defined as a quadrilateral with four right angles. If any angle is not a right angle, the quadrilateral cannot be classified as a rectangle | 677.169 | 1 |
Question 1. Say True or False : (a) Each angle of a rectangle is a right angle. (b) The opposite sides of a rectangle are equal in length. (c) The diagonals of a square are perpendicular to one another. (d) All the sides of a rhombus are of equal length. (e) All the sides of a parallelogram are of equal length. (f) The opposite sides of a trapezium are parallel.
Solution : (a) True (b) True (c) True (d) True (e) False (f) False.
Question 2. Give reasons for the following : (a) A square can be thought of as a special rectangle. (b) A rectangle can be thought of as a special parallelogram. (c) A square can be thought of as a special rhombus. (d) Squares, rectangles, parallelograms are all quadrilaterals. (e) Square is also a parallelogram.
Solution : (a) A rectangle with all sides equal becomes a square. (b) A parallelogram with each angle a right angle becomes a rectangle. (c) A rhombus with each angle a right angle becomes a square. (d) All these are four-sided polygons made of line segments. (e) The opposite sides of a square are parallels, so it is a parallelogram.
Question 3.
A figure is said to be regular, if its sides are equal in length and angles are equal in measure. Can you identify the regular quadrilateral?
Solution :
A square is a regular quadrilateral.
We hope the NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes Ex 5.7 help you. If you have any query regarding NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes Ex 5.7, drop a comment below and we will get back to you at the earliest. | 677.169 | 1 |
What does the secant line represent?
What does the secant line represent?
A secant line, also simply called a secant, is a line passing through two points of a curve. In geometry, a secant line commonly refers to a line that intersects a circle at exactly two points (Rhoad et al. 1984, p. 429).
What is the slope of the secant line through the points?
The slope of the secant line is also referred to as the average rate of change of f over the interval [x,x+Δx].
Which angles are formed by two secant lines?
If two lines intersect outside a circle , then the measure of an angle formed by the two lines is one half the positive difference of the measures of the intercepted arcs . In the circle, the two lines ↔AC and ↔AE intersect outside the circle at the point A .
How to calculate the secant?
How to calculate secant First, you need to determine what information is available to you. The next step is to plug these values into the above equation so sec x = 5/4 = 1.25. The last step would be to use the actual angle to check this. To do this you would simply need to enter the angle into the formula 1/cosx. Analyze the results.
What is the difference between secant and cosine?
As nouns the difference between secant and cosine is that secant is (geometry) a straight line that intersects a curve at two or more points while cosine is (trigonometry) in a right triangle, the ratio of the length of the side adjacent to an acute angle to the length of the hypotenuse symbol: cos.
What is secant the inverse of?
The secant is the inverse of the cosine. The reciprocal of the abscissa of the endpoint of an arc of a unit circle centered at the origin of a Cartesian coordinate system, the arc being of length x and measured counterclockwise from the point (1, 0) if x is positive or clockwise if x is negative.
Is secant 1 over cosine?
Let's recall that secant is 1 over cosine theta and so it inherits a lot of properties from cosine theta. For example cosine theta is even so is secant and it's easy to show that. Secant of negative theta is 1 over cosine of negative theta and that's of course cosine theta. | 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
Results 6-10 of 88
Page 22 ... given point to draw a straight line equal to a given straight line . F H G E Let A be the given point , and BC the given straight line : it is required to draw from A a straight line = BC . Join AB , Post . 1 and on it describe the ...
Page 23 ... given point be ? 2. AB is a given straight line ; show how to draw from A any number of straight lines equal to AB . 3. AB is a given straight line ; show how to draw from B any number of straight lines equal to AB . 4. AB is a given ...
Page 24 ... point B instead of the point A. 2. Produce the less of two given straight lines so that it may be equal to the greater . 3. If from AB ( fig . 1 and fig . 2 ) there be cut off AD and BE , each equal to C , prove AE = BD . Fig . 1 . Fig ...
Page 36 ... given straight line . 10. Find a straight line equal to half the sum of two given straight lines . 11. Find a straight line equal to half the difference of ... given point were situated at either end 36 [ Book I. EUCLID'S ELEMENTS .
Page 37 Euclides John Sturgeon Mackay. 2. If the given point were situated at either end of the given straight line , what additional construction would be necessary in order to draw a perpendicular ? 3. At a given point in a given straight ...
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
Page 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 |
8-3 skills practice special right triangles worksheet answers
8-3 Practice Special Right Triangles Worksheet – Triangles are among the fundamental shapes in geometry. Understanding triangles is crucial for studying 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 determine the perimeter and area of a triangle, and also provide examples of each. Types of Triangles There are three types for triangles: Equal, isosceles, and scalene. Equilateral triangles include three … Read more | 677.169 | 1 |
Which Lengths Would Form A Right Triangle
Do the following lengths form a right triangle ?
Which Lengths Would Form A Right Triangle. Web which side lengths form a right triangle? Web which side lengths form a right triangle?
Do the following lengths form a right triangle ?
