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Horizontal Staff
From Inventions
Contents
Inventor
Thales of Miletus (7th-6th cent. B.C.)
Euclid (3rd-2nd cent. B.C.)
Historic Period
7th-6th cent. B.C.
Description
In Theorem XXI of Euclid's Optics, the verge is positioned horizontally to measure an unknown length. This is a problem amply discussed in gromatic codices (F. Blume and others, 1967, p. 193) up to and beyond the abacus treatises. From the gromatic tradition of the fluminis varatio (measuring the width of rivers), linked to bridge-building, derives the use of a horizontal verge attached to another vertical one to form a square. This variant is found in the Geometria incerti auctoris (G. d'Aurillac, 972-1003, III, 11 and 24) and in other treatises on practical geometry (D. da Chivasso, 1346, I, 1; C. di Gherardo Dini, 1442, p. 95; F. di Giorgio Martini, c. 1480, fol. 31r). | 677.169 | 1 |
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An isosceles triangle with a vertex angle 306∘ 306 ∘. 2. A scalene triangle with one …Unit 1 The Ice Hotel. 1. I would like to stay in the Ice Hotel. I think it is. Pre-Reading (answers will vary) a unique place. 2. The most unusual place that I have heard of is this Ice Hotel. 3. An interesting place I have visited is Bangkok.Displaying all worksheets related to - Right Triangles And Trigonometry Gina Wilson. Worksheets are Trigonometry quiz gina wilson, Gina wilson all things algebra right angles and trigonometry, Right triangle trigonometry, Test review right triangle trigonometry answer key, Right triangle trig missing sides and angles, Gina wilson all things algebra …We would like to show you a description here but the site won't allow us. 0.7500. 0.7500. 18) tan 22°. 0.4040. 20) sin 77°. 0.9744. 22) cos 87°. 0.0523. Create your own worksheets like this one with Infinite Algebra 1.
11. Yes, 45-45-90. 12. Yes, 30-60-90. 13. The four sides of a square are congruent and the angles are right angles. When you cut the square in half, you create two right triangles, each with two congruent sides. Therefore, in each triangle the two non-right angles must be congruent and the triangles must be 45-45-90 triangles.. Episode 1 cast | 677.169 | 1 |
Answered question
Answer & Explanation
oppturf
Skilled2021-08-16Added 94 answers
The given trigonometry expression can be expressed as difference formula of tan angle which is given as:
tan(A−B)=tanA−tanB1+tanAtanB
where A and B are in degree.
Comparing with the given expression, we get:
A=78∘ and B=18∘
Put these value in the tan angle formula as:
tan(78∘−18∘)=tan78∘−tan18∘1+tan78∘tan18∘ tan(60∘)=3
The exact value of given expression can be calculated as:
tan78∘−tan18∘1+tan78∘tan18∘=4.7046−0.32492.5286 =4.37972.5286 =1.73206
Therefore, the exact value of given expression is 1.73206. | 677.169 | 1 |
How to measure the angle in theodolite?
1 Answer
1. Place the theodolite on a tripod and level it to make sure the instrument is stable and secure. 2. Set up a horizontal line by extending two rods, one at each end of the line. 3. Turn the theodolite so that the horizontal circle is parallel with the line. 4. Sight the rods at each end of the lineRead more
1. Place the theodolite on a tripod and level it to make sure the instrument is stable and secure.
2. Set up a horizontal line by extending two rods, one at each end of the line.
3. Turn the theodolite so that the horizontal circle is parallel with the line.
4. Sight the rods at each end of the line.
5. Rotate the theodolite until the vertical crosshair is aligned with the second rod.
6. Read the angle from the horizontal circle and record the results. | 677.169 | 1 |
What is a figure formed by 2 rays with a common endpoint?
What is a figure formed by 2 rays with a common endpoint?
Angle. A geometric figure consisting of the union of two rays that share a common endpoint.
What is two rays with a common endpoint and form a line?
An angle is formed by two rays with a common endpoint. Each ray is called an arm of the angle. The common endpoint is called the vertex of the angle.
What happens when two rays share a common endpoint?
An angle is the union of two rays with a common endpoint. The common endpoint of the rays is called the vertex of the angle, and the rays themselves are called the sides of the angle.
What is the common endpoint of the two rays of an angle called?
vertex of the angle
An angle is the union of two rays that share a common endpoint. The rays are called the sides of the angle, and the common endpoint is the vertex of the angle.
What is a figure formed by two rays with a common endpoint quizlet?
A figure formed by two rays with a common endpoint called the vertex. On an angle or polygon, the point where two sides intersect. An angle that measures less than 90 degrees.
Is an angle made of 2 rays?
In Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in the plane that contains the rays.
Is the ray with endpoint and intersect?
Definition. An angle is the intersection of two noncollinear rays at a common endpoint. The rays are called sides and the common endpoint is called the vertex.
What are 2 rays?
Explanation: The two sides of an angle are the two rays that compose it. Each of these rays begins at the vertex and proceeds out from there. In naming a ray, we always begin with the letter of the endpoint (where the ray starts) followed by another point on the ray in the direction it travels.
What is it called when you have 2 rays that share a vertex?
An angle is formed when two rays share a common endpoint. That common endpoint is called the vertex and the two rays are called the sides of the angle: [Figure 1]
When two rays originate from a common point is formed?
An angle is a figure in which two rays emerge from a common point. This point is called the vertex of the angle and the two rays forming the angle are called its arms or sides.
What is the common endpoint?
The common endpoint is called the vertex of the angle. The measure of an angle is a number representing the spread of the two arms of the angle.
Which type of angles have a common endpoint and share a common ray?
Adjacent angles are the angles with a common arm(side) and a common vertex. An angle is formed by two rays meeting at a common endpoint. For example, two pizza slices next to each other in the pizza box form a pair of adjacent angles when we trace their sides.
What is the name of two rays that share a common endpoint?
Opposite rays can be defined as a figure formed by two collinear rays with a common endpoint, since the two rays lie on the same line. that Q is between P and S. 5. Similarly, an angle can be defined as a figure formed by two rays with a common endpoint.
Which is the common endpoint of an angle?
Similarly, an angle can be defined as a figure formed by two rays with a common endpoint. The two rays are called the sides of the angle. The common endpoint is called the vertex.
How are two rays connected to form an angle?
An angle is created when two rays connect at a common point. You can see that the two rays are connected at a common endpoint, called a vertex. This forms the angle. An angle is named by points on the rays. What are two non collinear rays? An angle is the union of two noncollinear rays with a common endpoint.
Which is an example of a figure formed by two rays?
Similarly, an angle can be defined as a figure formed by two rays with a common endpoint. The two rays are called the sides of the angle. The common endpoint is called the vertex. 6. The figure formed by opposite rays is often referred to as a straight angle. Straight angles have a degree measure of 180 degrees. | 677.169 | 1 |
Angle Sum Worksheet
Angle Sum Worksheet. Use this worksheet when learning about angles and the angle sum of triangles. Web show that the measure of an exterior angle is equal to the sum of the related remote interior angles.
Triangle Angle Sum Worksheet Answers from
You can choose between interior and exterior angles, as well as an algebraic expression for the unknown. This worksheet contains 20 problems that focuses on finding the missing angle given the other two angles (no algebra involved). Here is one proof of the triangle sum theorem.
Source: zipworksheet.com
Web featuring myriad workout routines, this set of angles in a triangle worksheets help be taught the applying of angle sum property and exterior angle theorem to search. You must be able to identify these types of.
Source:
Web these triangle worksheets make use of the triangle sum theorem which states that the sum of the interior angles of a triangle is always 180 degrees. Web featuring myriad workout routines, this set of angles in a triangle worksheets help be taught the applying of angle sum property and exterior angle theorem to search.
Source:
A worksheet on evaluating the third angle in a triangle given the other two using the fact. Here is one proof of the triangle sum theorem.
Source:
Web a collection of three worksheets on the following topics: More details on how to.
Figure \ (\Pageindex {2}\) Given:
Web featuring myriad workout routines, this set of angles in a triangle worksheets help be taught the applying of angle sum property and exterior angle theorem to search. Web these triangle worksheets make use of the triangle sum theorem which states that the sum of the interior angles of a triangle is always 180 degrees. Use this worksheet when learning about angles and the angle sum of triangles.
Find the measure of angle x. You must be able to identify these types of. Web show that the measure of an exterior angle is equal to the sum of the related remote interior angles.
Web Angle Sums And Differences Worksheets.
You can do the exercises online or download the worksheet as pdf. Web a collection of three worksheets on the following topics: Web the purpose of this quiz and worksheet combo is to measure how much you know about types of angles and angle sum problems.
A Worksheet On Evaluating The Third Angle In A Triangle Given The Other Two Using The Fact.
Web triangle angle sum worksheet answers. This worksheet contains 20 problems that focuses on finding the missing angle given the other two angles (no algebra involved). Sum of angles in a triangle: | 677.169 | 1 |
where xk,yk,k=1,2,3, are the three vertices of the triangle. For the representative triangle in Figure 6.5.3(a), the centroid has coordinates a+b/3,c/3 | 677.169 | 1 |
Draw a circle with three random points on the circle. Connect the points to form a triangle. Display the angles in the triangle. Use the mouse to drag a point along the perimeter of the circle. As you drag it, the triangle and angles are redisplayed dynamically as shown in Figure 1.
You will need to create a DrawTriangle class (not main class) which takes some appropriate parameters in one of its methods to draw the triangle within the circle as shown in Figure 1. Your will not get full marks if DrawTriangle class is not created nor used as it as part of Object-Oriented Software Development57:012021-02-26 06:57:01Draw a circle with three random points on the circle. Connect the points to form a triangle | 677.169 | 1 |
44.
Σελίδα 10 ... equal to AC , and BC is common to both , 2. Therefore the two sides DB , BC , are equal to the two sides AC , CB ... CD . Demonstration . - 1 . Because AC is A assumed to be equal to AD . ( hyp . ) D 2. The triangle ADC is an isosceles ...
Σελίδα 11 ... CD . Demonstration . - 1 . Because AC is assumed to be equal to AD . ( hyp . ) 2. The triangle ADC is an isosceles triangle , and the angles ECD , FDC , upon the other side of its base CD , are equal A to one another . ( I. 5. ) תי E B | 677.169 | 1 |
Demystifying GRE Math: Mastering Polygons and Angles
Are you preparing for the GRE and feeling a little stuck on the quant section? Don't worry - we've got your back! In this article, we'll break down a tricky question about polygons and angles...
Mục lục
Are you preparing for the GRE and feeling a little stuck on the quant section? Don't worry - we've got your back! In this article, we'll break down a tricky question about polygons and angles to help you gain a deeper understanding of this topic. So, let's dive in and conquer those math problems together!
Surveying the Question
Before we jump into finding a solution, let's take a moment to analyze the problem. Understanding what the question is testing will guide us in applying the right math knowledge to solve it. Look for math-specific terms and any special characteristics of the given numbers. Mark down these details on your paper to keep them fresh in your mind.
What Do We Know?
Now that we have surveyed the question, let's carefully read through it and make a list of the information we already know. In this case, we are dealing with a regular 9-sided polygon and we need to find the value of an external angle shown in the figure.
Developing a Plan
To solve this question, we need to find the interior angle of the polygon and then subtract that from 180° to get the value of the external angle. Fortunately, we can use a simple approach to find the interior angle of any polygon. By dividing the polygon into triangles, we can take advantage of the fact that the sum of angles in any triangle is 180°. Let's start drawing triangles on our figure to make this process easier.
Solving the Question
After drawing triangles starting from one vertex of our polygon, we can see that the sum of all the internal angles can be represented by seven triangles. To find the value of the interior angle, we can multiply the number of triangles by 180° and then divide by the number of vertices (which is 9 in this case).
So, the interior angle of the polygon is (180° * 7) / 9, which simplifies to 140°. Now, since the external angle and one interior angle lie on the same side of a straight line, their sum must be 180°. Therefore, the value of the external angle (x) is 180° - 140°, which equals 40°.
The correct answer is 40°!
What Have We Learned?
By tackling this question, we have gained a solid understanding of how to find the interior angle of any regular polygon. You can simply divide the polygon into triangles, calculate the sum of the interior angles by multiplying the number of triangles by 180°, and then divide this sum by the number of vertices (which is also equal to the number of sides).
Now that you have mastered polygons and angles, you're one step closer to acing the GRE math section. Keep practicing and honing your skills, and you'll be well-prepared for any math question that comes your way.
Remember, if you want expert GRE prep tailored to your needs, sign up for our five-day free trial of the PrepScholar GRE Online Prep Program. Access personalized study plans, interactive lessons, and over 1600 GRE questions to take your preparation to the next level.
If you have any questions or need further assistance, feel free to leave a comment or email us at [email protected] We're here to help you succeed | 677.169 | 1 |
2 comments:
Let the mid points of A1A2,B1B2 and C1C2 are Ma,Mb and Mc. Draw perpendicular MaD to BH and MbE to AH. We get MaD/MbE=BMa/AMb ......(1) Let Angle HA2A1=x and Angle HB1B2=y, Then Angle MbHE=Angle MbHA1+ Angle A1HE=x+y Similarly Angle DHMa = Angle DHA2 + Angle MaHA2=x+y Hence Triangle HMbE is similar to triangle HMaD, We get MaD/MbE=HMa/HMb,using eq 1, we have BMa/AMb=HMa/HMb Similarly we get CMb/BMc=HMb/HMc, and AMc/CMa=HMc/HMa, We get (AMc/BMc)x(BMa/CMa)x(CMb/AMb)=1, hence by Menelaus Theorem Ma,Mb and Mc must be collinear.
Using results of Problem 1253, circumcircles of Triangle HA1A2, HB1B2,HC1C2 and ABC, pass through point P. Since HP is common chord of 3 circles, its centers will lie on perpendicular bisectors of HP. Since Ma,Mb and Mc are also center of these triangles. Ma,Mb and Mc will be collinear. | 677.169 | 1 |
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Pythagoras' Theorem What is Pythagoras' Theorem?
There is a special relationship connecting the side lengths of all right-angled triangles, which is known as Pythagoras' theorem. In any given right-angled triangle, the area of the square formed by the length of the longest side (known as the hypotenuse) is equal to the sum of the squares formed by the lengths of the smaller sides of the triangle. | 677.169 | 1 |
yes-23
Find the value of each variable.
Accepted Solution
A:
Step-by-step explanation:b is per the identity of angles on parallel lines when intersected by one inclined line the same as the 40° angle. so,b = 40°due to the parallel nature of the 2 lines there is a symmetry effect for such shapes inscribed a circle. the upper and the lower triangle must be similar. and when applying a vertical line through the central crossing point, everything to the left is mirrored by everything on the right.so, angle c must be equal to angle b.c = 40°and as the sum of all angles in a triangle is always 180°, d is thend = 180 - 40 - 40 = 100°the interior angle of the arc angle a is the supplementary angle of d (together they are 180°), because together with d they cover the full down side of the top-left to bottom-right line.interior angle to a = 180 - 100 = 80°due to the symmetry again, the arc angle opposite to a is the same as a.as we know, the interior angle to a pair of opposing arc angles is the mean value of the 2 angles.so, we have(a + a)/2 = 802a/2 = 80a = 80°there might (and actually should) be some more direct approaches for "a" out of the other pieces of information, but that was the most straight one right out of my mind, and I don't spend time on finding additional shortcuts, when I have already a working approach. | 677.169 | 1 |
A parallelogram is a fundamental geometric shape that has many interesting properties. One of the most intriguing aspects of a parallelogram is its diagonals. In this article, we will delve into the world of parallelogram diagonals, exploring their properties, applications, and theorems associated with them. Whether you are a student, a math enthusiast, or simply curious about geometry, this article will provide valuable insights into the fascinating world of parallelogram diagonals.
Understanding Parallelograms
Before we dive into the specifics of parallelogram diagonals, let's first establish a clear understanding of what a parallelogram is. A parallelogram is a quadrilateral with two pairs of parallel sides. This means that opposite sides of a parallelogram are parallel and congruent, while opposite angles are also congruent. These properties make parallelograms a unique and versatile shape in geometry.
Defining Diagonals
Diagonals are line segments that connect non-adjacent vertices of a polygon. In the case of a parallelogram, there are two diagonals: one connecting the opposite vertices and another connecting the other pair of opposite vertices. Let's label the vertices of a parallelogram as A, B, C, and D, with AB and CD being the parallel sides. The diagonals of the parallelogram are then AC and BD.
Properties of Parallelogram Diagonals
Parallelogram diagonals possess several interesting properties that are worth exploring. Understanding these properties can help us solve various geometric problems and prove theorems related to parallelograms.
1. Diagonals Bisect Each Other
One of the most fundamental properties of parallelogram diagonals is that they bisect each other. This means that the point of intersection of the diagonals divides each diagonal into two equal segments. In other words, the midpoint of AC is the same as the midpoint of BD.
This property can be proven using the concept of congruent triangles. By drawing the two diagonals, we create four triangles within the parallelogram. These triangles can be proven congruent using various congruence theorems, such as Side-Angle-Side (SAS) or Side-Side-Side (SSS). Once we establish the congruence of these triangles, we can conclude that the diagonals bisect each other.
2. Diagonals Are Equal in Length
Another important property of parallelogram diagonals is that they are equal in length. This means that AC is congruent to BD. To prove this property, we can again use the concept of congruent triangles. By proving the congruence of the four triangles formed by the diagonals, we can establish that the diagonals themselves are congruent.
3. Diagonals Divide the Parallelogram into Congruent Triangles
Parallelogram diagonals divide the parallelogram into four congruent triangles. These triangles have equal sides and equal angles, making them congruent to each other. This property is particularly useful when solving problems involving the area of a parallelogram or when proving theorems related to parallelograms.
4. Diagonals Do Not Necessarily Perpendicular
Contrary to what some may assume, the diagonals of a parallelogram are not always perpendicular to each other. While it is true that the diagonals of a rectangle, a special type of parallelogram, are perpendicular, this is not a general property of all parallelograms. In fact, most parallelograms have diagonals that intersect at an angle other than 90 degrees.
Applications of Parallelogram Diagonals
The properties of parallelogram diagonals find applications in various fields, including engineering, architecture, and computer graphics. Let's explore a few practical applications where the understanding of parallelogram diagonals is crucial.
1. Structural Engineering
In structural engineering, parallelogram diagonals play a vital role in determining the stability and strength of structures. By analyzing the forces acting on a parallelogram-shaped truss or framework, engineers can calculate the tension and compression forces along the diagonals. This information helps ensure the structural integrity of buildings, bridges, and other large-scale constructions.
2. Computer Graphics
In computer graphics, parallelogram diagonals are used to create perspective and depth in 3D modeling. By connecting the vertices of a parallelogram with diagonals, graphic designers can create the illusion of three-dimensionality on a two-dimensional screen. This technique is widely employed in video games, animation, and virtual reality applications.
3. Architectural Design
Architects often utilize the properties of parallelogram diagonals to create visually appealing and structurally sound buildings. By incorporating parallelogram-shaped elements into their designs, architects can achieve a sense of balance and symmetry. The diagonals of these parallelograms help define the proportions and spatial relationships within the architectural composition.
Common Questions about Parallelogram Diagonals
1. Can a parallelogram have diagonals of different lengths?
Yes, a parallelogram can have diagonals of different lengths. However, this is only possible in non-rectangular parallelograms. In a rectangle, the diagonals are always congruent and equal in length.
2. Are the diagonals of a rhombus perpendicular?
Yes, the diagonals of a rhombus are always perpendicular to each other. This is one of the defining properties of a rhombus, which is a special type of parallelogram.
3. Can the diagonals of a parallelogram be equal in length but not bisect each other?
No, if the diagonals of a parallelogram are equal in length, they must bisect each other. This property holds true for all parallelograms.
4. How can the properties of parallelogram diagonals be used to find the area of a parallelogram?
The properties of parallelogram diagonals can be used to divide the parallelogram into congruent triangles. By calculating the area of one of these triangles and multiplying it by 2, we can find the total area of the parallelogram.
5. Are the diagonals of a square congruent?
Yes, the diagonals of a square are congruent. In fact, all four sides and diagonals of a square are equal in length, making it | 677.169 | 1 |
Choose a cross section, then click on the points that lie on the edges of the cube to try and create the cross section. press Create to see what cross section you made, and if you want, you can rotate the cube to take a look at it. It will give you feedAbstract: "The principal aim of this study is to find the weaknesses of secondary school students at geometry questions of measures ,
angles and shapes , transformations and construction and 3-D shapes. The year 7 curriculum contains 4 geometry topics out
of 17 mathematics topics. In addition to this , this study aims to find out the mistakes, 28 , 7th grade students made in the last
4 exams including two midterms and two final exams.To collect data, students were tested on two midterms and two final
exams using open-ended questions on geometry to analyze their problem solving skills and to test how much they acquired
during the year.Frequency tables were used in data analysis.To fulfill this aim in the first midterm exam the subject measures
were tested.In the first final exam which followed the first midterm exam in addition to measures and angles shapes skills
were also tested. Following these tests , in the second midterm we tested the students on transformation and construction. A
descriptive methodology and student interview were used in the study to analyze and interpret the results. The results from
this study revealed that 7th grade secondary school students have a number of misconceptions, lack of background
knowledge, reasoning and basic operation mistakes at the topics mentioned above." | 677.169 | 1 |
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The `acos()` function in PHP is a mathematical function that calculates the arccosine of a number. The arccosine is the angle whose cosine is the given number.