It is used to determine the. 2003 aime ii problem 7. 5, \sqrt6, \sqrt31 5, 6, 31 a 5, \sqrt6, \sqrt31 5, 6, 31 \sqrt5, \sqrt5, 50 5, 5,50 b \sqrt5, \sqrt5, 50 5, 5,50 9, 12,. Three points defining a circle. {eq}a = 22\rm{cm}, b = 8\rm{cm}, c = 14\rm{cm} {/eq} step 1: Circumcenter of a right triangle. Web which side lengths form a right triangle? Web there are certain types of right triangles whose ratios of side lengths are useful to know. Choose all answers that apply: Web if a triangle has side lengths such that the set of sides comprise a pythagorean triple, the triangle is a right triangle.
2003 aime ii problem 7. Web using the triangle length calculator. The relation between the sides and other angles of the right triangle is the basis for So, those lengths form a right. Web there are certain types of right triangles whose ratios of side lengths are useful to know. Circumcenter of a right triangle. Our right triangle has a hypotenuse equal. O 9, 12, and 14 o 24,7, and 26 o 21, 16, and 12 o 30, 24, mathematicsmiddle school which lengths would form a. In a triangle of this type, the lengths of the three sides are. It is used to determine the. {eq}a = 22\rm{cm}, b = 8\rm{cm}, c = 14\rm{cm} {/eq} step 1:
If the lengths of the sides of a rightangle triangle class 9 maths CBSE
The default option is the right one. Web mathematics which lengths would form a right triangle? Three points defining a circle. Web which side lengths form a right triangle? Also, if the side lengths of a right triangle are all integers, they. In a triangle of this type, the lengths of the three sides are. Again, in option b, the given side lengths are 7, 24, and 25 identify if the following side lengths form a right triangle: Web if a triangle has side lengths such that the set of sides comprise a pythagorean triple, the triangle is a right triangle.
Special Right Triangles (SSS & AAA) Examples Included
Web mathematics which lengths would form a right triangle? 2003 aime ii problem 7. O 9, 12, and 14 o 24, 7, and 26 о 21, 16, and 12 o 30, 24, and 18 it's 30,24,18 report flag outlined. Choose all answers that apply: Web learn about the pythagorean theorem. Also, if the side lengths of a right triangle are all integers, they. Web assume that we have two sides, and we want to find all angles. In a triangle of this type, the lengths of the three sides are. So, those lengths form a right. Look at the given three side lengths and identify the longest side.
How To Find A Right Triangle With 3 Numbers The other angle, 2x, is 2
Also, if the side lengths of a right triangle are all integers, they. Our right triangle has a hypotenuse equal. The default option is the right one. It is used to determine the. So, those lengths form a right. Web home mathematics which lengths would form a right triangle? Web learn about the pythagorean theorem if all three sides of a right triangle have lengths that are integers, it is known as a pythagorean triangle. The relation between the sides and other angles of the right triangle is the basis for | 677.169 | 1 |
Geometric-mean Sentence Examples
Mention may also be made of his chapter on inequalities, in which he proves that the arithmetic mean is always greater than the geometric mean.
0
0
The process consists in taking the geometric mean of numbers above and below 5, the object being to at length arrive at 5 000Ooo.
0
0
To every geometric mean in the column of numbers there corresponds the arithmetical mean in the column of logarithms. The numbers are denoted by A, B, C, &c., in order to indicate their mode of formation.
0
0
The geometric mean of a continuous variable is calculated by taking the antilog of the mean of the logged original values.
A line is defined as a one-dimensional geometric figure with length but no width. It extends infinitely in either direction with no ends, and the equation of a straight line is ax + b = 0. Keep reading to learn about the five main types of lines in geometry with line examples in your everyday life. | 677.169 | 1 |
Geometry/Polygon
A Polygon is a two-dimensional figure, meaning all of the lines in the figure are contained within one plane. They are classified by the number of angles, which is also the number of sides.
One key point to note is that a polygon must have at least three sides. Normally, three to ten sided figures are referred to by their names (below), while figures with eleven or more sides is an n-gon, where n is the number of sides. Hence a forty-sided polygon is called a 40-gon.
For a list of n-gon names, go to [1] and scroll to the bottom of the page.
Polygons are also classified as convex or concave. A convex polygon has interior angles less than 180 degrees, thus all triangles are convex. If a polygon has at least one internal angle greater than 180 degrees, then it is concave. An easy way to tell if a polygon is concave is if one side can be extended and crosses the interior of the polygon. Concave polygons can be divided into several convex polygons by drawing diagonals. Regular polygons are polygons in which all sides and angles are congruent. | 677.169 | 1 |
1. If the angles of elevation of the top of a tower from two points at the distance of a m and b m from the base of tower and in the same straight line with it are complementary, then the height of the tower (in m) is
(a) √(a/b)
(b) √ab
(c) √(a + b)
(d) √(a – b)
None
2.
2. If the length of the shadow of a tower is increasing, then the angle of elevation of the sun
(A) is also increasing
(B) is decreasing
(C) remains unaffected
(D) Don't have any relation with length of shadow
None
3.
3. From a point on a bridge across a river the angle of depression of the banks on opposite sides of the river are 30° and 45° respectively. If the bridge is at the height of 30 m from the banks, the width of the river is
(a) 30(1 + √3) m
(b) 30(√3 – 1) m
(c) 30√3 m
(d) 60√3 m
None
4.
4. The angle of elevation of the top of a tower is 30°. If the height of the tower is doubled, then the angle of elevation of its top will
(A) also get doubled
(B) will get halved
(C) will be less than 60 degree
(D) None of these
None
5.
5. The ratio of the height of a tower and the length of its shadow on the ground is √3 : 1. The angle of elevation of the Sun is
(a) 30°
(b) 45°
(c) 60°
(d) 75°
None
6.
6. If the height of a tower and the distance of the point of observation from its foot,both, are increased by 10%, then the angle of elevation of its top
(A) increases
(B) decreases
(C) remains unchanged
(D) have no relation.
None
7.
7. A tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground making an angle of 30° with the ground. The distance between the foot of the tree to the point where the top touches the ground is 8 m. The height of the tree is
(a) 4√3 m
(b) 8√3 m
(c) 6√3 m
(d) 16√3 m
None
8.
8. A ladder 15 metres long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, then the height of the wall will be
(A) 7.5m
(B) 7.7m
(C) 8.5m
(D) 8.8m
None
9.
9. The angle of elevation of the top of a tower is 30°. If the height of the tower is doubled, then the angle of elevation of its top will be
(a) Greater than 60°
(b) Equal to 30°
(c) Less than 60°
(d) Equal to 60°
None
10.
10. An observer 1.5 metres tall is 20.5 metres away from a tower 22 metres high.Determine the angle of elevation of the top of the tower from the eye of the observer.
(A) 30°
(B) 45°
(C) 60°
(D) 90°
None
11.
11. The angles of elevation of the top of a tower from two points distant s and t from its foot are complementary. Then the height of the tower is:
(A) st
(B) s2t2
(C) √st
(D) s/t
None
12.
12. The shadow of a tower standing on a level plane is found to be 50 m longer when Sun's elevation is 30° than when it is 60°. Then the height of tower is:
(A) 20√3
(B) 25√3
(C) 10√3
(D) 30√3
None
13.
13. If a man standing on a platform 3 metres above the surface of a lake observes a cloud and its reflection in the lake, then the angle of elevation of the cloud is
(A) equal to the angle of depression of its reflection.
(B) double to the angle of depression of its reflection
(C) not equal to the angle of depression of its reflection
(D) information insufficient
None
14.
14. If a pole 6m high casts a shadow 2√3 m long on the ground, then the sun's elevation is
(A) 60°
(B) 45°
(C) 30°
(D) 90°
None
15.
15. The angle of elevation of the top of a tower from certain point is 30°. If the observer moves 20 metres towards the tower, the angle of elevation of the top increases by 15°. Find the height of the tower
(A) 10 (√3 + 1)
(B) 5√3
(C) 5 (√3 + 1)
(D) 10√3
None
16.
16. The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is 60°, then the distance between the two towers is:
(A) 10√3 m
(B) 15√3 m
(C) 12√3 m
(D) 36 m
None
17.
17. The angle of elevation of the top of a vertical tower from a point on the ground is60°. From another point 10 m vertically above the first, its angle of elevation is45°. Find the height of the tower.
(A) 5 (√3 + 3) m
(B) (√3 +3) m
(C) 15 (√3 +3)
(D) 5√3
None
18.
18. There are two windows in a house. A window of the house is at a height of 1.5 m above the ground and the other window is 3 m vertically above the lower window. Ram and Shyam are sitting inside the two windows. At an instant, the angle of elevation of a balloon from these windows are observed as 45° and 30° respectively. Find the height of the balloon from the ground.
(A) 7.598m
(B) 8.269m
(C) 7.269m
(D) 8.598 m
None
19.
Question 19. If at some time, the length of the shadow of a tower is √3 times its height, then the angle of elevation of the sun, at that time is:
(a) 15°
(b) 30°
(c) 45°
(d) 60°
None
20.
Question 20. A ladder 15 m long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, then the height of the wall is:
(a) 15√3 m
(b) 15√32 m
(c) 152 m
(d) 15 m
None
21.
Question 21. At some time of the day, the length of the shadow of a tower is equal to its height. Then, the sun's altitude at that time is:
(a) 30°
(b) 60°
(c) 90°
(d) 45°
None
22.
Question 22. A person is flying a kite at a height of 30 m from the horizontal level. The length of string from the kite to the person is 60 m. Assuming that here is no slack in the string, the angle of elevation of kite to the horizontal level is:
(a) 45°
(b) 30°
(c) 60°
(d) 90°
None
23.
Question 23. The angle of depression of a car, standing on the ground, from the top of a 75 m high tower is 30°. The distance of the car from the base of tower (in m) is:
(a) 25√3
(b) 50√3
(c) 75√3
(d) 150
None
24.