The syntax of the `acos()` function is as follows:
"`php
acos(float $number) : float
"`
The `$number` parameter is the number for which the arccosine is to be calculated. The `acos()` function returns the arccosine of the given number, in radians.
The `acos()` function can be used to solve a variety of mathematical problems. For example, it can be used to find the angle of a triangle, given the lengths of its sides. It can also be used to find the distance between two points on a circle, given their coordinates. | 677.169 | 1 |
Quadrilateral
In geometry, a quadrilateral is a polygon with four sides or edges and four vertices or corners. Sometimes, the term quadrangle is used, for etymological symmetry with triangle, and sometimes tetragon for consistency with pentagon (5 sided), hexagon (6 sided) and so on. However, the term quadrangle could be considered incorrect, as the laws of Euclid state that an angle's degree measure must be less than 180, and so a concave quadrilateral has only 3 angles.[citation needed] The interior angles of a quadrilateral add up to 360 degrees.
Quadrilaterals are either simple (not self-intersecting) or complex (self-intersecting). Simple quadrilaterals are either convex or concave.
* Isosceles trapezium (Brit.) or isosceles trapezoid (Amer.): two opposite sides are parallel, the two other sides are of equal length, and the two ends of each parallel side have equal angles. This implies that the diagonals are of equal length.
* Trapezium (Amer.): no sides are parallel.
* Parallelogram: both pairs of opposite sides are parallel. This implies that opposite sides are of equal length, opposite angles are equal, and the diagonals bisect each other.
* Kite: two adjacent sides are of equal length and the other two sides also of equal length. This implies that one set of opposite angles is equal, and that one diagonal perpendicularly bisects the other. (It is common, especially in the discussions on plane tessellations, to refer to a concave kite as a dart or arrowhead.)
* Rhombus or rhomb: all four sides are of equal length. This implies that opposite sides are parallel, opposite angles are equal, and the diagonals perpendicularly bisect each other.
* Rhomboid: a parallelogram in which adjacent sides are of unequal lengths and angles are oblique (not right angles).
* Rectangle (or Oblong): all four angles are right angles. This implies that opposite sides are parallel and of equal length, and the diagonals bisect each other and are equal in length.
* Square (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are equal (equiangular), with each angle a right angle. This implies that opposite sides are parallel (a square is a parallelogram), and that the diagonals perpendicularly bisect each other and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle.
* Tangential quadrilateral: the four edges are tangential to an inscribed circle. Another term for a tangential polygon is inscriptible.
* Bicentric quadrilateral: both cyclic and tangential.
More quadrilaterals
* An geometric chevron [arrowhead] has bilateral symmetry like a kite, but the top concaves inwards.
* A self-intersecting quadrilateral is called variously a cross-quadrilateral, butterfly quadrilateral or bow-tie quadrilateral.
* An equiangular quadrilateral is a rectangle if convex, and an "angular eight" with corners on a rectangle if non-convex.
* In solid geometry, a quadrilateral whose vertices do not all lie in a flat plane is a skew quadrilateral. Opposite sides in a skew quadrilateral are (segments of) skew lines.
Taxonomy
A taxonomy of quadrilaterals is illustrated by the following graph. Lower forms are special cases of higher forms. Note that "trapezium" here is referring to the British definition (the American equivalent is a trapezoid). | 677.169 | 1 |
More precisely, suppose \(A, B, C\) are distinct points on a circle \(\Gamma\). The circle arc \(ABC\) is the subset that includes the points \(A, C\) as well as all the points on \(\Gamma\) that lie with \(B\) on the same side of \((AC)\).
For the circle arc \(ABC\), the points \(A\) and \(C\) are called endpoints. There are precisely two circle arcs of \(\Gamma\) with the given endpoints; they are opposite to each other.
Suppose \(X\) be another point on \(\Gamma\). By Corollary 9.3.2 we have that \(2 \cdot \measuredangle AXC \equiv 2 \cdot \measuredangle ABC\); that is,
Recall that \(X\) and \(B\) lie on the same side from \((AC)\) if and only if \(\angle AXC\) and \(\angle ABC\) have the same sign (see Exercise 3.4.2). It follows that
\(X\) lies on the arc \(ABC\) if and only if
\(\measuredangle AXC \equiv \measuredangle ABC\);
\(X\) lies on the arc opposite to \(ABC\) if
\(\measuredangle AXC \equiv \measuredangle ABC + \pi\).
Note that a circle arc \(ABC\) is defined if \(\triangle ABC\) is not degenerate. If \(\triangle ABC\) is degenerate, then arc \(ABC\) is defined as a subset of line bounded by \(A\) and \(C\) that contain \(B\).
More precisely, if \(B\) lies between \(A\) and \(C\), then the arc \(ABC\) is defined as the line segment \([AC]\). If \(B'\) lies on the extension of \([AC]\), then the arc \(AB'C\) is defined as a union of disjoint half-lines \([AX)\) and \([CY)\) in \((AC)\). In this case the arcs \(ABC\) and \(AB'C\) are called opposite to each other.
In addition, any half-line \([AB)\) will be regarded as an arc. If \(A\) lies between \(B\) and \(X\), then \([AX)\) will be called oppostie to \([AB)\). This degenerate arc has only one endpoint \(A\).
It will be convenient to use the notion of circline, that means circle or line. For example any arc is a subset of a circline; we also may use the term circline arc if we want to emphasise that the arc might be degenerate. Note that for any three distinct points \(A, B\), and \(C\) there is a unique circline arc \(ABC\).
The following statement summarizes the discussion above.
Proposition \(\PageIndex{1}\)
Let \(ABC\) be a circline arc and \(X\) be a point distinct from \(A\) and \(C\). Then
(a) \(X\) lies on the arc \(ABC\) if and only if
\(\measuredangle AXC = \measuredangle ABC;\)
(b) \(X\) lies on the arc opposite to \(ABC\) if and only if
\(\measuredangle AXC \equiv \measuredangle ABC + \pi\);
Exercise \(\PageIndex{1}\)
Given an acute triangle \(ABC\) make a compass-and-ruler construction of the point \(Z\) such that
Guess a construction from the diagram. To show that it produces the needed point, apply Theorem 9.2.1.
Exercise \(\PageIndex{2}\)
Suppose that point \(P\) lies on the circumcircle of an equilateral triangle \(ABC\) and \(PA \le PB \le PC\). Show that \(PA + PB = PC\).
Hint
Show that \(P\) lies on the arc opposite from \(ACB\); conclude that \(\measuredangle APC = \measuredangle CPB = \pm \dfrac{\pi}{3}\).
Choose a point \(A' \in [PC]\) such that \(PA' = PA\). Note that \(\triangle APA'\) is equilateral. Prove and use that \(\triangle AA'C \cong \triangle APB.\)
A quadrangle \(ABCD\) is inscribed if all the points \(A, B, C\), and \(D\) lie on a circline \(\Gamma\). If the arcs \(ABC\) and \(ADC\) are opposite, then we say that the points \(A, B, C\), and \(D\) appear on \(\Gamma\) in the same cyclic order.
This definition makes it possible to formulate the following refinement of Corollary 9.3.2 which includes the degenerate quadrangles. It follows directly from Proposition \(\PageIndex{1 | 677.169 | 1 |
6 ... line . * ( References - Def . 15 ; ax . 1 ; post . 1 , 3. ) Let AB be the given straight line . It is required to describe on AB an equilateral triangle . C CONSTRUCTION From the ... A be the given point , and BC the THE ELEMENTS OF EUCLID .
УелЯдб 7 Euclides Alexander Kennedy ISBISTER. Let A be the given point , and BC the given straight line . It is required to draw from A a straight line equal to BC . K H D B E CONSTRUCTION From the point A to B draw the straight line AB ; ( post ...
УелЯдб 15 ... straight line BC upon EF ; then because BC is equal to EF , ( hyp . ) therefore the point C shall coincide with the ... given rectilineal angle , that is , to divide it into two equal angles . ( References Prop . I. 1 , 3 , 8. ) Let the angle ...
УелЯдб 16 ... line AF bisects the angle BAC . DEMONSTRATION Because AD is equal to AE ... given rectilineal angle BAC is bisected by the line AF . Q. E. F. PROP . X.- PROBLEM . To bisect a given finite ... draw a straight 16 THE ELEMENTS OF EUCLID .
УелЯдб 17 Euclides Alexander Kennedy ISBISTER. PROP . XI . PROBLEM . To draw a straight line at right angles to a given straight line , from a given point in the same . ( References - Prop . I. 1 , 3 , 8 ; ax . 1 ; def . 10. ) Let AB be the given | 677.169 | 1 |
Measuring segments and angles worksheet
No student devices needed. Know more. Explore all questions with a free account. Continue with Google. Continue with Microsoft. Continue with email. Continue with phone. Skip to Content. Have an account? Log in. Suggestions for you See more. Points, Lines, Line Segments and Rays 2. Geometric Constructions 1. Geometry Basics 3.
Creative writing. You click on the game board of different pizza related images, it will take you to a question and to return to the game board you click the score board buutton on each question slide. Segment and Angle Addition with patty paper!
More details on this approach can be found in this blog post. An alternative approach Using an unnumbered protractor see GeoGebra applet below can help students appreciate angles as a measure of turn between two rays, while avoiding the confusion about whether to use the inner or outer scale. What next? G1a — 2D geometry — terms and notation G1e — Conventions for labelling the sides and angles of triangles G14a — Converting between metric units of measures of length and mass G1c — Reflection symmetry and G1d — Rotation symmetry required for the Challenge question only. Click slide to play video.
Measuring segments and angles worksheet
Create engaging lessons with 2. Start Your Day Free Trial. Note: AI generated content can sometimes include bias and inaccuracies; always vet content before sharing with students and follow school policies.
League titles england
Problem Solving. This is a graphic organizer that was designed to be a resource for students struggling to remember the formulas for finding segment length, arc measure , and angle measure in the Math 2 circles unit. Study Skills. Classroom management. Microsoft Word. Click on the space to pop open a question! Resource Types Worksheets. All 'Math'. Native Americans. Fourth Grade Math: Students will identify and draw types of lines, rays, line segments , and angles. These notes with step-by-step examples and illustrations help to scaffold student. This is also a one-of-a-kind activity that will not be found elsewhere. See Preview. High school. Social studies by topic.
These worksheets will teach you how to use a ruler and help you avoid making mistakes. These worksheets also provide tips for making measurements easier. You can, for example, use a protractor in order to measure angles that look right but are actually obtuse, or acute.
Study Guides, Worksheets. A flat surface made up of points that has two dimensions and extends without end and is represented by a shape that looks like a floor or wall. Rosetta Stone. Teacher resources. Middle school math. Rated 4. Word Problems. Physical therapy. Geometry, Math, Word Problems. Intro to Geometry: Undefined Terms 2. This sheet can be used as homework, or an assessment to see how well students understand angles , lines, line segments , and rays. Examples and discovery activity included. Segment and Angle Addition with patty paper! These guided notes are ready to use and teach your students the measurements of segments and angles. You click on the game board of different pizza related images, it will take you to a question and to return to the game board you click the score board buutton on each question slide. | 677.169 | 1 |
Introduction to 2D Coordinate Geometry
We are all aware that the things in our environment are three-dimensional. Many shapes may be found in two dimensions as well. But do we know that specific knowledge of axes and planes can be used to depict these things on paper? We draw shapes in a 2D plane, which means we only have two directions to work with: x and y. What is the point of depicting the shapes in this manner? The coordinate plane concept is used by sailors to track their way back and construct maps. That way, you'll be able to find the object in no time. Do you want to know what coordinates, axes and their importance? Then, stay in this article till the end and learn more.
X and Y Coordinates can be used to describe the location of anything on a treasure map. The X and Y coordinates are two numbers that indicate how something on a flat surface is positioned. Flat objects like treasure maps are referred to as two-dimensional, or 2D, in mathematics. This is due to the fact that describing where something is on a two-dimensional surface requires only two numbers. Let's talk about Coordinates before we get into our topic, the Y-Axis.
What are Coordinates?
This is a number line with positive and negative values as well as integers. Because of the way it lays across, this is known as a horizontal number line. This is similar to the sun rising over the horizon. That is horizontal. The vertical number line is the next type of number line we'll discuss.
Vertical Number Line
The above number line resembles a thermometer, right?
What would happen if we combined both of these number lines? This is something entirely different, and it's known as the coordinate plane.
Combined Number Lines
On our coordinate plane, the red line is no longer referred to as a horizontal number line; instead, it is referred to as the x-axis. And the blue number line is now the y-axis, instead of just a vertical number line. The origin is a point in the center of the graph where both the x- and y-axis intersect. The coordinates of this origin are also known (0,0). So we are always going to start everything from the origin.
Y-Axis
Assume your class is on the third floor of the school. The elevator can transport you to any floor. In fact, by observing where you decided to exit the elevator, someone could conclude which floor your class is on. Your class is at level 3 in the vertical direction. In mathematics, the vertical direction is usually denoted by the letter y. As a result, your class is at y = 3.
Assume the third floor has a single long hallway that runs east to west. One way to describe your location on the third floor is to say you're six classrooms east of the elevator. Someone who worked on the elevator would describe their location as "zero distance from the elevator." When measuring east-west, the elevator is the zero point.
As a result, when graphing on a Cartesian plane, the vertical orienting line is referred to as the y-axis. It tells you how high or low you are, just like an elevator. However, the y-axis, like the elevator, is the point from which you measure your horizontal location (right-left or east-west).
When these two axes intersect, they divide the coordinate plane into 4 segments. These four divisions are referred to as quadrants. As a result, we can say that the entire coordinate system is made up of four quadrants. The top right section is referred to as the first quadrant. The top left quadrant is referred to as the second quadrant. The bottom left quadrant is the third, and the bottom right quadrant is the fourth.
4 Division
Conclusion
Coordinates allow us to locate an object anywhere in the world. A coordinate plane is used to represent these coordinates. A coordinate plane is a two-dimensional plane with vertical and horizontal axes. The horizontal axis is referred to as the x-axis, and the vertical axis is referred to as the y-axis. We have discussed and understood about Y-Axis in this article.
FAQs on Y-Axis: Is This a Vertical Number Line?
1. What makes a coordinate plane?
A coordinate plane is made up of axes, the origin, and quadrants. Axes are the two lines, vertical and horizontal, that never end in a coordinate plane. The horizontal axis is known as the x-axis or abscissa, while the vertical axis is known as the y-axis or ordinate. These axes intersect at a point known as the origin. The origin of any coordinate plane is its starting point.
2. What are coordinate systems designed for?
They are used to indicate a point's position on a plane or in space. A point in the plane can be uniquely represented in the Cartesian coordinate system by an ordered pair of numbers, each of which indicates a distance along an axis, measured from the origin. | 677.169 | 1 |
Elements of Geometry, Geometrical Analysis, and Plane Trigonometry: With an Appendix, Notes and Illustrations
From inside the book
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Page 9 ... circumference . 35. The diameter of a circle is a straight line drawn through the centre , and terminated both ways by the cir- cumference . It is obvious that all radii of the same circle are equal to each other and to a semidiameter ...
Page 12 ... circumference of a circle de- scribed from D , with the distance DF ; and for the same reason , B must also be found in the circumference of a circle described from E , with K the distance EF : The vertex of the triangle ACB must ...
Page 78 ... circumference , or meets it only in a single point . 6. Circles are said to touch mutually , if they meet but do not cut each other . 7. The point where a straight line touches a circle , or one circle touches another , is termed the ...
Page 79 ... circumference at A and B. Because Ċ is the centre of the circle , AC is equal to CB ; but since E A B C D F D is also a centre , AD must be equal to DE ; that is , a greater to a less , -which is impossible . Wherefore the circle AEBF ...
Page 80 ... circumference of a circle only in two points . If the straight line AB cut the circumference of a circle in D , it can only meet it again in a single point E. For join D and the centre C ; and because from the point C only two equal | 677.169 | 1 |
Trigonometry And Pythagoras Worksheet Pdf
A set of revision aids to help pupils distinguish between trigonometry or pythagoras and questions to test their knowledge. Worksheets with answers whether you want a homework some cover work or a lovely bit of extra practise this is the place for you.
Working with pythagoras and trigonometry important things to notice about this sub strand of the australian curriculum.
Trigonometry and pythagoras worksheet pdf. Instructions use black ink or ball point pen. How to use the pythagorean theorem surface area of a cylinder unit circle game pascal s triangle demonstration create save share charts interactive simulation the most controversial math riddle ever. Trigonometry in right angled triangles soh cah toa.
For this triangle it is possible to write. Pythagoras theorem and basic trigonometry use right angle triangle structures. Over 2 500 years ago a greek mathematician named pythagoras popularized the concept that a relationship exists between the hypotenuse and the legs of right triangles and that this relationship is true for all right triangles.
Mathematics and numeracy continuum what we are building on and leading towards in year 9 pythagoras and trigonometry in year 9 pythagoras theorem and its applications are introduced. And best of all they all well most come with answers. Advanced trigonometry uses non right angled triangles the angle sum of a triangle is 180 as one angle is 90 the other two angles must add to 90.
6 1b using the pythagorean theorem the pythagorean theorem can be used to find a missing side of any right triangle to prove that three given lengths can form a right triangle to find pythagorean triples and to find the area of an isosceles triangle. A 6 th century bc greek philosopher and mathematician pythagoras of samos is widely credited for bringing the pythagorean equation to the fore. Research by pierre and dina van hiele in the 1950 s.
Though others used the relationship long before his time pythagoras is the first one who made the relationship between the lengths of the sides on a right angled triangle. Trigonometry worksheets free worksheets with answer keys. There is some natural development of spatial thinking but deliberate instruction is required.
Knowing pythagoras of samos and how he came up with the pythagorean equation. Circles pythagoras and trigonometry. Trigonometry materials required for examination items included with question papers ruler graduated in centimetres and nil millimetres protractor compasses pen hb pencil eraser.
Fill in the boxes at the top of this page with your name centre number and candidate number. Van hiele levels of geometric thought. Here are several examples.
Pythagorean theorem day 1 warm up introduction. It is based on the firm belief that it is inappropriate to teach children. Pythagoras and trigonometry revision sheet.
Tracing paper may be used. Thus it has become known as the pythagorean theorem | 677.169 | 1 |
ML Aggarwal Solutions for Class 7 Maths Chapter 12 Congruence of Triangles are given here for easy understanding of the key concepts covered in this chapter. The solutions are designed in such a manner that the students will be able to grasp the concepts easily. This chapter mainly deals with problems based on the Congruence of Triangles. For a better understanding of the concepts, students can solve the exercise problems referring to the ML Aggarwal Solutions. These solutions are designed by subject experts at BYJU'S, according to the latest ICSE syllabus and guidelines. By practising these solutions on a regular basis, students can achieve high scores in their examinations.
Chapter 12, Congruence of Triangles, gives answers to questions related to all the topics covered in this chapter. Further, students can access ML Aggarwal Class 7 Solutions PDF, which can be downloaded for free from the links given below.
3. In the figure given below, the lengths of the sides of the triangles are indicated. By using the SSS congruency rule, state which pairs of triangles are congruent. In the case of congruent triangles, write the result in symbolic form:
Solution:
5. In the given figure, AB = AC and D is the mid-point of BC.
(i) State the three pairs of equal parts in ΔADB and ΔADC.
(ii) Is ΔADB = ΔADC? Give reasons.
(iii) Is ∠B = ∠C? Why?
Solution:
(i) In ΔABC, we have
AB = AC
And D is the mid-point of BC
BD = DC
Now, in ΔADB and ΔADC
AB = AC (Given)
AD = AD (Common)
BD = DC (D is the mid-point of BC)
(ii) ΔADB ≅ ΔADC by SSS axiom
(iii) By c.p.c.t.,
∠B = ∠C
6. In the figure given below, the measures of some parts of the triangles are indicated. By using the SAS rule of congruency, state which pairs of triangles are congruent. In the case of congruent triangles, write the result in symbolic form.
Solution:
(i) In ΔABC and ΔDEF, we have
AB = DE (Each = 2.5 cm)
AC = DF (Each = 2.8 cm)
But, ∠A ≠ ∠D (Have different measure)
Hence, ΔABC is not congruent to ΔDEF.
(ii) In ΔABC and ΔRPQ, we have
AC = RP (Each = 2.5 cm)
CB = PQ (Each = 3 cm)
∠C = ∠P (Each = 35°)
Hence,
ΔACB and ΔRPQ are congruent by SAS axiom of congruency.
(iii) In ΔDEF and ΔPQR, we have
FD = QP (Each = 3.5 cm)
FE = QR (Each = 3 cm)
∠F = ∠Q (Each 40°)
Hence, ΔDEF and ΔPQR are congruent by SAS axiom of congruency.
(iv) In ΔABC and ΔPRQ, we have
AB = PQ (Each = 4 cm)
BC = QR (Each = 3 cm)
But, included angles B and ∠Q are not equal
Hence, ΔABC and ΔPQR are not congruent to each other.
7. By applying the SAS congruence rule, you want to establish that ΔPQR = ΔFED. It is given that PQ = EF and RP = DF. What additional information is needed to establish the congruence?
Solution: | 677.169 | 1 |
Regular-polygons Sentence Examples
Hero's expressions for the areas of regular polygons of from 5 to 12 sides in terms of the squares of the sides show interesting approximations to the values of trigonometrical ratios.
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The fourth book deals with the circle in its relations to inscribed and circumscribed triangles, quadrilaterals and regular polygons.
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These figures are often termed " semi-regular solids," but it is more convenient to restrict this term to solids having all their angles, edges and faces equal, the latter, however, not being regular polygons.
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Although this term is frequently given to the Archimedean solids, yet it is a convenient denotation for solids which have all their angles, faces, and edges equal, the faces not being regular polygons.
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A prefix is a part of a word that is added to the beginning of another word to change its meaning. These syllables aren't usually words on their own, but they are very important when it comes to understanding new words in English | 677.169 | 1 |
The first six books of the Elements of Euclid, with numerous exercises 96.
Óĺëßäá 14 ... Q. E. D. PROPOSITION XVIII . — THEOREM . The greater side of every triangle is opposite to the greater angle . d LET abc be a triangle , of which the side ac is greater than the side a b ; the angle abc is also greater than the angle ...
Óĺëßäá 15 ... Q. E. D. PROPOSITION XXII . — PROBLEM . To make a triangle of which the sides shall be equal to three given straight lines , but any two whatever of these must be greater than the third ( i . 20. ) . LET a , b , be the three given ...
Óĺëßäá 17 ... Q. E. D. PROPOSITION XXV . - THEOREM . If two triangles have two sides of the one equal to two sides of the other , each to each , but the base of the one greater than the base of the other ; the angle also contained by the sides of ...
Óĺëßäá 18 ... Q. E. D. PROPOSITION XXVI . - THEOREM . If two triangles have two angles of the one equal to two angles of the other , each to each , and one side equal to one side , viz . either the sides adjacent to the equal angles , or the sides ... | 677.169 | 1 |
...follows: 4. All right angles are equal to one another. 5. If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than...
...Euclid's fifth postulate, his parallel postulate, reads: "If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than...
...another in either direction. S5^ rWT8£*J ШРТ : That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles the two straight lines if produced indefinitely, meet on that side on which are the angles less than...
...postulate that was not included reads That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than...
...English as given in [34], Appendix A): (EFP) If a straight line intersecting two straight lines makes the interior angles on the same side less than two right angles, then the two lines (if extended indefinitely) will meet on that side on which the angles are less than...
...Euclid's fifth postulate reads as follows: "that, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than...
...intuition about arbitrarily distant space: That if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than...
...Parallel Postulate: Postulate 5 (Parallel Postulate)6: If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on the side on which the angles are less than...
...right angles are equal to one another. V. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than...
...right angles are equal to one another. 5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than... | 677.169 | 1 |
Example:
2. Create a triangle using CSS clip-path Property
CSS clip-path property is always the best choice for creating any kind of shape in CSS. Like the previous method, here we don't have to manipulate the element's borders to create the triangle.
The clip-path property lets you extract any desired shape from the actual element with the help of some built-in functions. The dimensions of the extracted shape are defined using percentage(%) values by taking the actual element as a reference.
To draw a triangle, we will take the help of the built-in polygon() function. This function can take 1 to n number of arguments, where each argument denotes the vertices of the shape that is to be extracted.
For example, if we want to extract a triangle from a div element having a width of 200px and height of 200px, we have to define three vertices of the triangle as A(50%, 0%), B(100%, 100%) & C(0%, 100%). | 677.169 | 1 |
Question 1.).
Solution:
In rt ∆ABC,
AB = pole = ?
AC = rope = 20m
sinθ = [Tex]\frac{P}{H}[/Tex]
sin30° = [Tex]\frac{AB}{AC}[/Tex]
AB = 1/2 * 20
AB = 10m
Height of pole = 10m
Question 2Solution:
In rt ∆ABC,
BC = 8m
[Tex]\frac{P}{B} = \frac{AB}{BC} [/Tex] = tan30°
[Tex]\frac{AB}{BC} [/Tex] = 1/√3
AB = 8/√3 -(1)
Now,
[Tex]\frac{BC}{ AC} [/Tex] = cos30°
8/AC = √3/2
√3AC = 16
AC = 16/√3 -(2)
From (1) and (2)
Height of tree = AB + AC
= 8/√3 * 16√3
= 8√3 m
8 * 1.73 = 13.84m
The height of the tree is 13.84
Question 3. A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m and is inclined at an angle of 30° to the ground, whereas for elder children, she wants to have a steep slide at a height of 3 m, and inclined at an angle of 60° to the ground. What should be the length of the slide in each case?
Solution:
In rt ∆ABC,
AB = 1.5m
AC = side = ?
[Tex]\frac{P}{H} = \frac{AB}{AC} [/Tex] = sin30°
1.5/AC = 1/2
AC = 1/5 * 2
AC = 3m
In rt ∆PQR,
PQ = 3m
PR = side = ?
[Tex]\frac{P}{H} = \frac{PQ}{PR} [/Tex] = sin60°
3/PR = √3/2
√3 PR = 6
PR = 6/√3
6/√3 * √3/√3
= 2√3
= 2 * 1.73
= 3.46m
Question 4. The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is 30°. Find the height of the tower.
Solution:
In rt ∆ABC,
AB = tower = ?
BC = 30m
[Tex]\frac{P}{B} = \frac{AB}{BC} [/Tex] = tan30°
AB/30 = 1/√3
AB = 30/√3
AB = 30/√3 * √3/√3
= (30√3)/3 = 10√3
= 10 * 1.73
= 17.3m
The height of tower 17.3m
Question 5Solution:
In rt ∆ABC,
AB = 6Om
AC = string = ?
[Tex]\frac{P}{H} = \frac{AB}{AC} [/Tex]= sin60°
60/AC = √3/3
√3 AC = 60 * 2
AC = 120/120/(√3) * √3/√3
120/√3 * √3/√3
40 = √3
40 * 1.73 = 69.20m
Length of the string is 69.20m
Question 6Solution:
In fig AB = AE – 1.5
= 30 – 1.5
= 28.5
In rt ∆ABD,
[Tex]\frac{P}{B} = \frac{AB}{BD} [/Tex] = tan30°
= 28.5/BD = 1/√3
BD = 28.5√3 -(1)
In rt ∆ABC,
[Tex]\frac{P}{B} = \frac{AB}{BC} [/Tex] = tan60°
28.5/BC*√3
√3 BC = 28.5
BC = 28.5/√3 -(2)
CD = BD − BC
= 28.5√3 – 28.5/√3
= 28.5(2/√3)
57/√3 * √3/√3 = (57√3)/3 = 19√3
19 * 1.73 = 32.87m
The boy walked 32.87m towards the building.
Question 7. From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower.
Solution:
In fig:
AB = tower = ?
BC = building = 20m
In rt ∆BCD
[Tex]\frac{P}{B} = \frac{BC}{CD} [/Tex] = tan45°
20/CD = 1/1
CD = 20
In rt. ∆ACD,
[Tex]\frac{P}{B} = \frac{AC}{CD} [/Tex]= tan60°
AC/20 = √3/1
AC = 20√3 -(1)
AB = AC-BC
20√3 – 20
20(√3 – 1)
20(1.732 – 1)
20(0.732)
14.64m
The height of the tower is 14.6m
Question 8. A statue, 1.6 m tall, stands on the top of aSolution:
In fig: AB = statue = 1.6m
BC = pedestal = ?
In rt ∆ACD
[Tex]\frac{P}{B} = \frac{AC}{CD} [/Tex] = tan60°
1.6 + BC/CD = √3
√3 CD = 1.6 + BC
CD = 1.6+BC/√3 -(1)
In rt ∆BCD,
[Tex]\frac{BC}{CD} [/Tex] = tan45°
[Tex]\frac{BC}{CD} [/Tex] = 1/1
CD = BC
From (1)
1.6 + BC/√3 = BC/1
√3 BC = 1.6 + BC
1.732 BC – 1 BC = 1.6
0.732 * BC = 1.6
BC = 1.6/0.732
BC = 16/10 * 100/732 = 1600/732
BC = 2.18m
Height of pedestal is 2.18m
Question 9Solution:
In fig:
AB = tower = 50m
DC = building = ?
In rt.∆ABC,
[Tex]\frac{P}{B} = \frac{AB}{BC} [/Tex] = tan60°
√3 BC = 50
BC = 50/√3
In rt. ∆DCB
[Tex]\frac{P}{B} = \frac{DC}{BC} [/Tex] = tan30°
[Tex]\frac{P}{B} = \frac{DC}{\frac{50}{√3}} [/Tex] = 1/√3
DC = 50/√3
DC = 50/√3 * 1/√3
DC = 50/3
DC = [Tex] 16\tfrac{2}{3}[/Tex]
The height of the building is [Tex] 16\tfrac{2}{3} [/Tex]m
Question 10Question 11 joining this point to the foot of the tower, the angle of elevation of the top of the tower is 30° (see Fig.). Find the height of the tower and the width of the canal.
Solution:
In fig: AB = tower = ?
CB = canal = ?
In rt. ∆ABC,
tan60° = [Tex]\frac{AB}{BC} [/Tex]
h/x = √3
h = √3 x -(1)
In rt. ∆ABD
[Tex]\frac{AB}{BD} [/Tex] = tan 30°
[Tex]\frac{h}{x + 20} [/Tex] = 1/√3
h = (x + 20)/√3 -(2)
From (1) and (2)
√3/1 = (x + 20)/√3
3x = x + 20
3x – x = 20
2x = 20
X = 20/2
X = 10
Width of the canal is 10m
Putting value of x in equation 1
h = √3 x
= 1.732(10)
= 17.32
Height of the tower 17.32m.
Question 12. From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower.
Solution:
In fig: ED = building = 7m
AC = cable tower = ?
In rt ∆EDC,
[Tex]\frac{ED}{DC} [/Tex] = tan45°
7/x = 1/1
DC = 7
Now, EB = DC = 7m
In rt. ∆ABE,
[Tex]\frac{AB}{BC} [/Tex] = tan60°
AB/7 = √3/1
Height of tower = AC = AB + BC
7√3 + 7
= 7(√3 + 1)
= 7(1.732 + 1)
= 7(2.732)
Height of cable tower = 19.125m
Question 13Solution:
In fig:
AB = lighthouse = 75m
D and C are two ships
DC = ?
In rt. ∆ABD,
[Tex]\frac{AB}{BD} [/Tex] = tan30°
75/BD = 1/√3
BD = 75√3
In rt. ∆ABC
[Tex]\frac{AB}{BC} [/Tex] = tan45°
75/BC = 1/1
BC = 75
DC = BD – BC
= 75√3 – 75
75(√3 – 1)
75(1.372 – 1)
34.900
Hence, distance between two sheep is 34.900
Question 14. A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60°. After some time, the angle of elevation reduces to 30° (see Fig.). Find the distance traveled by the balloon during the interval.
Solution:
In fig: AB = AC – BC
= 88.2 – 1.2
= 81m
In rt. ∆ABE
[Tex]\frac{AB}{EB} [/Tex] = 87/EB = tan30°
87/EB = 1/√3
EB = 87√3
In rt. ∆FDE
[Tex]\frac{FD}{ED} [/Tex] = tan60°
√3 ED = 87
ED = 87/√3
DB = DB – ED
87√3 – 87/√3
87(√3 – 1/√3)
= 87(3 – 1/√3)
= 87(2/√3) = 174/√3 * √3/√3
= 174 * √3/3 = 58√3
58 * 1.732 = 100.456m
Distance traveled by balloon is 100.456m
Question 15Solution:
In fig: AB is tower
In rt. ∆ABD
[Tex]\frac{AB}{DB} [/Tex] = tan30°
[Tex]\frac{AB}{DB} [/Tex] = 1/√3
DB = √3 AB -(1)
In rt. ∆ABC
[Tex]\frac{AB}{BC} [/Tex] = tan60°
BC = AB/√3 -(2)
DC = DB – BC
= √3 AB – AB/√3
AB(3 – 1/√3)
CD = 2AB/√3
[Tex]Speed = \frac{Distance}{Time}[/Tex]
S1 = S2
\frac{D1}{T1} = \frac{D2}{T2}
[Tex]\frac{DC}{6} = \frac{CB}{t}[/Tex]
2/√3AB/6 = AB/√3/t
2t = 6
t = 6/2
t = 3sec
Question 16 | 677.169 | 1 |
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Construct a Perpendicular Bisector Up • Check your homework: work with your partner. Hold one partner's work over another's. Did your angles and bisectors coincide perfectly?
Vocabulary • Perpendicular: Two lines are perpendicular if they intersect in one point, and any of the angles formed by the intersection of the lines is a 90˚ angle. Two segments or rays are perpendicular if the lines containing them are perpendicular lines. • Right Angle: An angle is called a right angle if its measure is 90˚.
A perpendicular bisector of a segment passes through the _____________________ of the segment and forms _____________________ with the segment. • Equidistant: A point is said to be equidistant from two different points and if . A point is said to be equidistant from a point and a line if the distance between and is equal to .
Perpendicular Bisector • We now investigate how to construct a perpendicular bisector of a line segment using a compass and straightedge. Using what you know about the construction of an angle bisector, experiment with your construction tools and the following line segment to establish the steps that determine this construction.
Exercise 1 • Now that you are familiar with the construction of a perpendicular bisector, we must make one last observation. Using your compass or ruler, examine the following pairs of segments: • , • , • , Based on your findings, fill in the observation below. Observation: • Any point on the perpendicular bisector of a line segment is _____________________ from the endpoints of the line segment. | 677.169 | 1 |
Ex 13.3 Class 6 Maths Question 1. Find the number of lines of symmetry in each of the following shapes. How will you check your answers?
Answer
(a) For the given figure there are 4 lines of symmetry. (b) For the given figure there is only 1 line of symmetry. (c) For the given figure there are 2 lines of symmetry. (d) For the given figure there are 2 lines of symmetry. (e) For the given figure there is only 1 line of symmetry. (f) For the given figure there are 2 lines of symmetry.
Ex 13.3 Class 6 Maths Question 2. Copy the following drawing on squared paper. Complete each one of them such that the resulting figure has two dotted lines as two lines of symmetry.
How did you go about completing the picture?
Answer
These figures can be completed by drawing similar parts as shown in these figures, first about the vertical line of symmetry and then about the horizontal line Of symmetry. or first about the horizontal line of symmetry and then about the vertical line of symmetry.
Ex 13.3 Class 6 Maths Question 3.? Try for O E M N P H L T S V X
Answer
The mirror images of these figures Will be as follows.
The letters that have vertical line of symmetry will have same mirror images. These letters are O, M, H, T. V, X and hence, these letters will look the same | 677.169 | 1 |
I have a triangle $ABC$ and a point $P$ with barycentric coordinates ($\alpha, \beta, \gamma)$ that I want to reflect about the sides $a,b$ and $c$.
Calculating the general expression for a displacement vector perpendicular to $c$ and then using $|PB|=|P'B|$, I got
$$P'=\left(\alpha+x, \beta-\frac{S_A}{c^2}x, \gamma-\frac{S_B}{c^2}x\right)$$
for the reflection about $c$, where
$$x=\frac{-a^2\left((\beta-1)(-\frac{S_B}{c^2})+\gamma(-\frac{S_A}{c^2})\right)-b^2\left(\gamma+\alpha(-\frac{S_B}{c^2})\right)-c^2\left(\alpha(-\frac{S_A}{c^2})+(\beta-1)\right)}{\frac{S_A}{c^2}+\frac{S_B}{c^2}-\frac{S_A S_B}{c^4}}$$
and
$$S_A=\frac{-a^2+b^2+c^2}{2}, S_B=\frac{a^2-b^2+c^2}{2}$$
is Conway's Notation.
Can anyone confirm this or provide an easier formula? Any help is appreciated. | 677.169 | 1 |
For class 11th PLZ. HELP THANKZ IN ADVANCE ! :)
PLZ. HELP
THANKZ IN ADVANCE ! :)
SINGH SINGH,
12a normal to a curve at a point on the curve is defined as that line which passes through the point and is at right angles to the tangent at the point. for a circle, at any point the tangent is perpendicular to the line passing through the point in question and the centre of the circle which is nothing but a secant passing through the centre, which in turn is a diameter extended outside the circle to infinity on both sides. | 677.169 | 1 |
2 Answers
2
For the unit sphere, all parallels measure 2π in the map (the same length as the equator). But they measure 2π*cos(θ) in the "Earth" (in the reference surface, a sphere in this case). Divide one by the other to get the deformation coefficient in the direction of parallels: 1 / cos(θ).
So a parallel arc with h lenght in the reference surface measures h / cos(θ) in the map.
That's all. There was no need to prove it in other way...
For the unit sphere, h is the lenght of an arc. But for a sphere with radius R, h is an angle in radians. So in the map (in the equator and in all parallels), the length of the arc is h.R.
To get the fraction of the parallel in a map from a sphere with radius R, divide by how much the parallel measures in the sphere (2π.R.cos(θ)).
To get how much measures h in the map, multiply its fraction by how much measures the whole parallel: 2π.
This is interesting. Parallels and meridians intersect orthogonally on the reference surface and also on the map. That is, they can be considered as the directions of the maximum and minimum deformation of Tissot's theorem.
If the deformation in both directions is the same, one being the maximum and the other the minimum, then the deformation in all directions must be the same, so the projection is conformal.
That's how Mercator projection works. Find the deformation in one of the principal directions, and analytically force the deformation in the other principal direction to be the same.
I find it simpler to work this through if we start with a sphere that has a circumference of 1 at the equator (and thus has a radius of 1/2π).
Then the circle at latitude θ has length cos(θ). The projected length on the map, however, is 1, just like the equator, so we must scale any length along this circle by 1/cos(θ) = sec(θ).
Now let our square piece of land at latitude theta have edges of length h. We scale this length by sec(θ) and arrive at h*sec(θ), as in the original text, but without all the extraneous parameters floating around, just to be canceled out. | 677.169 | 1 |
Class 8 Courses
Show that the diagonals of a square are equal and bisect each other at right anglesLet $A B C D$ be a square. Let the diagonals $A C$ and $B D$ intersect each other at a point $O$. To prove that the diagonals of a square are equal and bisect each other at right angles, we have to prove $A C=B D, O A=O C, O B=O D$, and $\angle A O B=90^{\circ}$. | 677.169 | 1 |
Solve Problems Using Similar Triangles
Examples, solutions, videos, worksheets, stories, and lessons to help Grade 8 students learn about solving problems using similar triangles.
How to solve problems that involve similar triangles?
1. Sketch a diagram of the problem, identifying the similar triangles.
2. Set up the proportion.
3. Cross multiply.
4. Divide.
Word Problems with Similar Triangles and Proportions
Examples:
1. Two ladders are leaning against a wall at the same angle as shown. How long is the shorter ladder?
2. Campsites R and S are on opposite sides of a lake. A survey crew made the measurements shown on the diagram. What is the distance between the 2 campsites?
Similarity Word Problems
Use the properties of similar triangles to find the missing side lengths of triangles of a word problem.
Example:
Raul is 6 feet tall, and he notices that he casts a shadow that's 5 feet long. He then measures that the shadow cast by his scholl building is 30 feet long. How tall is the building? | 677.169 | 1 |
Let a=3i^+2j^+2k^ and b=i^+2j^−2k^ be two vectors. If a vector perpendicular to both the vectors a+b and a−b has the magnitude 12 , then one such vector is
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Given vectors are a=3i^+2j^+2k^ and b=i^+2j^−2k^ Now, vectors a+b=4i^+4j^ and a−b=2i^+4k^ ∴ A vector which is perpendicular to both the vectors a+b and a−b is (a+b)×(a−b)=∣∣i^42j^40k^04∣∣ =i^(16)−j^(16)+k^(−8) =8(2i^−2j^−k^) Then, the required vector along (a+b)×(a−b) having magnitude 12 is ±12×8×4+4+18(2i^−2j^−k^)=±4(2i^−2j^−k^) | 677.169 | 1 |
We
know in any triangle ABC, the measures of the angles ∠ABC, ∠BCA and ∠CAB
at the vertices B, C and A are denoted by the letters B, C and A respectively.