Question 24. A man at the top of a 100 m high tower sees a car moving towards the tower at an angle of depression of 30°. After some time, the angle of depression becomes 60°. The distance travelled by the car during this time interval is:
(a) 10√3 m
(b) 100√33 m
(c) 200√33 m
(d) 200√3 m
None
25.
Question 25. The angle of elevation of the top of a 15 m high tower at a point 15 m away from the base of tower is: | 677.169 | 1 |
Find an answer to your question 👍 "What is the difference between arithmetic and geometric ..." in 📗 Mathematics if the answers seem to be not correct or there's no answer. Try a smart search to find answers to similar questions. | 677.169 | 1 |
The Elements of Euclid with Many Additional Propositions and Explanatory Notes
therefore as the solid AK is to the solid CL, so is the solid FM to the solid HN (b).
[2.] Next, let the solid AK be to the solid CL, as the solid FM is to the solid HN: the straight line AB shall be to CD, as EF is to GH.
Take as AB is to CD, so is EF to ST, and from ST describe a solid parallelopiped SV similar and similarly situated to either of the solids FM, HN (e). And because AB is to CD, as EF is to ST, and that from AB, CD the solid parallelopipeds AK, CL are similarly described; and in like manner the solids FM, SV from the straight lines EF, ST; therefore AK is to CL, as FM is to SV; but, by the hypothesis, AK is to CL, as FM to HN; therefore HN is equal to SV (f): but it is likewise similar and similarly situated to SV; therefore the planes which contain the solids HN, SV are similar and equal, and their homologous sides GH, ST equal to one another: and because as AB is to CD, so is EF to ST, and that ST is equal to GH, therefore AB is to CD, as EF is to GH.
PROPOSITION XXXVIII.
THEOREM.-"If a plane (CD) be perpendicular to another plane (AB), and a straight line be drawn from a point (E) in one of the planes (CD) perpendicular to the other plane (AB), this straight line shall fall on the common section (AD) of the planes."
DEMONSTRATION.
E
"For if it does not, let it, if possible, fall elsewhere, as EF; and let it meet the plane AB in the point F; and from F draw, in the plane AB, a perpendicular FG to DA (a), which is also perpendicular to the plane CD (b); and join EG. Then, because FG is perpendicular to the plane CD, and the straight line EG which is in that plane, meets it, therefore FGE is a right angle (c): but EF is also at right angles to the plane AB, and therefore EFG is a right angle: wherefore two of the angles of the triangle EFG are equal together to two right angles; which is absurd (d); therefore the perpendicular from the point E to the plane AB, does not fall elsewhere than upon the straight line AD; that is, it therefore falls upon it."
(a) I. 12. XI. Def. 4. (c) XI. Def. 3. (d) I. 17.
PROPOSITION XXXIX.
THEOREM.—In a solid parallelopiped, if the sides of two of the opposite planes be divided, each into two equal parts, the common section of the planes passing through the points of division, and the diameter of the solid parallelopiped, cut each other into two equal parts.
B
K
M
E
DEMONSTRATION. Let the sides of the opposite planes CF, AH of the solid parallelopiped AF, be divided each into two equal parts in the points K, L, M, N; X, O, P, R; and join KL, MN, XO, PR: and because DK, CL are equal and parallel, KL is parallel to DC (a): for the same reason, MN is parallel to BA: and BA is parallel to DC; therefore, because KL, BA are each of them parallel to DC, and not in the same plane with it, KL is parallel to BA (b): and because KL, MN are each of them parallel to BA, and not in the same plane with it, KL is parallel to MN (6): wherefore KL, MN are in one plane. In like manner it may be proved, that XO, PR are in one plane. Let YS be the common section of the planes KN, XR; and DG the diameter of the solid parallelopiped AF: YS and DG shall meet, and cut one another into two equal parts.
(a) I. 33.
(e) I. 14.
(b) XI. 9.
(f) I. 15.
(c) I. 29.
(g) I. 26.
(d) I. 4.
Join DY, YE, BS, SG. Because DX is parallel to OE, the alternate angles DXY, YOE are equal to one another (c): and because DX is equal to OE, and XY to YO, and that they contain equal angles, the base DY is equal to the base YE (d), and the other angles are equal; therefore the angle XYD is equal to the angle OYE, and DYE is a straight line (e): for the same reason, BSG is a straight line, and BS equal to SG. And because CA is equal and parallel to DB, and also equal and parallel to EG, therefore DB is equal and parallel to EG (6): and DE, BG join their extremities; therefore DE is equal and parallel to BG (a): and DG, YS are drawn from points in the one, to points in the other, and are
therefore in one plane: whence it is manifest, that DG, YS must meet one another: let them meet in T. And because DE is parallel to BG, the alternate angles EDT, BGT are equal (c): and the angle DTY is equal to the angle GTS (f): therefore in the triangles DTY, GTS, there are two angles in the one, equal to two angles in the other, and one side equal to one side, opposite to two of the equal angles, viz. DY to GS, for they are the halves of DE, BG; therefore the remaining sides are equal, each to each (g): wherefore DT is equal to TG, and YT equal to TS.