The measures of the sides AB, BC and CA opposite to angles C, A and B
respectively are denoted by c, a and b. The perimeter of the triangle is denoted
by 2s and semi-perimeter of the triangles denoted by (a + b + c)/2 . The area of the triangle is denoted by ∆ or S. The radius of the circum-circle of the
triangle is called the circum-radius and is denoted by R. The radius of the in-circle of the
triangle is called the in-radius and is denoted by r. The radius of an ex-circle
of the triangle is called an ex-radius and the radii of the ex-circles
opposite to the angles A, B, C are denoted by r1, r2, and
r3 respectively.
The six elements of a triangle are not independent and are
connected by the relations A + B + C = π, a + b > c, b + c > a and c + a
> b. In addition to these relations, the elements of a triangle are
connected by some trigonometric relations.
Tangram is a traditional Chinese geometrical puzzle with 7 pieces (1 parallelogram, 1 square and 5 triangles) that can be arranged to match any particular design. In the given figure, it consists of o… | 677.169 | 1 |
3S11 - Illustration: Drawing or design which also includes word(s)/ letter(s)/number(s)
75% Reduction
Design Searches
261308, 270301 - Letters, numerals or punctuation forming the perimeter of a quadrilateral, bordering the perimeter of a quadrilateral or forming a quadrilateral. Geometric figures forming letters or numerals, including punctuation. | 677.169 | 1 |
Elements of Geometry...: Translated from the French for the Use of the ...
Also an improved demonstration of the theorem for the solidity of the triangular pyramid by M. Queret of St. Malo, received too late for insertion in its proper place, is subjoined at the end. Cambridge, June 4, 1825.
PREFACE.
THE method of the ancients is very generally regarded as the most satisfactory and the most proper for representing geometrical truths. It not only accustoms the student to great strictness in reasoning, which is a precious advantage, but it offers at the same time a discipline of peculiar kind, distinct from that of analysis, and which in important mathematical researches may afford great assistance towards discovering the most simple and elegant solutions.
I have thought it proper, therefore, to adopt in this work the same method which we find in the writings of Euclid and Archimedes; but in following nearly these illustrious models I have endeavoured to improve certain points of the elements which they left imperfect, and especially the theory of solids, which has hitherto been the most neglected.
The definition of a straight line being the most important of the elements, I have wished to be able to give to it all the exactness and precision of which it is susceptible. Perhaps I might have attained this object by calling a straight line that which can have only one position between two given points. For, from this essential property we can deduce all the other properties of a straight line, and particularly that of its being the shortest between two given points. But in order to this it would have been necessary to enter into subtile discussions, and to distinguish, in the course of several propositions, the straight line drawn between two points from the shortest line which measures the distance of these same points. I have preferred, in order not to render the introduction to geometry too difficult, to sacrifice something of the exactness at which I aimed. Accordingly I shall call a straight line that which is the shortest between two points, and I shall suppose that there can be only one between the same points. It is upon this principle, considered at the same time as a definition and an axiom, that I have endeavoured to established the entire edifice of the elements.
It is necessary to the understanding of this work that the reader should have a knowledge of the theory of proportions which is explained in common treatises either of arithmetic or algebra; he is supposed also to be acquainted with the first rules of algebra; such as the addition and subtraction of quantities, and the most simple operations belonging to equations of the first degree. The ancients, who had not a knowledge of algebra, supplied the want of it by reasoning and by the use of proportions which they managed with great dexterity. As for us, who have this instrument in addition to what they possessed, we should do wrong not to make use of it, if any new facilities are to be derived from it. I have accordingly not hesitated to employ the signs and operations of algebra, when I have thought it necessary, but I have guarded against involving in difficult operations what ought by its nature to be simple; and all the use I have made of algebra in these elements, consists as I have already said, in a few very simple rules, which may be understood almost without suspecting that they belong to algebra.
Besides, it has appeared to me, that, if the study of geometry ought to be preceded by certain lessons in algebra, it would be not less advantageous to carry on the study of these two sciences together, and to intermix them as much as possible. According as we advance in geometry, we find it necessary to combine together a greater number of relations, and algebra may be of great service in conducting us to our conclusions by the readiest. and most easy method.
This work is divided into eight sections, four of which treat of plane geometry, and four of solid geometry.
The first section, entitled first principles, &c. contains the properties of straight lines which meet those of perpendiculars, the theorem upon the sum of the angles of a triangle, the theory of parallel lines, &c.
The second section, entitled the circle, treats of the most simple properties of the circle, and those of chords, of tangents, and of the measure of angles by the arcs of a circle.
These two sections are followed by the resolution of certain problems relating to the construction of figures.
The third section, entitled the proportions of figures, contains the measure of surfaces, their comparison, the properties of a right-angled triangle, those of equiangular triangles, of similar
figures, &c. We shall be found fault with perhaps for having blended the properties of lines with those of surfaces; but in this we have followed pretty nearly the example of Euclid, and this order cannot fail of being good, if the propositions are well connected together. This section also is followed by a series of problems relating to the objects of which it treats.
The fourth section treats of regular polygons and of the measure of the circle. Two lemmas are employed as the basis of this measure, which is otherwise demonstrated after the manner of Archimedes. We have then given two methods of approximation for squaring the circle, one of which is that of James Gregory. This section is followed by an appendix, in which we have demonstrated that the circle is greater than any rectilineal figure of the same perimeter.
The first section of the second part contains the properties of planes and of solid angles. This part is very necessary for the understanding of solids and of figures in which different planes are considered. We have endeavoured to render it more clear and more rigorous than it is in common works.
The second section of the second part treats of polyedrons and of their measure. This section will be found to be very different from that relating to the same subject in other treatises; we have thought we ought to present it in a manner entirely new.
The third section of this part is an abridged treatise on the sphere and spherical triangles. This treatise does not ordinarily make a part of the elements of geometry; still we have thought it proper to consider so much of it as may form an introduction to spherical trigonometry.
The fourth section of the second part treats of the three round bodies, which are the sphere, the cone, and the cylinder. The measure of the surfaces and solidities of these bodies is determined by a method analogous to that of Archimedes, and founded, as to surfaces, upon the same principles, which we have endeavoured to demonstrate under the name of preliminary lemmas.
At the end of this section is added an appendix to the third section of the second part on spherical isoperímetrical polygons; and an appendix to the second and third sections of this part on the regular polyedrons. | 677.169 | 1 |
Nice question! I think you meant to ask why all the angle measurements in a triangle add to 180 degrees. The side measurements can add up to anything.
There is a geometry proof of the fact that the angles in a triangle must add to 180 degrees.
Draw a triangle with vertices at points A, B, and C, and draw a line through vertex C that is parallel to side AB. This diagram forms three non-overlapping angles at vertex C. Two of the angles are just outside the triangle and one of the angles is angle C inside the triangle.
These three non-overlapping angles formed at vertex C clearly add to 180 degrees (because together, they form a semicircle).
Because alternate interior angles are congruent when parallel lines are cut by another line (transversal), angle A in the triangle is congruent to one of the angles at C outside the triangle, and angle B in the triangle is congruent to the other angle at C outside the triangle.
It follows from these last two statements that angles A, B, and C in the triangle add to 180 degrees. | 677.169 | 1 |
Ideas in Geometry/Instructive examples/Lesson 6: Introduction to City Geometry
In City Geometry, we have points and lines, just like in Euclidean Geometry. However, since we can only travel on city blocks, the distance between points is computed in a bit of a strange way. We don't measure distance as the crow flies. Instead we use the Taxicab distance:
Definition: Given two points A=(ax, ay) and B=(bx, by), we define the Taxicab distance as dT(A,B) = │ax-bx│+│ay-by│
Triangles: Triangle look the same in City Geometry as they do in Euclidean Geometry. Also, you measure the angles in exactly the same way. However, there is one minor hiccup. The lengths of the sides of each of these triangles are a little odd.
Circles: Circle is the collection of all points equidistant from a given point. So it City Geometry, we must conclude that a circle of radius 2 would look like a diamond two points away from the center.
Example:
Will just bought himself a brand new gorilla suit. He wants to show it off at three parties this Saturday night. The parties are being held at his friends' house: The Antidisestablishment (A), Hausdorff (H), and the Wookie Loveshack (W). If she travels from party A to party H to Party W, how far does he travel this Saturday night?
Solution:
We need compute
dT (A,H) + dT (H,W)
Let's start by fixing a coordinate system and making A the origin. Then H is (2, -5) and W is (-10, -2). Then
dT (A,H) = │0-2│+│0-(-5)│
= 2+5
= 7
and
dT (H,W) = │2-(-10)│+│-5-(-2)│
= 12+3
= 15
Will must trudge 7 + 15 = 22 blocks in his gorilla suit.
Midset
Definition: Given two points A and B, their midset is the set of points that are equal distance away from both A and B. | 677.169 | 1 |
In the given figure, AB and CD are two equal chords of a circle, with centre O. If P is the mid-point of chord AB, Q is the mid-point of chord CD and ∠POQ=150∘ find angleAPQ
Video Solution
Text Solution
Verified by Experts
The correct Answer is:an≥APQ=75∘
|
Answer
Step by step video & image solution for In the given figure, AB and CD are two equal chords of a circle, with centre O. If P is the mid-point of chord AB, Q is the mid-point of chord CD and anglePOQ = 150^(@) find angleAPQ by Maths experts to help you in doubts & scoring excellent marks in Class 9 exams. | 677.169 | 1 |
The Power of "cos a + cos b": Exploring the Mathematical Concept and its Applications
Mathematics is a language that allows us to understand and describe the world around us. One of the fundamental concepts in trigonometry is the addition of cosine functions, commonly denoted as "cos a + cos b." This mathematical expression holds significant importance in various fields, from physics and engineering to music and art. In this article, we will delve into the intricacies of "cos a + cos b," exploring its properties, applications, and real-world examples.
Understanding the Basics: Cosine Function
Before we dive into the addition of cosine functions, let's first establish a solid understanding of the cosine function itself. In trigonometry, the cosine of an angle is defined as the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. It is often abbreviated as "cos" and is represented by the Greek letter "θ" (theta).
The cosine function is periodic, meaning it repeats its values after a certain interval. The period of the cosine function is 2π radians or 360 degrees. It oscillates between the values of -1 and 1, with its maximum value of 1 occurring at 0 radians (or 0 degrees) and its minimum value of -1 occurring at π radians (or 180 degrees).
The Addition of Cosine Functions: cos a + cos b
When we add two cosine functions, such as "cos a + cos b," we obtain a new function that combines the properties of both individual functions. The resulting function represents the sum of the two original functions and exhibits unique characteristics.
To understand the addition of cosine functions, let's consider two specific examples:
Example 1: cos x + cos x
In this case, we have two identical cosine functions, "cos x + cos x." When we add these functions, the resulting function can be simplified using trigonometric identities. Using the identity "cos a + cos b = 2 cos((a + b)/2) cos((a – b)/2)," we can rewrite the expression as:
cos x + cos x = 2 cos((x + x)/2) cos((x – x)/2)
= 2 cos(x) cos(0)
= 2 cos(x)
Therefore, the sum of two identical cosine functions is equal to twice the cosine of the original angle. This result demonstrates that the amplitude of the resulting function is doubled compared to the individual cosine functions.
Example 2: cos x + cos (x + π/2)
In this example, we have two cosine functions with different angles, "cos x + cos (x + π/2)." To simplify this expression, we can again utilize the trigonometric identity mentioned earlier:
Therefore, the sum of two cosine functions with different angles results in a new function with an amplitude of √2 and a phase shift of π/4. This demonstrates that the addition of cosine functions can lead to a change in amplitude and phase.
Applications of "cos a + cos b"
The addition of cosine functions, "cos a + cos b," finds applications in various fields. Let's explore some of the key areas where this mathematical concept is utilized:
1. Physics and Engineering
In physics and engineering, the addition of cosine functions is often used to model and analyze periodic phenomena. For example, in the study of waves, the superposition principle states that the displacement of a medium at any point and time is the sum of the individual displacements caused by each wave. By representing waves as cosine functions and adding them together, scientists and engineers can accurately predict and understand complex wave behaviors.
Moreover, in electrical engineering, the addition of cosine functions is crucial in analyzing alternating current (AC) circuits. AC circuits involve sinusoidal voltages and currents, which can be represented as cosine functions. By adding these functions, engineers can determine the overall behavior of the circuit, including voltage drops, power dissipation, and resonance.
2. Music and Sound Engineering
The addition of cosine functions plays a significant role in music and sound engineering. In music theory, the concept of harmonics is closely related to the addition of cosine functions. Harmonics are multiples of a fundamental frequency that contribute to the overall sound of a musical note. By adding different harmonics, musicians and sound engineers can create complex and rich sounds.
For example, when playing a musical instrument, such as a guitar or piano, the sound produced is a combination of various harmonics. Each harmonic can be represented as a cosine function, and their addition results in the unique timbre and quality of the instrument's sound.
3. Image and Signal Processing
In image and signal processing, the addition of cosine functions is utilized in various algorithms and techniques. One such application is the Fourier Transform, which decomposes a signal or image into its constituent frequencies. The Fourier Transform represents the signal or image as a sum of sine and cosine functions, allowing for analysis and manipulation in the frequency domain.
By adding cosine functions with different frequencies and amplitudes, image and signal processing algorithms can enhance images, remove noise, and extract valuable information. This concept is widely used in fields such as computer vision, telecommunications, and audio processing.
Real-World Examples
To further illustrate the practical applications of "cos a + cos b," let's explore a few real-world examples:
Example 1: Ocean Waves
Ocean waves are a classic example of the addition of cosine functions. When multiple waves propagate through the ocean, their individual displacements add up to create the overall wave pattern. By understanding the addition of cosine functions, scientists can study and predict wave behaviors, such as interference, diffraction, and resonance, which are crucial for coastal engineering, marine navigation, and offshore structures.
Example 2: Musical Instruments | 677.169 | 1 |
How to Use LaTeX Arccosine Symbol
To write the Arccosine (arccos) in LaTeX, use the LaTeX command \arccos. It will add arccos symbol in the text.
In this article, we will discuss how to use Arccosine (arccos) in the LaTeX document and its applications in trigonometric calculations and mathematical concepts.
Symbol Overview
Symbol: Arccosine
Unicode: U+220F
Type: Trigonometric
Package Requirement: None (built-in symbol)
Argument: None (no additional arguments needed)
LaTeX Command: \arccos
Example: arccos
Description
The Arccosine is a built-in symbol available in LaTeX, and one of the fundamental trigonometric functions used in mathematics. Arccosine is used in mathematics to find the angle whose sine is a given value.
Syntax
The LaTeX command \arccos is used to display the Arccosine symbol.
\[ \arccos \]
Let's understand using Arccosine symbol in various domains like mathematics, trigonometry with the help of examples. | 677.169 | 1 |
Equilateral Triangle
Use our extensive free resources below to learn about Equilateral Triangle Equilateral Triangles?
Trigonometry (the study of triangles) forms a huge component of the National 5 course, and it plays an equally important role in all levels of maths you can pursue beyond National 5.
One of the most fundamental geometric facts about triangles is:
The interior angles of any triangle always sum to \boldsymbol{180^\circ}.
It is also important to understand the various types of triangles that can be constructed:
You will also be familiar with right-angled triangles. These can be either isosceles or irregular as these examples show:
SQA Past Paper Questions:
Equilateral Triangle
Learn About Equilateral Triangle | 677.169 | 1 |
Math in Focus Grade 8 Course 3 B Chapter 8 Review Test Answer Key
State whether a rotation, translation, or a combination of both is involved in each activity.
Question 1.
A turning blade of a windmill
Answer:
A turning blade of a windmill is rotation. Because the blade of a windmill rotates in a horizontal axis.
Question 2.
Pressing the keys on a computer keyboard
Answer:
Pressing the keys on a computer keyboard is a translation. Because the keyboard translates the number into binary data.
Question 3.
A printer head moving left and right
Answer:
A printer head moving left and right is a translation. Because it is moving from left to right and right to left it means translating from left to right and right to left.
Question 4.
Wheels on a moving bicycle
Answer:
Wheels on a moving bicycle is a rotation. Because the bicycle wheel is rotated in clockwise and anticlockwise.
Describe the translations.
Question 5.
Climbing up 8 steps of a staircase (assume horizontal and vertical distances of each step are the same)
Answer:
The translation is a type of transformation that moves each point in a figure with the same distance in the same direction. Climbing up 8 steps of a staircase is a translation.
Question 6.
Taking an elevator from level 2 to level 5 of a building
Answer: Taking an elevator from level 2 to level 5 of a building is translation. Because the elevator is a device that moves up and down.
Write an equation of the line(s) of reflection.
Question 7.
Answer:
Reflection in x axis is (x,y) = (x,-y)
Reflection in y axis is (x,y) = (-x,y)
In the above figure consider the left image as ABCD and right image as PQRS
The reflection in y axis is
P(1,1) reflection is A(-1,1)
Q(3,1) reflection is B(-3,1)
R(4,-1) reflection is C(-4,-1)
S(2,-1) reflection is D(-2,-1)
Each diagram shows a figure and its line of reflection. On a copy of the graph, draw the image.
Question 11.
Answer:
Reflection in x axis is (x,y) = (x,-y)
Reflection in y axis is (x,y) = (-x,y)
Consider the image below the x axis is ABCD.
A(-3,-3) reflection to the x axis is (-3,3)
B(0,-3) reflection to the x axis is (0,3)
C(-1,-4) reflection to the x axis is (-1,4)
D(-2,-4) reflection to the x axis is (-2,4)
Question 12.
Answer:
Solve on graph paper. Show your work.
Question 13.
\(\overline{\mathrm{AB}}\) is dilated with center at the origin and scale factor 2. Draw \(\overline{\mathrm{AB}}\) and its image \(\overline{A^{\prime} B^{\prime}}\). Use 1 grid square on the horizontal axis to represent 1 unit for x interval from -4 to 10, and 1 grid square on the vertical axis to represent 1 unit for the y interval from -2 to 4.
a) A (2, 1) and B (5, 2)
Answer:
b) A (1, -1) and B (-2, 2)
Answer:
Question 14.
The table shows the coordinates for ∆XYZ and its images using two transformation. Use 1 grid square on both axes to represent 1 unit for the interval from -3 to 9.
a) ∆XYZ is mapped onto ∆X'Y'Z' and ∆X"Y"Z" by a dilation. Draw each triangle and its image on the same coordinate plane. Then mark and label D as the center of dilation.
Answer:
In the drawn graph ABC = XYZ, DEF = X'Y'Z', GHI = X"Y"Z"
b) ∆XYZ is mapped onto ∆ABC by a rotation 90° counterclockwise about the origin. Draw ∆ABC on the coordinate plane.
Answer:
c) ∆XYZ is mapped onto ∆PQR by a translation 3 units to the left and 4 units up. Draw ∆PQR on the coordinate plane.
Answer:
In the drawn graph XYZ = ABC and PQR = DEF.
In the graph ∆DEF is the translation.
d) Compare the transformations that mapped ∆XYZ onto ∆ABC and ∆PQR in terms of preservation of the shape and size of ∆XYZ.
Answer:
The transformations ∆XYZ onto ∆ABC is rotation it means the figure turns in the clockwise or counter clockwise but it doesn't change the shape and size.∆PQR onto ∆XYZ is a translation it means moving the figure on the coordinate plane without changing the shape and size.
Problem Solving
Solve. Show your work.
Question 15.
Jane had lunch with her friends from 1 P.M. to 2 P.M. Describe the geometric transformation of the hour hand of the clock.
Answer:
Question 16.
A scientist used a sensor to track the movement of a mouse. It moved from the point (-2, 3) to the point (8, 6). State the new coordinates of any point (x, y) under this translation.
Answer:
Given that,
The mouse moved from the point (-2,3) to the point (8,6)
(x1,y1) = (-2,3)
(x2,y2) = (8,6)
New coordinates are (x,y)
x coordinate = x2 – x1 = 8 – (-2) = 8 + 2 = 10.
y coordinate = y2 – y1 = (6-3) = 3.
Therefore, New coordinates are (x+10, y+3).
Question 17.
Mrs. Morales outlined a clover with four identical leaflets, as shown, on a coordinate plane. The center of the clover is at (1, 0).
a) How many lines of symmetry does the clover have? Sketch them on a copy of the clover leaf.
Answer:
b) Find an equation of each line of reflection.
Answer:
The equation of each line of reflection is x = 0, y = 0, y = -x and y = x
Therefore there are four symmetry.