PROPOSITION XL.
THEOREM.-If there be two triangular prisms of the same altitude, the base of one of which is a parallelogram, and the base of the other a triangle: if the parallelogram be double of the triangle, the prisms shall be equal to one another.
DEMONSTRATION. Let the prisms ABCDEF, GHKLMN be of the same altitude, the first whereof is contained by the two triangles ABE, CDF, and the three parallelograms AD, DE, EC; and the other by the two triangles GHK, LMN, and the three parallelograms LÍ, HN, NG; and let one of them have a parallelogram AF, and the other a triangle GHK, for its base: if the parallelogram AF be double of the triangle GHK, the prism ABCDEF shall be equal to the prism GHKLMN. Complete the solids AX, GO: and because the parallelogram AF is double of the triangle GHK, and the parallelogram HK double of the same triangle, therefore the parallelogram AF is equal to HK (a): but solid parallelopipeds upon equal bases, and of the same altitude, are equal to one another (b); there
B
E
G
(a) I. 34.
(b) XI. 31.
(c) XI. 28.
fore the solid AX is equal to the solid GO: and the prism ABCDEF is half of the solid AX (c): and the prism GHKLMN half of the solid GO (c): therefore the prism ABCDEF is equal to the prism GHKLMN.
THE
ELEMENTS OF EUCLID.
BOOK XII.
LEMMA I.
THEOREM.-If from the greater of two unequal magnitudes, there be taken more than its half, and from the remainder more than its half, and so on; there shall at length remain a magnitude less than the least of the proposed magnitudes.
A
K
F
DEMONSTRATION. Let AB and C be two unequal magnitudes, of which AB is the greater: if from AB there be taken more than its half, and from the remainder more than its half, and so on; there shall at length remain a magnitude less than C. For C may be multiplied so as at length to become greater than AB. Let it be so multiplied, and let DE its multiple be greater than AB, and let DE be divided into DF, FG, GE, each equal to C. From AB, take BH greater than its half; and from the remainder AH, take HK greater than its half, and so on, until there be as many divisions in AB as there are in DE. And because DE is greater than AB, and that EG taken from DE is not greater than its half, but BH taken from AB is greater than its half, therefore the remainder GD is greater than the remainder HA. Again, because GD is greater than HA, and that GF is not greater than the half of GD, but HK is greater than the half of HA; therefore the remainder FD is greater than the remainder AK. And FD is equal to C, therefore C is greater than AK; that is, AK is less than C.
COROLLARY. And if only the halves be taken away, the same thing may, in the same way, be demonstrated.
SCHOLIUM. This is the first proposition in the 10th book, and being nocessary to some of the propositions of this book, it is here inserted.
PROPOSITION I.
THEOREM. Similar polygons (ABCDE, FGHKL) inscribed in circles, are to one another as the squares on their dia
BAE, GFL having an
V. Def. 10 and 22.
VI. 20.
angle in one, equal to an angle in the other, and the sides about the equal angles proportionals, are equiangular; and therefore the angle AEB is equal to the angle FLG: but AEB is equal to AMB, because they stand upon the same circumference (b): and the angle FLG is, for the same reason, equal to the angle FNG: therefore also the angle AMB is equal to FNG; and the right angle BAM is equal to the right angle GFN (c); wherefore the remaining angles in the triangles ABM, FGN are equal, and they are equiangular to one another; therefore as BM is to GN, so is BA to GF (d); and therefore the duplicate ratio of BM to GN, is the same with the duplicate ratio of BA to GF (e): but the ratio of the square on BM to the square on GN, is the duplicate ratio of that which BM has to GN; and the ratio of the polygon ABCDE to the polygon FGHKL is the duplicate of that which BA has to GF (ƒ): therefore as the square on BM is to the square on GN, so is the polygon ABCDE to the polygon FGHKL. | 677.169 | 1 |
(Solved):
11. If the central angle of a circle is \( \frac{5 \pi}{6} \) radians and the arc length is \( 12 ...
11. If the central angle of a circle is \( \frac{5 \pi}{6} \) radians and the arc length is \( 12 \mathrm{~cm} \), find the radius of the circle.
12. A wheel is turning at a rate of 48 revolutions per minute. a) Find the angular velocity, in radians per second. | 677.169 | 1 |
Hexagon - Angles, Diagonals, Properties, Area, and Perimeter
What is a Hexagon?
A hexagon is a 2D geometric figure enclosed by six straight edges. In the below-given picture, you can see that the six edges of a hexagon make six vertices and six angles. Are you wondering why this shape is termed hexagon? Well, a six-sided polygon is termed a hexagon as "hex" stands for six and "gonia" stands for corners. Hexagons can be broadly classified as regular hexagons and irregular hexagons on the basis of the length of their sides. Also, hexagons can be classified into convex and concave on the basis of the measure of their angles.
Can we Find the Hexagon Pattern Around Us?
The hexagon shape can be found in various objects around us, for example, a honeycomb. Have you ever noticed a honeycomb? Well, let us take a closer look at the below-given picture. Each little unit in a honeycomb has six sides, making it a hexagon.
Each little unit of a honeycomb resembles a hexagon shape
We can find the shape of a hexagon in a floor tile, on a clock, on hexagonal specs, on the cross-section of a pencil, etc.
Hexagonal pencil surface
What is a Regular Hexagon?
When the lengths of all hexagonal sides are equal, it is known as a regular hexagon. Also, the angles of a regular hexagon are equal. Interestingly, a regular hexagon can be divided into six equilateral triangles.
A regular hexagon divided into six equilateral triangles
What is an Irregular Hexagon?
When the sides of a hexagon are unequal in length, it is known as an irregular hexagon. The angles of an irregular hexagon are not equal. An irregular hexagon cannot be divided into six equilateral triangles.
An irregular hexagon
Convex and Concave Hexagons
Each vertex of a convex hexagon is pointed outwards, and all interior angles are less than 1800. The sides of a convex hexagon can be equal or unequal.
In a concave hexagon, one or more interior angles measure greater than 1800. Also, one or more vertex points inwards in a concave hexagon.
Convex and Concave hexagons
Sides and Angles of a Hexagon
As stated above, a hexagon, regular or irregular, has six sides. The hexagon sides are straight, forming closed 2-D shapes. Adding up the lengths of all sides of a hexagon gives its perimeter. If the perimeter of a regular hexagon is given, the length of its every side can be determined by dividing the perimeter by 6.
Length of each side of a regular hexagon = Perimeter/6
However, the same is not applicable for an irregular hexagon. Since two or more sides of an irregular hexagon are unequal, the length of its sides cannot be determined from its perimeter.
Perimeter of a regular hexagon
A hexagon has 6 interior angles and 6 exterior angles. The sum of the interior angles of a hexagon is 7200. As the interior angles of a regular hexagon are equal, the measure of each can be determined as 7200/6. Therefore, each interior angle of a regular hexagon measures 1200.
The sum of the exterior angles of a hexagon is 3600. Each exterior angle of a regular hexagon can be determined as 3600/6. Therefore, each exterior angle of a regular hexagon measures 600.
Interior and exterior angles of a regular hexagon
Diagonals of Hexagons
A line segment connecting any two non-adjacent vertices of a hexagon is referred to as the diagonal of the hexagon.
A diagonal of a regular hexagon
A hexagon has 9 diagonals connecting its non-adjacent vertices. Of these, 3 diagonals pass through the centre of the hexagon. The diagonals of a hexagon can be classified as long diagonals and short diagonals. The 3 diagonals of a hexagon that pass through its centre happen to be greater in length than the rest of the diagonals. Hence, these are referred to as the long diagonals.
Length of a long diagonal of a regular hexagon = 2 x length of each side of the hexagon.
The 6 diagonals of a hexagon that do not pass through its centre are referred to as the short diagonals.
Length of a short diagonal of a regular hexagon = 3 x length of each side of the hexagon.
Regular Hexagon Diagonals
Properties of Hexagons
There are different types of hexagons, each with a definite set of properties. Some hexagons may have equal sides, whereas some may have their vertices pointing inwards. So, let us sum up the basic properties of hexagons in general.
A hexagon ("hex": six and "gonia": corners) has 6 sides, 6 angles, and 6 vertices.
A regular hexagon is symmetrical.
The opposite sides of a regular hexagon run parallel to each other.
The sides of regular hexagons have the same lengths.
The sum of the interior angles of all hexagons is 7200.
The sum of the exterior angles of all hexagons is 3600.
There are 9 diagonals, 3 long and 6 short, in a regular hexagon.
All regular hexagons are convex.
Some irregular hexagons are concave.
Area and Perimeter of Regular Hexagons
As a regular hexagon can be divided into 6 equilateral triangles, the area of a regular hexagon is calculated by the following formula:
Area: 3 \[ \frac{\sqrt{3}}{2} \] a2
(where \[ \frac{\sqrt{3}}{2} \] a2is the area of an equilateral triangle and a is the length of each side of the hexagon)
The perimeter of a hexagon can be calculated by adding the lengths of all sides of the hexagon. As the sides of a regular hexagon are equal in length, its perimeter can be calculated by the following formula:
Perimeter = 6a
(where a is the length of each side of the hexagon)
Solved Examples
The below-given examples will help you to understand the application of the above-given formulas of area and perimeter of hexagons.
Each side of a regular hexagon is equal to 5 cm. Find its perimeter and area correct up to two decimal places.
Solution:
Length of each side of the regular hexagon = 5 cm
{ Since perimeter of hexagon= 6a }
Therefore, the perimeter of the hexagon = (6 x 5) cm
= 30 cm
{ Since Area of a hexagon= \[ \frac{\sqrt{3}}{2} \] a2}
The area of the hexagon = \[ \frac{\sqrt{3}}{2} \] 52 cm2
= 64.95 cm2
The perimeter of a hexagonal field is 96 metres. Find its area correct up to two decimal places.