Question 18.
A circular mold in a Petri dish had a diameter of \(\frac{1}{2}\) inch. The diameter grew by 32% in a day.
a) What is the scale factor of dilation?
Answer:
Given that the circular in a Petri dish has a diameter of ½ inch.
The diameter grew in a day = 32%
The formula for the scale factor of dilation is = dimensions of a new shape/dimensions of a old shape
= 32%/1/2 = 0.32/0.5 = 0.64
b) Find the diameter of the mold after a day.
Answer:
The circular mold in a Petri dish has a diameter of 1/2
The diameter grew in a day = 32%
= 2 × 0.32 = 0.64
Therefore the diameter of the mold after a day = 0.64.
Question 19.
A figurine of the Statue of Liberty is 10 inches tall. The height of the Statue of Liberty is 150 feet. What is the scale factor of the dilation if the figurine is the image of the statue?
Answer:
Given that,
Figurine of the statue of Liberty is 10 inches tall
The height of the statue of Liberty is 150 feet.
The scalar factor of the dilation if the figurine is the image of the statue = figurine of the statue of Liberty/height of the statue of Liberty
Here 1 feet = 12 inches
150 inches = 150 × 12 = 1800
= 10/1800
= 1/180
= 0.005
Therefore, the scale factor of the dilation is 0.005 inches.
Question 20.
A spotlight is placed 2 feet from a 1-foot tall vase. A shadow 5 feet tall is cast on a wall as shown in the diagram. Find the distance of the vase from the wall.
Answer:
Given that,
The spotlight is placed 2 feets away from a 1 foot tall vase.
A shadow 5 feet tall is cast on a Wall.
Consider the distance from the vase to the wall is d.
The distance of a vase from the wall = distance of shadow from the spot light/distance of vase from the spotlight
= 2+d/2 = 5
= 2+d = 10
d = 10 – 2
d = 8
Therefore the distance from the vase to the wall is 8 feet. | 677.169 | 1 |
The Element of Geometry
Im Buch
Ergebnisse 6-10 von 41
Seite 27 ... fore the base CG is equal ( 4. 1. ) to the base DF , and the triangle GBC is equivalent ( 4. 1. ) to the triangle DEF ; therefore the angle GCB is equal to the angle DFE : but DFE is , by the hypothesis , equal to the angle ACB ...
Seite 30 ... produced , the perpendiculars will be continually increased ; and there- fore they may be produced until the perpendicular is greater than any given straight line . PROP . XIII . THEOR . If a straight line 30 THE ELEMENT.
Seite 31 ... fore , if a straight line meet two straight lines , so as to make the two interior angles on the same side of it , taken together , less than two right angles , these straight lines being continually produced , shall at length meet upon ...
Seite 33 ... fore ABED is a parallelogram ; whence AB is equal ( 9. 2. ) to DE , and AD to BE ; but AB is equal to AD ; therefore the four straight lines AB , AD , BE , DE , are equal to one another , and the parallelogram D ABED is equilateral ...
Seite 35 ... fore the squares of AD , AC are equal to the squares of AB , AC ; but the square of CD is equal ( 17. 2. ) to the squares of AD , AC , be- cause DAC is a right angle ; and the square of BC , by hypothesis , is equal to the squares of AB ...
Beliebte Passagen
Seite 25 - If a straight line be divided into two equal parts, and also into two unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line.
Seite 29 - In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side...
Seite 90Seite 87 - ... 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.
Seite 13 - If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles. | 677.169 | 1 |
you could also chose the cardinal points on the nodes for examples 0 deg, 45 deg -- represented as box-a.0, box-a.45, etc and the corresponding points on box-b.180, box-b.135, etc -- see the answer below | 677.169 | 1 |
Ray(JR Cuevas)
He is a graduate of Bachelor of Science in Civil Engineering specializing in Structural Engineering and now a Licensed Civil Engineer. He enjoys writing articles related to education especially mathematics, science, and health is a guide to solving for centroids and centers of gravity of different compound shapes using the method of geometric decomposition. Learn how to obtain the centroid from different examples provided.
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Age and mixture problems are tricky questions in Algebra. It requires deep analytical thinking skills and great knowledge in creating mathematical equations. Practice these age and mixture problems with solutions in Algebra.
This is a guide to solving for centroids and centers of gravity of different compound shapes using the method of geometric decomposition. Learn how to obtain the centroid from different examples provided.
Spearman Rank Correlation and Pearson Correlation can be both used to identify the relationship between two sets of data. This a brief comparison between the two measures of correlation and there are examples provided to explain how to solve them.
A Love so Beautiful is a Web Series from China. It is a love story between two young people who grew up together trying to overcome different challenges in life. Here are the eight love lessons that you can learn from the series.
Estimation of painting works follows a pattern. Learn how to estimate the painting for your walls, ceilings, roofs, doors, windows, and wood surfaces and have an idea of the cost. Steps provided are easy to learn and understand | 677.169 | 1 |
Elements of Geometry: Containing the First Six Books of Euclid, with a ...
is the figure BLPQ; and because the plane ACDF is Book III. that in which are the parallels AC, FDOR, and in which also is the figure CAOR:, and the plane CBKE is that in which are the parallels CB, RQEK, and in which also is the figure CBQR; therefore the figures ALPO, CBQR are in parallel planes. But the planes ACBL, ORQP are also parallel; therefore the solid CP is a parallelepiped. Now the solid parallelepiped CM is equal to the solid parallelepiped CP; b 5. 3. Sup. because they are upon the same base, and their insisting straight lines AF, AO, CD, CR; LM, LP, BH, BQ are terminated in the same straight lines FR, MQ; and the solid CP is equal: Therefore the solid CM is equal to the solid CN. Wherefore solid parallelepipeds, &c. Q. E. D.
b
Supplement
b 14. 1.
PROP. VII. THEOR.
Solid parallelepipeds which are upon equal bases, and of the same altitude, are equal to one another.
Let the solid parallelep be in a straight line; therefore the straight line LM, which is at right angles to the plane in which the bases are, in a 11.2.Sup. the point L, is common to the two solids AE, CF; let the other insisting lines of the solids be AG, HK, BE; DF, OP, CN; and first, let the angle ALB be equal to the angle CLD; then AL, LD are in a straight line". Produce OD, HB, and let them meet in Q, and complete the solid parallelep the base CD to the same LQ; and because the solid parallelepiped AR is cut by the plane LMEB, which is parallel to the opposite planes AK, DR; as the base AB is
e 7. 5.
.3.3. Sup. to the base LQ, so is the solid AE to the solid LR: for the same reason, because the solid parallelepiped CR is
cut by the plane LMFD, which is parallel to the oppo- Book III. site planes CP, BR; as the base CD to the base LQ; so is the solid CF to the solid LR: but as the base AB to the base LQ, so the base CD to the base LQ, as has been proved therefore, as the solid AE to the solid LR, so is the solid CF to the solid LR; and therefore the solid AE is equal to the solid CF.
e
f
€ 9. 5.
But let the solid parallelepipeds SE, CF be upon equal bases SB, CD, and be of the same altitude, and let their insisting straight lines be at right angles to the bases; and place the bases SB, CD in the same plane, so that CL, LB be in a straight line; and let the angles SLB, CLD be unequal; the solid SE is also in this case equal to the solid CF. Produce DL, TS until they meet in A, and from B draw BH parallel to DA; and let HB, OD produced meet in Q, and complete the solids AE,' LR : therefore the solid AE, of which the base is the parallelogram LE, and AK the plane opposite to it, is equal to f 5. 3. Sup. the solid SE, of which the base is LE, and SX the plane opposite; for they are upon the same base LE, and of the same altitude, and their insisting straight lines, viz. LA, LS, BH, BT; MG, MU, EK, EX are in the same straight lines AT, GX: and because the parallelogram AB is equal to SB, for they are upon the same base g 35 1. LB, and between the same parallels LB, AT; and because the base SB is equal to the base CD; therefore the base AB is equal to the base CD; but the angle ALB is equal to the angle CLD; therefore, by the first case, the solid AE is equal to the solid CF; but the solid AE is equal to the solid SE, as was demonstrated; therefore the solid SE is equal to the solid CF.
g
CASE 2. If the insisting straight lines AG, HK, BE,
Supplement LM; CN, RS, DF, OP be not at right angles to the bases AB, CD; in this case likewise the solid AE is equal to the solid CF. Because solid parallelepipeds on the b 6. 3. Sup. same base, and of the same altitude, are equal h, if two solid parallelepipeds be constituted on the bases AB and CD of the same altitude with the solids AE and CF, and with their insisting lines perpendicular to their bases, they will be equal to the solids AE and CF; and, by the first case of this proposition, they will be equal to one another; wherefore the solids AE and CF are also equal. Wherefore, solid parallelepipeds, &c. Q. E. D.
PROP. VIII. THEOR.
Solid parallelepipeds which have the same altitude are to one another as their bases.
Let AB, CD be solid parallelepipeds of the same altitude: they are to one another as their bases; that is, as the base AE to the base CF, so is the solid AB to the solid CD.
To the straight line FG apply the parallelogram FH a Cor. 45. 1. equal a to AE, so that the angle FGH be equal to the
angle LCG; and complete the solid parallelepiped GK upon the base FH, one of whose insisting lines is FD, whereby the solids CD, GK must be of the same altitude. b 7. 3. Sup. Therefore the solid AB is equal to the solid GK, because they are upon equal bases AE, FH, and are of the same altitude and because the solid parallelepiped CK is cut by the plane DG which is parallel to its opposite
c
c 3. 3. Sup.
planes, the base HF is to the base FC, as the solid HD Book III. to the solid DC: But the base HF is equal to the base AE, and the solid GK to the solid AB: therefore, as the base AE to the base CF, so is the solid AB to the solid CD. Wherefore solid parallelepipeds, &c. Q. E. D.
COR. 1. From this it is manifest, that prisms upon triangular bases, and of the same altitude, are to one another as their bases. Let the prisms BNM, DPG, the bases of which are the triangles AEM, CFG, have the same altitude; complete the parallelograms AE, CF, and the solid parallelopipeds AB, CD, in the first of which let AN, and in the other let CP be one of the insisting lines. And because the solid parallelepipeds AB, CĎ have the same altitude, they are to one another as the base AE is to the base CF; wherefore the prisms, which are their halves d are to one another, as the base AE to d 4. 3. Sup. the base CF; that is, as the triangle AEM to the triangle CFG.
COR. 2. Also a prism and a parallelepiped, which have the same altitude, are to one another as their bases; that is, the prism BNM is to the parallelepiped CD as the triangle AEM to the parallelogram LG. For by the last Cor. the prism BNM is to the prism DPG as the triangle AME to the triangle CGF, and therefore the prism BNM is to twice the prism DPG as the triangle AME to twice the triangle CGF; that is the prism BNM is e 4. 5. to the parallelepiped CD as the triangle AME to the parallelogram LG. | 677.169 | 1 |
6
Seite 6 ... it is manifest that the angles at any point in a straight line , on both sides of it , or all the angles round a point , are together equal to four right angles . PROP . XIV . THEOR . If two straight lines 6 EUCLID'S ELEMENTS .
Seite 15 ... equal to two - thirds of a right angle . COR . 4. - All the internal angles of any rectilinear figure are equal to twice as many right angles ( de- ducting four ) as the figure has sides . B E COR . 5. - All the external angles of any ...
Seite 24 ... equal to the squares of the parts , together with twice the rectangle contained by the parts . C E D + H K G F B COR ... equal to four times the square of the half . PROP . V. THEOR . If a straight line be 24 EUCLID'S ELEMENTS .
Seite 26 ... one of its parts is equal to four times the rectangle contained by the whole line and that part , together with the square of the other part . M K G I H 10 N L Р R PROP . IX . THEOR . If a straight line 26 EUCLID'S ELEMENTS .
Seite 60 ... equal to an angle of the other , have the sides about the equal angles reciprocally pro- portional and ... four straight lines 60 EUCLID'S ELEMENTS . | 677.169 | 1 |
CLASS-8 GEOMETRY-TRIANGLE-SOME-IMPORTANT-TERMS
Some Important Terms –
Altitude
An altitude of a triangle is considered via drawing perpendicular from any vertex of the triangle to the opposite side. It is considered that, any triangle should have three altitudes. In the figure, AL, BM, and CN are the altitudes of said ∆ ABC.
The three altitudes of a triangle pass through a common point is considered or called the 'orthocenter' of any triangle.
Median -
The line segment which is joining a vertex of a triangle to
the midpoint of the opposite side is to be considered or called a median of the triangle. A triangle is considered
to have three medians. In the figure, it has been seen that L, M, and N are the
mid-points of the sides BC, CA, and AB respectively of the said triangle. So, AL, BM, and CN are considered
as the medians of ∆ ABC.
The three medians of the said triangle
intersect at the point is called or known as the 'centroid' or 'centre of
gravity of the said triangle'.
Incircle
The circle that lies inside a
triangle and touches its three sides then will be called or considered as
the incircle. The centre of the incircle is called or to be considered as the incentre. The incentre is the point at which the three
(internal) bisectors of the angles of the triangle is observed to meet each other.
In the figure, we can observe that AI, BI, and CI bisect ∠A, ∠B, and ∠C respectively and meet at point I, which is to be considered incentre.
Circumcircle
The circumcircle of a triangle is the circle which passes through its three
vertices as we can observe from the given picture. Its centre is called the circumcentre, the point at which the perpendicular bisectors of the sides of the
triangle when meets each other then this point is to be considered as circumcentre. | 677.169 | 1 |
Nov 25, 2019 · We call this the direction of positive torque. Putting it together, the torque vector is the cross product of the force F F times the moment arm d (length of the wrench arm from the center of rotation to the point of application of force) or. T …To get the most from your health insurance, you need to make sure that your see providers who are in the Anthem Blue Cross and Blue Shield network. Here are the steps you need to t...Jan 7, 2024 · Defining the Cross Product. The dot product represents the similarity between vectors as a single number: For example, we can say that North and East are 0% similar since ( 0, 1) ⋅ ( 1, 0) = 0. Or that North and Northeast are 70% similar ( cos ( 45) = .707, remember that trig functions are percentages .) The similarity shows the amount of one ...: Get the latest Southern Cross Media Group stock price and detailed information including news, historical charts and realtime prices. Indices Commodities Currencies StocksApr 7, 2023 · Simply take the inverse sine of the cross product and magnitudes to find the angle between the vectors. Using your calculator, find the arcsin or sin-1 function. Then, enter in the cross product and magnitude. In our example, enter "arcsin(√1539 / √14 * √110) into your calculator to get θ = 88.5º.In the previous example, we computed the vector moment of a planar force about a point using the formula 𝑀 = ⃑ 𝑟 × ⃑ 𝐹. We can see that the resulting vector of the cross product only contained a ⃑ 𝑘 component, and the ⃑ 𝑖 and ⃑ 𝑗 components vanished. This is not surprising if we consider the geometric property of a cross product. In Aug 29, 2021 · Cross Learn The vector multiplication or the cross-product of two vectors is shown as follows. vector perpendicular to the plane ...The prospect of contacting a satellite to send a text may soon be an effortless reality as startups go from proof of concept to real product. The prospect of contacting a satellite...Instead we can imagine that we already know n ′ =v 1 ×v2→. Then this becomes: x3,y3,z3 ×n ′ = y3n′z −z3n′y,z3n′x −x3n′z,x3n′y −y3n′x . Now, setting this equal to the second argument of the original cross product, we have a set of linear equations for three unknowns ( x3,y3,z3 ):Angle between vectors given cross and dot product. 2. Angle in Rodrigues' rotation formula. 1. Length of vector resulting from cross product. 1. Confusion regarding cross product formula. 1. Test into the book Halliday-Resnick on scalar product and cross product = a1, a2, a3 and B = b1, b2, b3 . Cross-Product Magnitude. It is a straightforward exercise to show that the cross-product magnitude is equal to the product of the vector lengths times the sine of the angle between them: B.21. (B.16) (Recall that the vector cosine inner product norms 454 ].) To derive Eq. Advertisement The American Red Cross is made up of 769 regional or city-based chapters. Every chapter is officially chartered by the national Board of Governors. The directors of t...Cross product refers to a binary operation on two vectors in three-dimensional Euclidean vector space. The right-hand rule is used to calculate the cross product of two vectors. The right-hand rule is mainly the result of any two vectors which are perpendicular to the other two vectors. The magnitude of the resulting vector can also be ...The American Red Cross is on the ground in Houston providing hurricane relief. Here's what to know about donating to the organization. By clicking "TRY IT", I agree to receive news... …Solution. Since i = (1, 0, 0) and j = (0, 1, 0), then. i × j = ((0)(0) − (0)(1), (0)(0) − (1)(0), (1)(1) − (0)(0)) = (0, 0, 1) = k. Similarly it can be shown that j × k = i and k × i = j. Figure 1.4.1. In the above example, the … Cross Product. The cross product is a binary operation on two vectors in three-dimensional space. It again results in a vector which is perpendicular to both vectors. The cross product of two vectors is calculated by the right-hand rule. The right-hand rule is the resultant of any two vectors perpendicular to the other two vectors. The lower chamber of the French parliament passed a bill that aims to introduce some new requirements for social media influencers. The lower chamber of the French parliament, the PNW is an ideal winter destination for hitting the trails. Here are the best snowshoeing and cross-country skiing trails in Washington. When the snow falls, you can't go wrong ...I have a vectorial equation where there is a cross product, and the unknown 'x' is within the cross product. The equation is simply the cross product between two vectors, which is equal to the torque being applied to a rotating system. The code is the following: P = [126.7611; -118.5356; 331.2583]; % Point P, at which force is applied. A = …Learn how to compute the cross product of two vectors, a vector operation that is perpendicular to both vectors and measures how far apart they are. See the right …Jun 16, 2014 · The overdot notation I used here is just a convenient way of not having to write out components while still invoking the product rule. When you differentiate a product in single-variable calculus, you use a product rule. When you differentiate a product of vectors, there is a vector extension of the product rule. Seems sensible to me. $\endgroup$La Crosse Technology is a renowned company that specializes in manufacturing and distributing high-quality weather stations, clocks, and other consumer electronics. With a wide ran...Two vectors |a→| = 5.39 and |b→| = 4.65 | a → | = 5.39 a n d | b → | = 4.65 intersect and make a 120° angle. Find |a→ × b→| | a → × b → |. Now I tried to solve this problem for too much time and since I have the solution I've seen that the result is −12.5 − 12.5 and in particular −12.5 =|a→| ⋅| b→| ⋅ cos 120 − ...6 others. contributed. The cross product is a vector operation that acts on vectors in three dimensions and results in another vector in three dimensions. In contrast to dot product, which can be defined in both 2-d and 3-d space, the cross product is only defined in 3-d space. Another difference is that while the dot-product outputs a scalar ... …Need a cross platform mobile app development company in Poland? Read reviews & compare projects by leading cross platform app developers. Find a company today! Development Most Pop...While some may equate perfectionism with detail-oriented output, it also takes a serious toll on your happiness—and, it turns out, your overall productivity. Avoid falling into the...Mar 30, 2023 · Cross-multiplying reduces these two fractions to one simple equation, allowing you to easily solve for the variable in question. It's also a useful method to know when you're adding and subtracting unlike fractions and comparing ratios and proportions. Keep reading and follow along as we take you through the steps of cross-multiplication.Jan 16, 2023 · Figure 1.4.8. For vectors v = v1i + v2j + v3k and w = w1i + w2j + w3k in component form, the cross product is written as: v × w = (v2w3 − v3w2)i + (v3w1 − v1w3)j + (v1w2 − v2w1)k. It is often easier to use the component form for the cross product, because it can be represented as a determinant. The prospect of contacting a satellite to send a text may soon be an effortless reality as startups go from proof of concept to real product. The prospect of contacting a satellite...The Equator passes through three of the seven continents: South America, Africa and Asia. Although it does not pass through the mainland of Asia, it does run through Indonesia and The scalar product of a vector with itself is the square of its magnitude: →A2 ≡ →A · →A = AAcos0° = A2. 2.28. Figure 2.27 The scalar product of two vectors. (a) The angle between the two vectors. (b) The orthogonal projection A ││ of vector →A onto the direction of vector →B .Another way of starting is to substitute the given x in a × x, and then use the properties of the cross product (linearity etc) to simplify the equation, and see if you get what you want. Let x be a solution of the equation. a × x = b ⇒ a ⋅ (a × x) = x ⋅ (a × a) = 0 = (a ⋅ b) In this case, if there is a solution that verifies the ... TheJan 29, 2024 · 1. There are two main ways to express the equation of a plane - vector format or Cartesian equation. The vector format, in simplest form, is usually written →n. (→r − r0) = 0 where →n is a normal vector to the plane, →r is the variable vector (typically (x, y, z) in 3-space) and r0 is some given initial point. The equation for the red plane is x-2y+z=-6 and the equation for the blue plane is x-2y+z=0. This means that the planes are parallel with the red one is shifted down. ... then we can take the cross-product of those two vectors to find out a normal to this blue plane, and then use that information to actually figure out the equation for the blue ...Vector rotational kinematic quantities. In the previous section, we defined angular quantities to describe the motion of a particle about the \(z\) axis along a circle of radius \(R\) that lies in the \(xy\) plane. By using vectors, we can define the angular quantities for rotation about an axis that can point in any direction.Given an axis of rotation, the path of any particle …Shipping your car cross-country can be expensive. In this article, we're sharing the top five cheap ways to ship a car cross-country so you can save more money. Expert Advice On Im...4 days ago · simplify the right side of the equation. The result will be a vector a×b = c1i + c2j + c3k. A set of two vectors must occupy three-dimensional space to have a Sep product is defined as the vector product of two vectors, and is denoted by the symbol . Cross product is perpendicular to both input vectors, and its Using In today's digital age, where technology plays an integral role in our daily lives, it is essential to have tools that enhance productivity and streamline tasks. One such tool that...Determinants and the Cross Product. Using Equation 2.9 to find the cross product of two vectors is straightforward, and it presents the cross product in the useful component …Learn how to boost your finance career. The image of financial services has always been dominated by the frenetic energy of the trading floor, where people dart and weave en masse ...Calculate the cross product of two given vectors. Use determinants to calculate a cross product. The Cross Product and Its Properties The dot product is a multiplication of …
The formula for vector cross product can be derived by using the following steps: Step 1: Firstly, determine the first vector a and its vector components. Step 2: Next, determine the second vector b and its vector components. …. Warren peay
The Hong Kong-Zhuhai-Macao Bridge is an engineering marvel. HowStuffWorks looks at this amazing structure. Advertisement The world's longest sea-crossing bridge has opened, connect...The …It only took 26 days. Marvel's critically acclaimed Black Panther continues to roar and pounce. The movie crossed the $1 billion mark on March 10, Variety reported. Black Panther h...18 hours ago · The properties of a cross product can vary depending on the type of cross-product formula that is used. 1. General Properties of a Cross Product. Length of two vectors to form a cross product. \ [\left | \vec {a}\times \vec {b} \right |= \left | a \right |\left | b \right |sin\theta\] This length is equal to a parallelogram determined by two ... Around 300,000 people cross the northern border with Canada each day, which equates to annual approximates of 39,254,000 crossings by Canadians into the United States (in 2009) and...The cross product of two parallel vectors is 0, and the magnitude of the cross product of two vectors is at its maximum when the two vectors are perpendicular. There are lots of other examples in physics, though. Electricity and magnetism relate to each other via the cross product as well. The Hong Kong-Zhuhai-Macao Bridge is an engineering marvel. HowStuffWorks looks at this amazing structure. Advertisement The world's longest sea-crossing bridge has opened, connect...The cross product of two parallel vectors is 0, and the magnitude of the cross product of two vectors is at its maximum when the two vectors are perpendicular. There are lots of other examples in physics, though. Electricity and magnetism relate to each other via the cross product as well. ForGiven three points that lie in a plane, we can find the equation of the plane passing through those three points. We'll use a cross product to find the slope in the x, y, and z directions, and then plug those slopes and the three points into the formula for the equation of the plane. About Pricing Login GET STARTED About Pricing Login. Step-by …Mastercard unveils Cross-Border Services Express, offering easy setup of international payments for SMEs and consumers in a digital-first experience. Mastercard has introduced Cros.... | 677.169 | 1 |
where a and b are the equatorial radii (along the x and y axes) and c is the polar radius (along the z-axis), all of which are fixed positive real numbers determining the shape of the ellipsoid.
More generally, an arbitrarily oriented ellipsoid, centered at v, is defined by the equation
where A is a positive definite matrix and x, v are vectors. In that case, the eigenvectors of A define the principal directions of the ellipsoid and the square root of the eigenvalues are the corresponding equatorial radii.
If all three radii are equal, the solid body is a sphere; if two radii are equal, the ellipsoid is a spheroid:
The points (a,0,0), (0,b,0) and (0,0,c) lie on the surface and the line segments from the origin to these points are called the semi-principal axes. These correspond to the semi-major axis and semi-minor axis of the appropriate ellipses.
Scalene ellipsoids are frequently called "triaxial ellipsoids",[1] the implication being that all three axes need to be specified to define the shape.
Any planar cross section passing through the center of an ellipsoid forms an ellipse on its surface, with the possible special case of a circle if the three radii are the same (i.e., the ellipsoid is a sphere) or if the plane is parallel to two radii that are equal.
Volume
Note that this equation reduces to that of the volume of a sphere when all three elliptic radii are equal, and to that of an oblate or prolatespheroid when two of them are equal. The volumes of the maximum inscribed and minimum circumscribed boxes are respectively:
Surface area
Unlike the surface area of a sphere, the surface area of a general ellipsoid cannot be expressed exactly by an elementary function.
An approximate formula is:
Where p ≈ 1.6075 yields a relative error of at most 1.061% (Knud Thomsen's formula [2]); a value of p = 8/5 = 1.6 is optimal for nearly spherical ellipsoids, with a relative error of at most 1.178% (David W. Cantrell's formula).
Exact formulae can be obtained for the case a = b (i.e., a circular equator):
Rotational equilibrium
Scalene ellipsoids and cuboids rotate stably along their major or minor axes, but not along their median axis. This can be seen experimentally by throwing an eraser with some spin. In addition, moment of inertia considerations mean that rotation along the major axis is more easily perturbed than rotation along the minor axis. One practical effect of this is that scalene astronomical bodies such as Haumea generally rotate along their minor axes (as does the Earth, which is merely oblate); in addition, because of tidal locking, scalene moons in synchronous orbit such as Mimas orbit with their major axis aligned radially to their planet.
A relaxed ellipsoid, that is, one in hydrostatic equilibrium, has an oblateness a − c directly proportional to its mean density and mean radius. Ellipsoids with a differentiated interior—that is, a denser core than mantle—have a lower oblateness than a homogeneous body. Over all, the ratio (b–c)/(a−c) is approximately 0.25, though this drops for rapidly rotating bodies.[3]
Fluid properties
The ellipsoid is the most general shape for which it has been possible to calculate the creeping flow of fluid around the solid shape. The calculations include the force required to translate through a fluid and to rotate within it. Applications include determining the size and shape of large molecules, the sinking rate of small particles, and the swimming abilities of microorganisms.[4]
Linear transformations
An invertible linear transformation applied to a sphere produces an ellipsoid, which can be brought into the above standard form by a suitable rotation, a consequence of the polar decomposition (also, see spectral theorem). If the linear transformation is represented by a symmetric 3-by-3 matrix, then the eigenvectors of the matrix are orthogonal (due to the spectral theorem) and represent the directions of the axes of the ellipsoid: the lengths of the semiaxes are given by the eigenvalues. The singular value decomposition and polar decomposition are matrix decompositions closely related to these geometric observations.
The intersection of an ellipsoid with a plane is either empty, a single point, or an ellipse (including a circle).
One can also define ellipsoids in higher dimensions, as the images of spheres under invertible linear transformations. The spectral theorem can again be used to obtain a standard equation akin to the one given above.
Egg shape
Oval
The shape of a chicken egg is approximately that of half each a prolate and a roughly spherical (potentially even minorly oblate) ellipsoid joined at the equator, sharing a principal axis of rotational symmetry.[5] Although the term egg-shaped usually implies a lack of reflection symmetry across the equatorial plane, it may also refer to true prolate ellipsoids. It can also be used to describe the 2D figure that, revolved around its major axis, produces one of the 3D surfaces described above. See also oval.
Look at other dictionaries:
Ellipsoid — El*lip soid, n. [Ellipse + oid: cf. F. ellipsoide.] (Geom.) A solid, all plane sections of which are ellipses or circles. See {Conoid}, n., 2 (a) . [1913 Webster] Note: The ellipsoid has three principal plane sections, a, b, and c, each at right… … The Collaborative International Dictionary of English
ellipsoid — [e΄lip soid′ le lip′soid΄, ilip′soid] n. [Fr ellipsoïde: see ELLIPSE & OID] Geom. 1. a solid formed by rotating an ellipse around either axis: its plane sections are all ellipses or circles 2. the surface of such a solid adj. of or shaped like an … English World dictionary | 677.169 | 1 |
Math resources created with the student in mind! My math resources include interactive notebooks, activities, foldables, assessments, exit slips, videos, and so much more!
Area of Polygons in the Coordinate Plane Capture the Easter Bunnies Puzzle
Students will plot points on the coordinate plane to create 10 polygons around the Easter bunnies. They will calculate the area of the polygons then glue each puzzle piece according to the area. The polygons include triangles, squares, rectangles, parallelograms, and trapezoids. In addition, there are 2 composite figures (rectangle and triangle together).
THERE ARE 2 VERSIONS: (1) First Quadrant Only and (2) All Quadrants.
This resource is great for self-assessment and looks amazing in math interactive notebooks. | 677.169 | 1 |
Construction
From Trigonometry
A basic pentagon can be created from the basic code
;
REPEAT5[FD2RT72]
note: the internal angle of a regular pentagon is 108o, making the external angle, which we are are turning, 72o
from this initial pentagon, we need to create pentagons connecting to our 5 initial edges. To do this, we need to consider what angle these are to be connected at. The dihedral angle of a dodecahedron is116.56505° = arccos(-1/√5), meaning that we need to tilt down 63.43495o
this process needs to be repeated several times to create all edges of our dodecahedron. The final code would look something like.
By the time I attempted the dodecahedron, I had already worked my way through the simpler platonic solids. This allowed me to familiarise myself with the coding whilst creating solids which I found easy to understand. There were still a lot of false starts and editing before I succesfuly completed the dodecahedron. | 677.169 | 1 |
Reflectional symmetry involves breaking the two dimensional square into two halves by using a one dimensional line. If you think about breaking a three dimensional square into two halves, you'll see that you need to use a two dimensional plane:
1. A cube has 9 different planes of reflectional symmetry. See if you can find them all!
Rotational Symmetry of a Cube
A square has rotational symmetry about its central point, as every time you turn it 90 \deg, you come back to the same shape. We call this 4-fold symmetry, since it comes back to the shape 4 times when you spin by degrees(360).
In the same way, a cube has rotational symmetry. However, instead of rotating about a zero-dimensional point, cubes rotate about one-dimensional lines. The easiest way to think of this is to imagine a kebab with cube shaped pieces, with the skewer as the axis of rotation:
As you spin the skewer, the pieces rotate. If they are skewed down the middle, as shown in the diagram below, they will end up being the same shape after spinning by 90 \deg degrees. Just like for squares, we call this 4-fold symmetry.
1. There are 13 axes of rotational symmetry in total. See how many you can find! It may help to build a model of a cube and skewer so that you can try spinning it yourself. You can make a cube using a net or find out how you can do it using origami.
2. Of those that you found, how many are 2-fold? How many are 3-fold? How many are 4-fold?
Symmetry of a Tetrahedron
Now that we've had a look at the cube, it's up to you to figure out the case for the tetrahedron. A tetrahedron is a pyramid made up of equilateral triangles:
1. There are 6 planes of reflectional symmetry for a tetrahedron. How many can you find?
2. There are 7 axes of rotational symmetry for a tetrahedron. How many can you find? How many are 2-fold? How many are 3-fold?
Again, you can use a net or origami to make your own tetrahedron, which may help you in visualising the symmetries.
More Sides, More Symmetries?
Both the cube and the tetrahedron are examples of "regular polyhedra", meaning that they are made up of shapes which have all sides equal. We saw that as we went from triangular faces with three sides to square faces with four sides, the number of symmetries increased. Does this mean that as we look at regular polyhedra with even more sides, the number of symmetries will increase?
1. See if you can use the internet to research how many planes of reflectional symmetry and axes of rotational symmetry a dodecahedron has. A dodecahedron is a twelve-sided regular polyhedron made from regular pentagons (pictured above).
2. Of the axes of rotational symmetry, how many are 2-fold, how many are 3-fold, how many are 4-fold and how many are 5-fold?
If you want to check for yourself, you can once again make it with nets or origami (Good luck!). | 677.169 | 1 |
Survival Shelters and Triangles: Building Sturdy Structures in the Wilderness
In the vast expanse of wilderness, where nature dictates the rules, constructing a survival shelter is not merely an act of finding refuge. It's a profound engagement with geometry, specifically the strength and stability offered by triangular structures. The geometry of survival isn't just about angles and lines; it's about leveraging the most durable shapes known to both man and nature. This article delves into the fascinating interplay between survival shelters and triangles, exploring how this simple geometric principle can mean the difference between vulnerability and security in the wild.
The Geometry of Survival: Embracing Triangular Rigor
The triangle, with its three sides and three angles, is celebrated in engineering and architecture for a reason: it epitomizes strength and resilience. This geometric shape is inherently rigid; unlike rectangles or squares, which can easily distort unless supported, a triangle holds its form under pressure. This principle is paramount in survival situations where resources are scarce, and the need for a sturdy shelter is non-negotiable. But why is this shape so uniquely suited for survival shelters? The answer lies in its ability to distribute weight evenly, reducing the likelihood of collapse under harsh environmental conditions.
In the wilderness, embracing the triangular rigor means not just understanding its stability but also recognizing its versatility. Triangles can be scaled up or down, adapted to different materials, and configured in various ways to suit specific environmental challenges. For instance, a simple A-frame shelter, essentially two large triangles leaning against each other, can be a quick and effective refuge against wind and rain. This adaptability makes the triangle an indispensable tool in the survivalist's arsenal, allowing for the construction of shelters that can withstand unpredictable elements.
Moreover, the psychological impact of constructing a geometrically sound shelter should not be underestimated. In survival scenarios, the certainty that comes from knowing you have built something structurally solid can be a significant morale booster. The triangle, in its simplicity and strength, offers not just physical protection but also mental reassurance, a crucial element in the harsh unpredictability of the wilderness.
Constructing Durable Shelters: Triangles in Practice
Implementing triangles in the construction of survival shelters requires both creativity and a practical understanding of the environment. The first step is identifying the materials at your disposal: branches, leaves, snow, or even the terrain itself can serve as components of a triangular shelter. The key is to think in terms of angles and supports, ensuring that each element contributes to the overall rigidity of the structure. For example, longer branches can form the skeleton of the shelter, while foliage and snow can act as insulating materials, filling in the gaps and providing protection from the elements.
The process of constructing a triangular shelter is both an art and a science. It involves not just the physical assembly of materials but also the strategic planning of dimensions and orientation. The angle at which the sides of the triangle meet can influence the stability and interior space of the shelter, requiring careful consideration based on the intended use and occupancy. Furthermore, the orientation of the shelter relative to prevailing winds and potential hazards can significantly impact its effectiveness and safety. This careful balance between geometric principles and environmental factors is what makes constructing durable shelters in the wilderness a complex, yet rewarding challenge.
Finally, the innovation and improvisation that come with applying triangles in practice highlight the resourcefulness required for survival. From the snow-laden landscapes of the Arctic to the rainforests of the Amazon, the triangle remains a universal symbol of strength. Whether it's a lean-to, a teepee, or a snow cave, the underlying triangular structure provides a foundation upon which survivalists can build and adapt. This is a testament to the triangle's unparalleled ability to offer security and shelter in the most adverse conditions, reinforcing its status as the cornerstone of survival architecture.
The interplay between survival shelters and triangles is a compelling narrative of geometry applied under the most extreme conditions. It underscores a fundamental truth: that the principles of design and construction hold firm, even (and especially) in the wilderness. This article has not only explored the why and how of triangular shelters but has also celebrated the ingenuity and resilience of those who construct them. As we venture into the wild, let us carry with us the knowledge that in the geometry of survival, the triangle is our most steadfast ally, a symbol of strength, stability, and the enduring human spirit to prevail against the odds.
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I believe there might be a bit of confusion with the request. An excerpt for an article should be longer than 40 to 60 characters as that range is more akin to a headline or a very short tagline. Excerpts typically provide a bit more detail to give readers an insight into the content of the article. However, if you meant a headline or title for an article, here is one within the character limit: "Semantic Kernel: Fusing AI with Coding" If you actually meant an excerpt, I can provide a short passage instead that goes beyond the 40 to 60 character range: "The emergence of the Semantic Kernel marks a groundbreaking advancement in the synthesis of artificial intelligence and programming languages. This innovative interface serves as a bridge between the abstract complexities of code and the intuitive capabilities of AI. By offering a more natural and semantic approach to coding, developers are empowered to communicate with machines in unprecedented ways, opening up new realms of possibility in the field of software development. This article delves into how Semantic Kernels are reshaping the future of programming and AI interaction."
Introducing Icepure's Compatible Water Filters: A Gateway to Cleaner Water | 677.169 | 1 |
How to construct a Regular Pentagon using just a compass and a straightedge
Any pentagon has: Sum of Interior Angles of 540° 5 diagonals; Make a Regular Pentagon. You can make a regular pentagon with a strip of paper! Start
Start The definition of pentagon shape is taken from the Greek. "Penta" denotes five and "gon" denotes angle.
In geometry, a pentagon (from the Greek πέντε pente and γωνία gonia, meaning five and angle) is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting regular pentagon (or star pentagon) is called a pentagram
A pentagram (sometimes known as a pentalpha, pentangle, pentacle or star pentagon) is the shape of a five-pointed star polygon.. Pentagrams were used symbolically in ancient Greece and Babylonia, and are used today as a symbol of faith by many Wiccans, akin to the use of the cross by Christians.
SACRED GEOMETRY SPIRITUAL MEANING OF THE PENTAGON AND THE PENTAGRAM INTRODUCTION The key dimension of both geometrical objects is five (5). CONNECTION TO THE GOLDEN RATIO (PHI) AREAS WITHIN THE GEOMETRY Count of enclosed spaces (Areas) Inside Pentagram = 6, Outside Pentagram = 5, Within Pentagon = 11. ANGLES WITHIN THE GEOMETRY | 677.169 | 1 |
Parallel and Perpendicular Line Calculator
Parallel and Perpendicular Line Calculator
Slope of Line 1:Slope of Line 2:
Navigating Parallel and Perpendicular Paths: Unveiling the Potential of the Calculator
Geometry and algebra hinge on the foundational concepts of parallel and perpendicular lines. These principles are integral to numerous mathematical and real-world scenarios, be it in the academic realm or professional fields such as geometry, architecture, or engineering. Mastering the understanding of these lines is pivotal, and to facilitate this, the Parallel and Perpendicular Line Calculator steps into the spotlight.
Unveiling Parallel Lines
Parallel lines, coursing in the same direction, stand as entities that will forever remain unacquainted, refusing intersection despite their infinite extension. The linchpin in identifying these lines lies in their shared slope or gradient, a consistent attribute that sets them apart.
Decoding Perpendicular Lines
Contrary to their parallel counterparts, perpendicular lines converge at a 90-degree angle, creating right angles at their meeting point. The relationship between the slopes of perpendicular lines is fascinatingly mathematical—their slopes are negative reciprocals of each other, culminating in a product of -1.
Maneuvering with the Parallel and Perpendicular Line Calculator
Entering the Equation Terrain: The initial step involves inputting the equations of the lines in question. Typically presented in the slope-intercept format, y = mx + b, where 'm' embodies the slope, and 'b' signifies the y-intercept.
Crunching the Numbers: The calculator, a virtual mathematician, swiftly computes the slopes of the specified lines, driven by the equations you furnish.
Delineating the Connection: Post-equation entry, the calculator assumes the role of a relationship arbitrator, decisively categorizing lines as parallel, perpendicular, or neither.
Calculator Advantages
Velocity in Identification: Instantaneous responses from the calculator accelerate the process of discerning the connection between two lines, salvaging precious time.
Precision Personified: By automating the identification of parallel and perpendicular lines, the calculator acts as a bulwark against errors that may infiltrate manual computations.
Educational Ally: For students navigating the labyrinth of mathematical concepts, the calculator evolves into a potent educational tool, unraveling the intricacies of line properties and relationships.
In conclusion, the Parallel and Perpendicular Line Calculator emerges as an indispensable ally for anyone immersing themselves in the realm of lines within geometry and algebra. It transforms the intricate task of ascertaining whether two lines share parallel or perpendicular trajectories into a seamless and efficient process. Be it the pursuit of mathematical knowledge or the practical application of these principles in professional spheres, this calculator stands as a beacon, illuminating the path to enhanced understanding and efficiency in handling lines. | 677.169 | 1 |
A Supplement to the Elements of Euclid
on the same side of it not less than two right angles, these lines shall never meet on that side, if produced ever so far.