Solution:
The perimeter of the field = 96 m
{Since perimeter of hexagon= 6a }
Therefore, the length of each side of the hexagonal field = 96/6 m
=16 m
Therefore, the area of the field = \[ \frac{\sqrt{3}}{2} \] 162 m2
= 665.11 m2
Hexagons form one of the most fundamental concepts of geometry. With practice, kids develop a thorough understanding of the properties of hexagons. Most kids enjoy the hexagon drawing or construction, with given hexagon angles and sides, as they get a better insight of this polygon during the activity. So, encourage your kids to solve various interactive worksheets and activities on hexagons for a seamless learning experience.
FAQs on Hexagon
1. What is a convex hexagon?
If all the vertices of a given hexagon point in the outward direction, it is referred to as a convex hexagon. The regular hexagons have all the vertices pointing outward, hence they are convex hexagons. Also, some irregular hexagons that do not have equal sides can be convex.
2. How many diagonals are there in a regular hexagon?
There are a total of 9 diagonals in a regular hexagon. The lengths of the diagonals are not equal and three diagonals are longer than the rest. These are called the long diagonals of a regular hexagon and run through its centre. The other six diagonals of the regular hexagon are called its short diagonals. | 677.169 | 1 |
What is a 2D shape definition?
2D shapes have sides and corners, and are completely flat. Watch the video to learn all about 2D shapes, like circles, triangles, squares, rectangles, pentagons, hexagons and octagons!
What 2D shape has?
2-dimensional shapes are flat and only have two dimensions: length and width. They include squares, rectangles, circles, triangles, and more. Read on to learn all about 2D shapes and the differences between 2D and 3D shapes.
What is meant by 2D?
In geometry, a two-dimensional shape can be defined as a flat plane figure or a shape that has two dimensions – length and width. Two-dimensional or 2-D shapes do not have any thickness and can be measured in only two faces.
Is a pizza 2D or 3D?
2D Shapes. Two dimensional, or 2D, shapes are flat shapes. Circles are round 2D shapes with no corners. Pizza pies, clocks and bike tires are all real-world examples of circles.
Why is a circle 2D?
Is Circle a 2D Shape? Yes, a circle is a 2D shape because it exists on a plane with no depth. It is a curved shape that has no corners or edges.
Are humans 2D or 3D
What is 2D example?
A circle, triangle, square, rectangle and pentagon are examples of two-dimensional shapes. A point is zero-dimensional, while a line is one-dimensional, for we can only measure its length.
Is a clock a 2D or 3D shape?
The common shapes of a clocks are 𝗰𝗶𝗿𝗰𝗹𝗲, 𝗼𝘃𝗮𝗹, 𝘁𝗿𝗶𝗮𝗻𝗴𝗹𝗲, 𝗼𝗿 𝘀𝗾𝘂𝗮𝗿𝗲. It is only after a clear knowledge on 2D shapes we shall start to learn the 3D shapes. Most of the objects we use in our daily life are 3D objects.
Is a prism 2D?
A prism shape is a 3D shape which has a constant cross-section. Both ends have the same 2D shape, and they're connected by rectangular sides.
Do ants see in 2D?
Because the ant can only perceive her two dimensions, she does not realize that her world is curved. From her point of view, space stretches out flat in front of her,, like the Midwest (flat and endless).
Do we think in 2D?
We are 3D creatures, living in a 3D world but our eyes can show us only two dimensions. The miracle of our depth perception comes from our brain's ability to put together two 2D images in such a way as to extrapolate depth. This is called stereoscopic | 677.169 | 1 |
Before I try and answer the question, note that a Vectorcannot pass through a point. A Vector is merely a direction and magnitude, not a line. Because of this, I assume you are tryign to solve the shortest distance from a line to a point problem.
OK, time for some maths using Unity functions:
Lets say we have the line that passes through the two points A, C and we want to find the vector from that line to B which has the closest distance. By simple logic we can assume that that vector must be perpendicular to the vector AC:
@Benproductions1 thanks a lot for this great deduction. Not only it is very well explained, but I couldn't find anything similar on Internet.
However, after implementing and plotting the results, I believe there is still a little error in the calculation of vector V (second last equation), which causes V having the opposite direction (wrong sign).
You wrote:
V = P - B
but it should be:
V = B - P
Consequently, the following (and last) equation becomes:
V = B - A - (O · D)/(D · D)*D
Please let me know if this sounds correct to you and, in that case, I agree with @aldonaletto in saying that your original answer deserves to be fixed, as it might be helpful for many others. | 677.169 | 1 |
Shape in Year 4
Turning clockwise and anticlockwise will be revisited in Year 4, as will classifying angles into acute, obtuse and right angles.