For, if it be possible, let two straight lines meet, which make, with another straight line, the two interior angles, on the same side, not less than two right: Then it is plain, that the three straight lines will thus include a ▲, two of which are not less than two right angles; which (E. 17. 1.) is absurd. Wherefore, the two straight lines cannot meet, on that side of the straight line, on which they make the two interior not less
15. COR. Two straight lines, which are both perpendicular to the same straight line, are parallel to each other.
PROP. IX.
16. THEOREM. The three sides of a triangle taken together, exceed the double of any one side, and are less than the double of any two sides.
For, since (E. 20. 1.) any two sides of a ▲ are > the third, if the third side be added both to those two and to itself; it is evident that the three sides are, together, > the double of the third.
Again, since (E. 20. 1.) any side of a ▲ is < the other two, if the other two be added both to that side, and to themselves, it is evident, that the
three sides are, together,< than the double of the other two.
PROP. X.
17. THEOREM. Any side of a triangle is greater than the difference between the other two sides.
If, the A be equilateral, or isosceles, the proposition is manifestly true. But let it be a scalene A: Then, since (E. 20. 1.) any two sides of the ▲ are the third, if either of those two be taken from that third side, it is plain that the remaining side is greater than the difference of the other
two.
PROP. XI.
18. THEOREM. Any one side of a rectilineal figure is less than the aggregate of the remaining sides.
Then since (hyp.) BD=DC, and (constr.) ADDE, the two sides BD, DE, of the ▲ BDE, are equal to the two sides AD, DC of the A ADC;
and (E. 15. 1.) the BDE=ADC; ... (E. 4. 1.) BE=AC. But (E. 20. 1.) AB + BE > AE ; but AC has been proved to be equal to BE, and AE is (constr.) the double of AD; ... AB+ AC> 2 AD.
PROP. XIII.
20. THEOREM. The two sides of a triangle are, together, greater than the double of the straight line drawn from the vertex to the base, bisecting the vertical angle.
Let ABC be any given A, and let AD be drawn from the vertex A, to the base BC, bisecting the vertical BAC: Then, AB + AC>2 AD.
If the given ▲ be isosceles, the straight line which bisects the vertical is (E. 4. 1.) L to the base; and since (E. 17. 1. and E. 19. 1.) each of the equal sides is greater than the perpendicular, the proposition, is, in this case, manifestly true.
But, let ABC be a scalene A, and let the side AB be less than AC: Then, of the segments into which AD, bisecting the 4 BAC, divides the
21. COR. From the demonstration it is manifest, that of the segments into which the straight line bisecting the vertical of a scalene A, divides the base, that which is adjacent to the less side, is the less.
PROP. XIV.
22. THEOREM. If a trapezium and a triangle stand upon the same base, and on the same side of it, and the one figure fall within the other, that which has the greater surface shall have the greater perimeter. | 677.169 | 1 |
Practice A Bisectors In Triangles Answers
1 Practice A Bisectors In Answers Free PDF ebook Download: Practice A Bisectors In Answers Download or Read Online ebook practice a bisectors in triangles answers in PDF Format From The Best User Guide Database Lesson 5-2 Bisectors in Bisectors in. play a key role in relationships involving perpendicular bisectors and angle bisectors. Pra ctice 5-1 Midsegments of 10. a is a midsegment of ALMN.. Bisectors in. Practice 5-2. Use the gure at the right for Exercises Honors Geometry Chapter 5; Relationships Within I Lesson 5-2 Perpendicular and Angle Bisectors. Lesson 5-3 Bisectors in incenter. Bisectors of. Fill in the perpendicular bisectors of a triangle are concurrent.. You may use a compass and straightedge or a protractor. LESSON Some Books Bellow will present you all related to practice a bisectors in triangles answers! 5-2 Bisectors in 5-2 Bisectors In Lesson 5-2 Bisectors in Bisectors in. play a key role in relationships involving perpendicular bisectors and angle bisectors. This PDF book provide lesson 5 2 bisectors of triangle document. To download free 5-2 bisectors in triangles
6 Special Right Side 1: Practice 8.3 (Mixed Exercises) Special Right Side 1: Practice 8.3 (Mixed Exercises) Special Right. Side 1: Practice 8.3 (Mixed Exercises). 1. x = 2; y = sqrt(3). 2. a = 4; b = 4sqrt(3). 3. x = 3sqrt(3); y = x = 8sqrt(2). 5. y = 14sqrt(2). 6. This PDF book contain mixed practice with right triangles conduct. To download free special right triangles side 1: practice 8.3 (mixed exercises) Facts Practice Using Addition/Subtraction Fact Facts Practice Using Addition/Subtraction Fact Fact triangles are a type of flash card that group together families of related other person now identifies the missing number and the four facts in that fact family. This PDF book incorporate missing number fact triangles information. To download free facts practice using addition/subtraction fact triangles Practice CPC Exam Note: All answers have options of A-D answers Practice CPC Exam Note: All Answers Have Options Of A-D Answers written permission of HCPro, Inc. ( ). No claim b Per CPT Coding Guidelines for debridement in multiple wounds, sum the surface area of those. This PDF book provide hcpro coding module guide. To download free practice cpc exam note: all answers have options of a-d answers you Practice Questions and Answers from Lesson I-5: Efficiency Practice Practice Questions And Answers From Lesson I-5: Efficiency Practice Practice Questions and Answers from Lesson I-5: Efficiency. The following questions practice these skills: Compute consumer surplus from willingness to pay, This PDF book provide suena sam answers information. To download free practice questions and answers from lesson i-5: efficiency practice Perpendicular and Angle Bisectors Perpendicular And Angle Bisectors Reteach. Perpendicular and Angle Bisectors Theorem. Example. Perpendicular Bisector Theorem. If a point is on the perpendicular bisector of a segment This PDF book incorporate reteach perpendlicar and angle bisector conduct. To download free perpendicular and angle bisectors you 1-5 Segment and Angle Bisectors 1-5 Segment And Angle Bisectors To solve real-life problems, of a segment is the point that divides, or the segment into two congruent segments. In this book, matching red congruence marks. has coordinates.,. y1 + y2. 2 x1 + x2. 2. THE MIDPOINT FORMULA. Using. This PDF book incorporate real life story problems using coordinate points document. To download free 1-5 segment and angle bisectors 5-1 Perpendicular and Angle Bisectors 5-1 Perpendicular And Angle Bisectors Holt Geometry. Reteach. Perpendicular and Angle Bisectors. The Converse of the Perpendicular Bisector Theorem is also true. If a point is equidistant from the This PDF book include reteach perpendlicar and angle bisector information. To download free 5-1 perpendicular and angle bisectorsNew York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students
AG geometry domain Name: Date: Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (1 of 36 ) 1. Amy drew a circle graph to representCircles, Chords, Diameters, and Their Relationships Student Outcomes Identify the relationships between the diameters of a circle and other chords of the circle. Lesson Notes Students are asked to construct
High School - Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,
1st Nine Weeks Experiment with transformations in the plane G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point,
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Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the widthBlue Pelican Geometry Theorem Proofs Copyright 2013 by Charles E. Cook; Refugio, Tx (All rights reserved) Table of contents Geometry Theorem Proofs The theorems listed here are but a few of the total in
Instruction Goal: To provide opportunities for students to develop concepts and skills related to proving right, isosceles, and equilateral triangles congruent using real-world problems Common Core Standards
Sec 1.6 CC Geometry Triangle Proofs Name: POTENTIAL REASONS: Definition of Congruence: Having the exact same size and shape and there by having the exact same measures. Definition of Midpoint: The point
39 Symmetry of Plane Figures In this section, we are interested in the symmetric properties of plane figures. By a symmetry of a plane figure we mean a motion of the plane that moves the figure so that
lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:Heron s Formula Lesson Summary: Students will investigate the Heron s formula for finding the area of a triangle. The lab has students find the area using three different methods: Heron s, the basic formula,
dvanced Euclidean Geometry What is the center of a triangle? ut what if the triangle is not equilateral?? Circumcenter Equally far from the vertices? P P Points are on the perpendicular bisector of a lineSan Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS Recall that the bisector of an angle is the ray that divides the angle into two congruent angles. The most important results about angle bisectors
Name Period CHAPTER 9 Right Triangles and Trigonometry Section 9.1 Similar right Triangles Objectives: Solve problems involving similar right triangles. Use a geometric mean to solve problems. Ex. 1 UseHigh School Geometry Test Sampler Math Common Core Sampler Test Our High School Geometry sampler covers the twenty most common questions that we see targeted for this level. For complete tests and breakChapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are
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Chapter 9 Circles Objectives A. Recognize and apply terms relating to circles. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates,Name: lass: _ ate: _ I: SSS Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Given the lengths marked on the figure and that bisects E, use SSS to explain
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Chapter 11 Areas of Plane Figures You MUST draw diagrams and show formulas for every applicable homework problem! Objectives A. Use the terms defined in the chapter correctly. B. Properly use and interpret. Sum of Measures of Interior ngles Geometry 8-1 ngles of Polygons 1. Interior angles - The sum of the measures of the angles of each polygon can be found by adding the measures of the angles of a triangle.
Math The Arrow Way Free PDF ebook Download: Math The Arrow Way Download or Read Online ebook solving math the arrow way in PDF Format From The Best User Guide Database 4 SYSTEMS OF LINEAR EQUATIONS AND
GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,
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PRACTICAL GEOMETRY 57 Practical Geometry CHAPTER 4 4.1 Introduction You have learnt how to draw triangles in Class VII. We require three measurements (of sides and angles) to draw a unique triangle. Since | 677.169 | 1 |
How to Navigate in the Woods Without a Compass or Map
As an eighth grade math teacher, a lot of the stuff we teach kids makes no sense. Students rarely get a chance to apply mathematics in the real world. We're too busy pushing through the state mandated curriculum to get our hands dirty applying the concepts being taught.
A little dirt time in the woods or a homestead would go a long way in helping students (and teachers) trade theory for action. So put on your boots. School of the Woods is in session!
Like any other skill, estimating distance takes practice. The method I used in the video below is based on the Pythagorean Theorem → a² + b² = c². Don't freak out about the formula. We won't even use it!
Here's the cool thing about this method…
There's no math calculations involved! No square roots, no dividing, no multiplication, no algebra. If you can walk a straight line and count simple steps, you can use this method to estimate distance. In fact, all you really need is a stick.
Estimating Distance with Right Triangles
Estimations are more than guessing. They are based on calculations and useful for many tasks in bushcraft, homesteading, and outdoor self-reliance.
Here's a quick refresher on geometry terms we'll be using. A right triangle has two short sides called legs (a & b). The long side of the triangle is the hypotenuse (c).
What if you needed to ford a river, build a fence, or erect a foot bridge over a creek in the woods? I've never seen any of my woodsmen friends pull out a 100 foot measuring tape from their pack. But you can get an accurate estimation of width without a measuring device. | 677.169 | 1 |
If the sum of the lengths of the hypotenuse and one side of a right-angled triangle is given, the area of the triangle is maximum when the angle between these sides is
A
60∘
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B
90∘
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C
30∘
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D
120∘
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Open in App
Solution
The correct option is A60∘ Let ABC be a right-angled triangle in which side BC=x(say) and hypotenuse AC=y(say).
Given x+y=k(const.) ⇒y=k−x
Now, the area of the triangle ABC is given by A=12BC⋅AB=12x√(y2−x2)=12x√[(k−x)2−x2]
Let u=A2=14x2(k2−2kx) ⇒dudx=12k(kx−3x2) and d2udx2=12k(k−6x)
For maximum or minimum of u,dudx=0⇒x=k3(∵x≠0)
when x=k3,d2udx2=−12k2<0 ⇒u i.e., A is maximum when x=k3 and y=k−x=2k3.
Now, cosθ=BCAC=xy=12⇒θ=π3.
Hence, the required angle is π3. | 677.169 | 1 |
Groupe de soutien
How to Use the Pythagorean Theorem to Solve Lesson 6 Homework Problems
If you are struggling with lesson 6 homework practice on using the Pythagorean theorem, don't worry. In this article, we will explain what the Pythagorean theorem is, how to use it, and how to check your answers with a key. By the end of this article, you will be able to solve any right triangle problem with confidence.
Lesson 6 Homework Practice Use The Pythagorean Theorem Answer Key
What is the Pythagorean Theorem?
The Pythagorean This theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, often called the Pythagorean equation:
a + b = c
The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proven numerous times by many different methods possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.
How to Use the Pythagorean Theorem?
The Pythagorean theorem can be used to find any missing side length of a right triangle, as long as you know two other side lengths. To use the theorem, you need to follow these steps:
Identify which side is the hypotenuse and which sides are the legs of the right triangle.
Plug in the known side lengths into the Pythagorean equation and solve for the unknown side length.
Simplify your answer and write it with the appropriate units.
Let's look at an example:
In this right triangle, we know that a = 3 cm and b = 4 cm. We want to find c, which is the hypotenuse.
We plug in the known values into the Pythagorean equation: a + b = c
We get: 3 + 4 = c
We simplify: 9 + 16 = c
We get: 25 = c
We take the square root of both sides: c = 25
We simplify: c = 5
We write our answer with units: c = 5 cm
The Pythagorean theorem can also be used to check if a triangle is a right triangle or not. If a triangle has sides with lengths that satisfy the Pythagorean equation, then it must be a right triangle. Conversely, if a triangle does not have sides with lengths that satisfy the Pythagorean equation, then it cannot be a right triangle.
How to Check Your Answers with a Key?
To check your answers with a key, you need to compare your solutions with the correct ones provided by the key. You can do this by:
Making sure you have used the correct formula and followed the correct steps.
Making sure you have simplified your answers and written them with correct units.
Making sure you have rounded your answers to the appropriate decimal places or fractions if needed.
Making sure you have labeled your answers clearly and correctly.
If your answers match with the key, then you have done well. If your answers do not match with the key, then you need to review your work and find where you made a mistake. You can also ask for help from your teacher or peers if you are stuck.
How to Solve Lesson 6 Homework Practice Problems?
Now that you know how to use the Pythagorean theorem and check your answers with a key, you are ready to tackle some lesson 6 homework practice problems. Here are some tips to help you:
Read the problem carefully and identify what you are given and what you need to find.
Draw a diagram of the right triangle and label the sides with the given information.
Use the Pythagorean theorem to write an equation and solve for the unknown side length.
Check your answer by plugging it back into the equation and making sure it is true.
Compare your answer with the key and see if it matches.
Let's look at some examples:
Example 1: A ladder leans against a building. The base of the ladder is 8 feet from the building and the top of the ladder is 15 feet above the ground. How long is the ladder?
We are given that a = 8 ft and b = 15 ft. We need to find c, which is the length of the ladder.
We use the Pythagorean theorem to write an equation: a + b = c
We plug in the given values: 8 + 15 = c
We simplify: 64 + 225 = c
We get: 289 = c
We take the square root of both sides: c = 289
We simplify: c = 17
We write our answer with units: c = 17 ft
We check our answer by plugging it back into the equation: 8 + 15 = 17
We simplify: 64 + 225 = 289
We get: 289 = 289, which is true.
We compare our answer with the key and see that it matches.
Example 2: A rectangular rug has a length of 10 feet and a diagonal of 13 feet. What is the width of the rug?
We are given that b = 10 ft and c = 13 ft. We need to find a, which is the width of the rug.
We use the Pythagorean theorem to write an equation: a + b = c
We plug in the given values: a + 10 = 13
We simplify: a + 100 = 169
We subtract 100 from both sides: a = 69
We take the square root of both sides: a = 69
We approximate: a 8.3
We write our answer with units: a 8.3 ft
We check our answer by plugging it back into the equation: (8.3) + 10 13
We simplify: 68.9 + 100 169
We get: 168.9 169, which is close enough.
We compare our answer with the key and see that it matches.
How to Find More Practice Problems and Answer Keys?
If you want to practice more problems on using the Pythagorean theorem, you can find many online resources that offer worksheets, quizzes, games, and videos on this topic. Some of these resources are:
Quizlet: This website provides free solutions for Glencoe MATH Course 3, Volume 2, which includes lesson 6 on the Pythagorean theorem. You can also create your own flashcards and study sets on this topic.
Khan Academy: This website offers free videos, exercises, and quizzes on the Pythagorean theorem and its applications. You can also track your progress and earn badges as you learn.
ChiliMath: This website provides free worksheets and answer keys on the Pythagorean theorem practice problems. You can also find detailed explanations and examples on how to solve them.
These are just some of the online resources that you can use to practice and improve your skills on using the Pythagorean theorem. You can also ask your teacher or peers for more help if you need it.
How to Apply the Pythagorean Theorem to Real-World Problems?
The Pythagorean theorem is not only useful for solving geometry problems, but also for understanding and modeling many real-world situations. For example, you can use the Pythagorean theorem to:
Find the distance between two points on a map or a coordinate plane.
Find the height of a building, a tree, a mountain, or any other object by using its shadow and the angle of elevation.
Find the diagonal of a screen, a box, a room, or any other rectangular object.
Find the length of a ramp, a ladder, a slide, or any other inclined plane.
Find the missing side of a right triangle in trigonometry, physics, engineering, or any other field that involves angles and forces.
Let's look at some examples:
Example 3: A soccer field is 100 meters long and 60 meters wide. How far does a player run when he goes from one corner to the opposite corner of the field?
We are given that a = 100 m and b = 60 m. We need to find c, which is the distance that the player runs.
We use the Pythagorean theorem to write an equation: a + b = c
We plug in the given values: 100 + 60 = c
We simplify: 10000 + 3600 = c
We get: 13600 = c
We take the square root of both sides: c = 13600
We approximate: c 116.6
We write our answer with units: c 116.6 m
Example 4: A person is flying a kite that is attached to a string of length 50 feet. The string makes an angle of 40 degrees with the ground. How high is the kite above the ground?
We are given that c = 50 ft and A = 40. We need to find b, which is the height of the kite.