Children will be expected to know the names of, and recognise, some quite tricky 3D shapes. These include prisms, cuboids, cylinders and pyramids. As well as this, the term prism can be used for a variety of shapes, including triangular prism, hexagonal prism and octagonal prism. Unlike basic shapes these are not terms that are really used in everyday life. If children are to become familiar with them they will need a lot of practice naming the shapes and recognising their properties.
Triangles form a major part of the work in Year 4, with the terms scalene, isosceles and equilateral introduced. Right angled triangles will also be looked at.
Again, the language of 2D shape is quite tricky, with 4 sided shapes such as a trapezium, a rhombus and a parallelogram being introduced. Children will also consider polygons and whether they are regular or irregular.
Understanding of symmetry is also extended to include any line of symmetry in any direction. | 677.169 | 1 |
angle properties of triangles and quadrilaterals worksheets
Angle Properties Of Triangles And Quadrilaterals Worksheets – Triangles are among the most fundamental forms in geometry. Understanding triangles is important for getting more advanced concepts in geometry. In this blog post we will look at the different types of triangles with triangle angles. We will also discuss how to determine the size and perimeter of a triangle, and provide illustrations of all. Types of Triangles There are three types of triangles: equal, isosceles, and scalene. | 677.169 | 1 |
Emma wrote the following paragraph proof showing that rectangles are parallelograms with congruent diagonals. According to the given information,...
Emma wrote the following paragraph proof showing that rectangles are parallelograms with congruent diagonals.According to the given information, quadrilateral RECT is a rectangle. By the definition of a rectangle, all four angles measure 90°. Segment ER is parallel to segment CT and ______________ by the Converse of the Same-Side Interior Angles Theorem. Quadrilateral RECT is then a parallelogram by definition of a parallelogram. Now, construct diagonals ET and CR. Because RECT is a parallelogram, opposite sides are congruent. Therefore, one can say that segment ER is congruent to segment CT. Segment TR is congruent to itself by the Reflexive Property of Equality. The Side-Angle-Side (SAS) Theorem says triangle ERT is congruent to triangle CTR. And because corresponding parts of congruent triangles are congruent (CPCTC), diagonals ET and CR are congruent.Which of the following completes the proof? | 677.169 | 1 |
Cos 145 Degrees
The value of cos 145 degrees is -0.8191520. . .. Cos 145 degrees in radians is written as cos (145° × π/180°), i.e., cos (29π/36) or cos (2.530727. . .). In this article, we will discuss the methods to find the value of cos 145 degrees with examples.
Cos 145°: -0.8191520. . .
Cos (-145 degrees): -0.8191520. . .
Cos 145° in radians: cos (29π/36) or cos (2.5307274 . . .)
What is the Value of Cos 145 Degrees?
The value of cos 145 degrees in decimal is -0.819152044. . .. Cos 145 degrees can also be expressed using the equivalent of the given angle (145 degrees) in radians (2.53072 . . .)
FAQs on Cos 145 Degrees
What is Cos 145 Degrees?
Cos 145 degrees is the value of cosine trigonometric function for an angle equal to 145 degrees. The value of cos 145° is -0.8192 (approx)
What is the Value of Cos 145° in Terms of Cosec 145°?
Since the cosine function can be represented using the cosecant function, we can write cos 145° as -[√(cosec²(145°) - 1)/cosec 145°]. The value of cosec 145° is equal to 1.74344.
How to Find the Value of Cos 145 Degrees?
The value of cos 145 degrees can be calculated by constructing an angle of 145° with the x-axis, and then finding the coordinates of the corresponding point (-0.8192, 0.5736) on the unit circle. The value of cos 145° is equal to the x-coordinate (-0.8192). ∴ cos 145° = -0.8192.
How to Find Cos 145° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 145° can be given in terms of other trigonometric functions as: | 677.169 | 1 |
JB/135/292/001
6.
Equality of lines — the
idea founded on that of
identity. Euclid aware
of this when he proved
the equality of two
by a circle:
method of making
a circle on sand.
Of the idea we have of equality as applied to
lines the foundation is the idea of identity.
Of this truth it appears that Euclid had a conception
more or less explicit when for proof of equality as
between two lines he made use of the figure of a circle
He delineated suppose upon paper a figure which he
called a circle. Instead of paper, the production of
comparatively modern times, it wd. have been better to
have taken a small piece of ground covered with sand
which had existence in all times: in any rate in all times
we have any acquaintance with. Now then having occasion
to exhibit the figure of a circle on this sand, what
is the course he wd. take? Settling, upon a rough estimate,
the size of the circle he proposed to make, he wd. take
the first dry twig he happened to meet with, and reducing
it to its intended length, he wd. apply it to the
purpose of forming his intended circle. A circle then in
what way was it to be formed? One way there was, & in
the nature of the case there wd. be but one: this was
the fixing on some part of its length, no matter what,
& keeping that part pinned down constantly to the
same spot, making the other parts move round till they
came back to the same position they maintained at
first. if this point were any other than one of the two
extremities he wd. be making 2 circles one within the other,
but what he was in need of was one circle & no more; he wd | 677.169 | 1 |
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