We use the Pythagorean theorem to write an equation: a + b = c
We use trigonometry to find a in terms of c and A: a = c cos A
We plug in the given values: (50 cos 40) + b = 50
We simplify: (38.3) + b = 2500
We subtract (38.3) from both sides: b = 1054.5
We take the square root of both sides: b = 1054.5
We approximate: b 32.5
We write our answer with units: b 32.5 ft
Conclusion
In this article, we have learned how to use the Pythagorean theorem to solve lesson 6 homework practice problems. We have also learned how to check our answers with a key and how to apply the Pythagorean theorem to real-world problems. The Pythagorean theorem is a powerful and versatile tool that can help us understand and model many situations involving right triangles. We hope that this article has helped you master the Pythagorean theorem and improve your math skills. d282676c82 | 677.169 | 1 |
The Proof. So, let's say we have two lines L1, and L2 intersected by a transversal line, L3, creating 2 corresponding angles, 1 & 2 which are congruent (∠1 ≅ ∠2, m∠1=∠2). We want to prove the L1 and L2 are parallel, and we will do so by contradiction. Assume L1 is not parallel to L2. Then, according to the parallel line axiom we started ...In today's lesson, we will prove the converse to the Base Angle theorem - if two angles of a triangle are congruent, the triangle is isosceles. We will use congruent triangles for the proof. From the definition of an isosceles triangle as one in which two sides are equal, we proved the Base Angles Theorem - the angles between the equal sides …If you're a fan of challenging platformer games, then you've probably heard of Geometry Dash. This popular game has gained a massive following due to its addictive gameplay and cat...Dec 16, 2020 ... Math Lesson: Converse of Pythagoras Theorem (Acute, Right or Obtuse)(With Examples) ... KutaSoftware: Geometry- The Pythagorean Theorem And Itsconverse: [verb] to have acquaintance or familiarity. to become occupied or engaged. Eucl of the 19th century, when non-Euclidean …Sep 23, 2021 ... ... examples. Equivalent propositions are explained by establishing the ... Converse, Inverse, and Contrapositive: Lesson (Geometry Concepts). CK ...Maybe you talk too much in conversation; maybe you clam up. Either way, communication skills don't come naturally for everyone. For a better conversational flow, use the 50/50 ratiSep 12, 2014 ... Comments30 ; Converse, Inverse, & Contrapositive - Conditional & Biconditional Statements, Logic, Geometry. The Organic Chemistry Tutor · 539K ...Home All Definitions Geometry Diameter Definition. Diameter Definition. Diameter is a line segment connecting two points on a circle or sphere which pass through the center. Diameter is also used to refer to the specific length of this line segment. More specifically, the diameter of a circle is the distance from a point on the circle to a point π radians …Converse : In Mathematical Geometry, a Converse is defined as the inverse of a conditional statement. It is switching the hypothesis and conclusion of a conditional statement. Geometry is an important subject for children to learn. It helps them understand the world around them and develop problem-solving skills. But learning geometry can be a challenge Mar 10, 2019 ... See here, the definitions of the word converse, as video and text. (Click show more below.) converse (verb) To keep company; ...Feb Correspond ExampleHere you'll learn how to find the converse, inverse and contrapositive of a conditional statement. You will also learn how to determine whether or not a statement is biconditional. ThisOct 29, 2021 · In today's geometry lesson, we'll prove the converse of the Alternate Interior Angles Theorem.. We have shown that when two parallel lines are intersected by a transversal line, the interior alternating angles and exterior alternating angles are congruent (that is, they have the same measure of the angle.) An elementary theorem in geometry whose name means "asses' bridge," perhaps in reference to the fact that fools would be unable to pass this point in their geometric studies. The theorem states that the angles at the base of an isosceles triangle (defined as a triangle with two legs of equal length) are equal and appears as the fifth …In geometry, the hinge theorem (sometimes called the open mouth theorem) states that if two sides of one triangle are congruent to two sides of another triangle, and the Optimize your conversion rate at Conversion Conference 2023 by learning some key aspects of conversion techniques in a digital world. Conversion rate optimization (CRO) is a core fSupplementary angles refer to the pair of angles that always sum up to 180°. The word 'supplementary' means 'something when supplied to complete a thing'. Therefore, these two angles are called supplements of each other. Let us learn more about the definition and meaning of supplementary angles along with some supplementary angles examples.Sep 12, 2014 ... Comments30 ; Converse, Inverse, & Contrapositive - Conditional & Biconditional Statements, Logic, Geometry. The Organic Chemistry Tutor · 539K ...Converse. The hypothesis and conclusion are switched. Inverse. The inverse is formed by negating the hypothesis and conclusion. Contrapositive. Where you switch and negate the hypothesis and conclusion. Bi Conditional Statement. When a conditional statement has the phrase "If and only If". Used when the conditional and its converse are both true. Geometry is an important subject that children should learn in school. It helps them develop their problem-solving skills and understand the world around them. To make learning geo...ATherefore, the converse of a statement P ⇒ Q is Q ⇒ P. It should be observed that P ⇒ Q and Q ⇒ P are converse of each other. In Geometry, we have come across the …Find 30 different ways to say CONVERSE, along with antonyms, related words, and example sentences at Thesaurus.com.Converse statements are often used in geometry to prove that a set of lines are parallel. Learn about the properties of parallel lines and how to use converse statements to prove …Geometry games are a great way to help children learn and practice math skills. Not only do they provide an enjoyable way to practice math, but they can also help children develop ...Definition Midpoint Definition. The midpoint of a line segment is a point that divides the line segment into two equal halves. In other words, the midpoint is in the exact middle of the line segment. An converse of the theorem is true as well. That is if a line is drawn through the midpoint of triangle side parallel to another triangle side then the line will bisect the third side of the triangle. The triangle formed by the three parallel lines through the three midpoints of sides of a triangle is called its medial triangle. ProofThe Consecutive Interior Angles Theorem states that the consecutive interior angles on the same side of a transversal line intersecting two parallel lines are supplementary (That is, their sum adds up to 180). Here we will prove its converse of that theorem. We will show that if the consecutive interior angles on the same side of a …Find 30 different ways to say CONVERSE, along with antonyms, related words, and example sentences at Thesaurus.com.Corresponding Angles Converse. If 2 lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel. Transitive Property of Parallel Lines. If 2 lines are parallel to the same line, then they are parallel to each other. Study with Quizlet and memorize flashcards containing terms like Alternate Interior ...Correspond CorrespondAngle bisector theorem states that an angle bisector divides the opposite side into two line segments that are proportional to the other two sides. Here, in $\Delta ABC$, the line AD is the angle bisector of $\angle A$. AD bisects the side BC in two parts, c and d. a and b are the lengths of the other two sides.The inventor of geometry was Euclid, and his nickname was The Father of Geometry. Euclid obtained his education at Plato's Academy in Athens, Greece and then moved to Alexandria.Here you'll learn how to find the converse, inverse and contrapositive of a conditional statement. You will also learn how to determine whether or not a statement is biconditional. This...In this Geometry lesson you will learn about how to create biconditional statements and definitions from conditional statements and their converse.Learn Oct 3, 2022 ... Inverse converse and contrapositive are examples of conditional statements and we will take a ... geometry #maths #logic. triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the L p spaces ( p ≥ 1 ), and inner product spaces .Converse of alternate interior angles theorem. The converse of the alternate interior angles theorem states that if two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel. Alternate interior angles examples. We can prove both these theorems so you can add them to your toolbox.Aug 11, 2014 · Discover more at you'll learn how to find the converse, inverse and cont...Angle Bisector. An angle bisector is defined as a ray, segment, or line that divides a given angle into two angles of equal measures. The word bisector or bisection means dividing one thing into two equal parts. In geometry, we usually divide a triangle and an angle by a line or ray which is considered as an angle bisector ConAn explanation and proof of the side splitter theorem and a discussion of its converse. This video is provided by the Learning Assistance Center of Howard Co This geometry video tutorial explains how to write the converse, inverse, and contrapositive of a conditional statement - if p, then q. This video also discusses the definition of a biconditional ... Jan 18, 2019 ... Converse, Inverse, & Contrapositive - Conditional & Biconditional Statements, Logic, Geometry. The Organic Chemistry Tutor•539K views · 5:43. Go&nbs...TransConverse geometry definition, weather 11234, a perfectly useless afternoon
The angle subtended by a chord (or two radii) at the center of a circle is two times the angle subtended by it on the remaining part of the circle. _\square . Let us now try to prove Thales' theorem with the help of the above theorem. According to the angle segment theorem, we have the following diagram: \angle AOB = 2 \angle ADB. ∠AOB = 2∠ADB.. Converse geometry definition
ToZeroGeometry Dash is an addictive rhythm-based platformer game that challenges players with its fast-paced levels and catchy soundtrack. With its online play feature, players can compe...Mar 10, 2019 ... See here, the definitions of the word converse, as video and text. (Click show more below.) converse (verb) To keep company; ConWe would like to show you a description here but the site won't allow us. People with ADHD have a hard time with conversation. They might get distracted and lose track of what the othe People with ADHD have a hard time with conversation. They might get d...Geometry is defined as the area of mathematics dealing with points, lines, shapes and space. Geometry is important because the world is made up of different shapes and spaces. Geom...Zero of a Function. A value of x which makes a function f (x) equal zero. In other terms a value of x such that f (x) = 0. A zero of a function may be a real or complex number. < All Applied Mathematics >. Browse our growing collection of algebra definitions. 61. 4.2K views 5 years ago High School Geometry Course. A review of the Corresponding angles postulate with an explanation of the Latin meaning of converse.ConGeometry Definitions. Browse our growing collection of geometry definitions: A B C E ABC ~ DEF D F. AA Similarity or angle angle similarity means when two triangles have …Home All Definitions Geometry Height of a Cylinder Definition. Height of a Cylinder Definition. The height or altitude of a cylinder is the distance between the bases of a cylinder. It is the shortest line segment between the (possibly extended) bases. Height can also be used to refer to the specific length of this segment.Sep 23, 2021 ... ... examples. Equivalent propositions are explained by establishing the ... Converse, Inverse, and Contrapositive: Lesson (Geometry Concepts). CK TransFlexBook Platform®, FlexBook®, FlexLet® and FlexCard™ are registered trademarks of CK-12 Foundation.Apr 15, 2011 ... Proof: Consecutive Interior Angles Converse. 15K views · 12 years ago ... 5 Tips to Solve Any Geometry Proof by Rick Scarfi. HCS Math Class by ...Don't underestimate the value of knowing how to start a conversation when networking in a business setting to make a long-lasting impression. Knowing what to say is a big part of b...In geometry, one might wonder what the definition of Converse is. Author has 3.8k responses and 3.3 million answer views, as of May 27, 2017. In geometry, a conditional statement is reversed from the premise "if p" and the conclusion "then q." If a polygon is a square, it has four sides. This statement is correct. Are you ready to dive into the exciting world of Geometry Dash? This addictive rhythm-based platformer has captivated gamers around the globe with its challenging levels and catchy...Try these one-liners to excuse yourself gracefully from awkward networking conversations. Trusted by business builders worldwide, the HubSpot Blogs are your number-one source for e...Segment addition postulate. If B is between A and C, then AB + BC= AC. Segment addition post. converse. If AB + BC= AC, then B is between A and C. Angel addition postulate. If P is in the interior of <RST, then m<RST=m<RSP + m<PST. Linear Pair postulate. if two angles form a linear pair, then they are supplementary. Parallel Postulate.Optimize your conversion rate at Conversion Conference 2023 by learning some key aspects of conversion techniques in a digital world. Conversion rate optimization (CRO) is a core f...Converse of the Perpendicular Bisector Theorem Example. You can prove or disprove this by dropping a perpendicular line from Point T through line segment HD. Where your perpendicular line crosses HD, call it Point U. If Point T is the same distance from Points H and D, then HU ≅ UD.Find 30 different ways to say CONVERSE, along with antonyms, related words, and example sentences at Thesaurus.com.Maybe you talk too much in conversation; maybe you clam up. Either way, communication skills don't come naturally for everyone. For a better conversational flow, use the 50/50 rati...DefinitionJan 5, 2015 ... Converse: Switch the order and the inverse: you negate and the contrapositive: you switch and you negate conditional. If m m is a prime number, then it is an odd number. contrapositive. If m m is not an odd number, then it is not a prime number. converse. If m m is an odd number, … ConBy definition, perpendicular lines are two lines that intersect at a single point that create four 90 ∘ angles. The most well-known set of perpendicular lines are the axes found on the ...The converse is also true. ... Geometry problems can be solved with the help of circle theorems. 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Next, let there be four magnitudes, A, B, C, and D, and other four, E, F, G, and H, which, taken two and two, in a cross order, have the same ratio, viz. A:B::G: H; B:C:: F: G, and C: D:: E: F, then A: D:: E: H. For, since A, B, C are three magnitudes, and F, G, H other three, which, taken two and two, in a cross order, have the same ratio, by the first case, A:C::F: H. But C: D:: E: F, therefore again, by the first case, A: D:: E: H. In the same manner may the demonstration be extended to any number of magnitudes.
PROP. XXIV. THEOR.
If the first has to the second the same ratio which the third has to the fourth; and the fifth to the second, the same ratio which the sixth has to the fourth; the first and fifth, together, shall have to the second, the same ratio which the third and sixth, together, have to the fourth.
1. Rectilinear figures are said to be similar, when they have their several angles equal, each to each, and the sides about the equal angles proportional.
2. Two sides of one figure are said to be reciprocally proportional to two sides of another figure, when one of the sides of the first is to one of the sides of the second, as the remaining side of the second is to the remaining side of the first.
3. A straight line is said to be cut in extreme and mean ratio, when the whole is to the greater segment, as the greater segment is to the less.
4. The altitude of any figure is the straight line drawn from its vertex perpendicular to its
base.
П
PROP. I. THEOR.
Triangles (CAB, EAD) and parallelograms (CKAB, EALD), having the same altitude, are to one another as their bases (CB, ED).
Since the triangles and parallelograms have the same altitude AP, their bases must lie in the same straight line CD. Produce CD both ways towards H and N; and in the
H
K A L
line produced from BC, take consecutively any number of parts, as CF, FG, GH, each equal to BC; and also in the line produced from ED, take any number of parts, as DM, MN, each equal to ED; and join AF, AG,
D M N
AH, AM, and AN. Then the tri- H G F C B PE angles EAD, DAM, MAN are all equal (i. Prop. 38); and consequently the triangle EAN is the same multiple of EAD, as the base EN is of ED. For like reasons, the triangle BAH is the same multiple of BAC as the base BH is of BC. But if the base EN=BH, then likewise the triangle EAN = BAH (i. Prop. 38); and consequently also, if EN BH, EAN BAH; and if EN ▼BH, EAN▼BAH. The same conclusion may be drawn, whatever multiples be assumed of the bases CB and ED; and therefore CAB: EAD::CB:ED (v. Def. 5).
COR.-Hence it is manifest that triangles and parallelograms, having equal altitudes, are proportional to their bases. For, if the figures be so placed that their bases may lie in the same straight line, then since their altitudes are equal, the straight line joining the vertices of the triangles will be parallel to that in which their bases lie (i. Prop. 33), and the demonstration proceeds as above.
PROP. II. THEOR.
If a straight (DE) be drawn parallel to one of the sides (BC) of a triangle (BAC), it will cut the other sides, or the other sides produced, proportionally: and if the sides (AC, AB), or the sides produced, be cut proportionally, the straight line (DE) which joins the points of section, will be parallel to the remaining side of the triangle.
Join BE, CD; and since DE is parallel to BC (Hyp.), the triangles BED, CDE are equal (i. Prop. 37); and therefore
have the same ratio to the triangle DAE (v. Prop. 7), or BED:DAE::CDE:DAE; but BED: DAE :: BD : DA, and also CDE: DAE::CE: EA (vi. Prop. 1); therefore BD: DA::CE: EA (v. Prop. 11), E or the sides of the triangle are cut proportionally by the line drawn parallel to the base.
But if the hypothesis be that the sides of the triangle, or those sides produced, are cut proportionally, then the line joining the points of section must be parallel to the base. For the
D
C
triangle BED : DAE :: BD : DA, and also CDE: DAE::CE: EA (vi. Prop. 1); but BD:DA ::CE: EA (Hyp.), and therefore BED: DAE B :: CDE: DAE (v. Prop. 11), and consequently BED=CDE (v. Prop. 9); but these equal triangles being upon the same base DE, must be between the same parallels (i. Prop. 39), and therefore DE joining the points of section of the sides or of the sides produced, is parallel to the base BC.
PROP. III. THEOR.
If an angle (BAC) of a triangle be bisected by a straight line (AD) which also cuts the base, the segments of the base shall be proportional to the other sides of the triangle (BD: DC:: BA: AC); or, if the segments of the base be proportional to the other sides of the triangle, the straight line (AD) drawn from the vertex to the point of section, shall bisect the vertical angle.
E
From the point C draw CE parallel to DA and meeting BA produced to E. Then, because DA and CE are parallel, ▲ DAC = ≤ ACE, and also / DAB = AEC (i. Prop. 29); but DAC = ≤ DAB (Hyp.), and therefore also ACE = ▲ AEC, and AE = AC (i. Prop. 6). But since DA is parallel to CE, BD: DC :: BA: AE (vi. Prop. 2); and since AE = AC, BA AC :: BA : AE (v. Prop. 7); therefore BD: DC ::BA: AC, or the segments of the base made by the line bisecting the opposite angle, are proportional to the sides containing that angle. | 677.169 | 1 |
Tan 74? Diagram's answer is the key.
Sure, here's a brief introduction for your blog post:
Welcome to Warren Institute! In this article, we'll explore the concept of finding the tangent of an angle using a diagram. Understanding trigonometric ratios like tangent (tan) is crucial in mathematics education, and this visual approach will help you grasp the concept with ease. We will delve into the specifics of how to utilize diagrams to determine tan 74 and the significance of this calculation in practical applications. Let's dive into the world of trigonometry and uncover the secrets hidden within diagrams.
The concept of tangent in trigonometry
Tangent is a fundamental concept in trigonometry that represents the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle. It is often used to solve for unknown angles or sides in a triangle. Understanding the concept of tangent is crucial for students to grasp the relationships between angles and sides in trigonometric functions.
Calculating the tangent of an angle
To calculate the tangent of an angle, such as tan 74°, students can use the formula tan θ = opposite/adjacent. In the case of tan 74°, students need to identify the length of the opposite and adjacent sides in the given triangle and then apply the tangent formula to find the value.
Interpreting the results
Once students have calculated the tangent of 74°, they should understand that the result represents the ratio of the length of the opposite side to the length of the adjacent side in the specific triangle. It's essential for them to interpret the result in the context of the given diagram to understand its significance in solving trigonometric problems.
Practical applications of tangent
Understanding tangent is not only important for solving mathematical problems but also has practical applications in various fields such as engineering, physics, and architecture. Students should be encouraged to explore real-world scenarios where the concept of tangent is used to solve practical problems, reinforcing the relevance of trigonometry in everyday life.
frequently asked questions
How can students use diagrams to understand and solve trigonometric problems?
Students can use diagrams to visually represent trigonometric problems, allowing them to visualize the relationships between angles and sides in a triangle. This visual aid can help them understand and solve trigonometric problems more effectively.
What strategies can help students accurately identify angles and calculate trigonometric values based on given diagrams?
Visual aids such as diagrams and interactive manipulatives can help students accurately identify angles and understand trigonometric values. Additionally, practice problems and real-world applications can enhance their understanding.
In what ways do educators incorporate visual aids, such as diagrams, to teach trigonometric concepts like tangent in the classroom?
Educators incorporate visual aids such as diagrams to teach trigonometric concepts like tangent in the classroom by using interactive whiteboards, geometric models, and graphing calculators to visually represent the relationships between angles and sides in right-angled triangles.
How does understanding the relationship between angles and trigonometric ratios enhance students' problem-solving skills when interpreting diagrams?
Understanding the relationship between angles and trigonometric ratios enhances students' problem-solving skills by enabling them to accurately interpret diagrams and calculate unknown quantities, leading to a deeper understanding of geometric concepts.
What are effective instructional approaches for guiding students to determine the value of tan 74 based on a given diagram?
Using visual aids and real-life examples can be effective instructional approaches for guiding students to determine the value of tan 74 based on a given diagram in Mathematics education.
In conclusion, according to this diagram, the value of tan 74 can be determined using the given information. It is crucial for students to understand the concept of trigonometric functions and how to apply them in various mathematical problems. This knowledge is essential for their success in Mathematics education and beyond | 677.169 | 1 |
... and beyond
How do you use a calculator to evaluate #tan3#?
1 Answer
Explanation:
At time your question sounds confuaing and not much from trignometry rather technology.
I'm more familiar with degrees than radians. So first going on with degrees. #tan 3°=0.0524077793#
All you need to do is too just tap on 'tan' and then input angle in degrees or radians ,whatever your calculator supports.
If you type this or commamd this to google assistant then it google calcultor generally accepts it in radians form (in case you haven't altered settings) you could choose between the 3 units of angle available there. | 677.169 | 1 |
Mathematics is a fascinating subject that encompasses a wide range of concepts and formulas. One such concept that holds great significance in trigonometry is "2 sin a sin b." In this article, we will delve into the depths of this mathematical expression, understand its meaning, explore its applications, and provide valuable insights to the reader.
Understanding "2 sin a sin b"
Before we dive into the applications of "2 sin a sin b," let's first understand what this expression represents. In trigonometry, the sine function (sin) relates the angles of a right triangle to the ratios of its sides. When we multiply two sine values, such as sin a and sin b, we obtain the product of their respective ratios.
The expression "2 sin a sin b" signifies the double angle formula for sine. It is derived from the trigonometric identity:
sin(2θ) = 2 sin θ cos θ
By substituting θ with a – b, we can rewrite the formula as:
sin(2(a – b)) = 2 sin(a – b) cos(a – b)
Expanding the right side of the equation, we get:
2 sin a cos b – 2 cos a sin b
Thus, "2 sin a sin b" is an equivalent expression for the double angle formula for sine.
Applications of "2 sin a sin b"
Now that we have a clear understanding of the mathematical concept, let's explore some practical applications where "2 sin a sin b" finds its utility:
1. Harmonic Analysis in Music
Harmonic analysis plays a crucial role in understanding the structure and composition of music. By applying "2 sin a sin b" in the context of music theory, we can analyze the harmonics present in a musical piece. The expression helps identify the fundamental frequency and its harmonics, enabling musicians and composers to create harmonious melodies.
For example, when analyzing the harmonics of a guitar string, "2 sin a sin b" can be used to determine the frequencies produced by plucking the string at different positions. This knowledge aids in tuning the instrument and achieving the desired musical notes.
2. Signal Processing and Fourier Analysis
In the field of signal processing, Fourier analysis is a fundamental technique used to decompose complex signals into simpler sinusoidal components. The expression "2 sin a sin b" finds its application in Fourier analysis, allowing us to analyze and manipulate signals in various domains.
For instance, in image processing, the Fourier transform is used to convert an image from the spatial domain to the frequency domain. By applying "2 sin a sin b" in the Fourier transform, we can extract specific frequency components from the image, enabling tasks such as noise removal, image compression, and pattern recognition.
3. Mechanical Engineering and Vibrational Analysis
In mechanical engineering, vibrational analysis is crucial for understanding the behavior of structures subjected to dynamic loads. The expression "2 sin a sin b" plays a significant role in analyzing and predicting the vibrations of mechanical systems.
For example, when studying the vibrations of a bridge, "2 sin a sin b" can be used to determine the natural frequencies at which the bridge tends to vibrate. This knowledge helps engineers design structures that can withstand external forces and minimize the risk of resonance-induced failures.
Q&A
1. What is the significance of the double angle formula for sine?
The double angle formula for sine allows us to express the sine of a double angle in terms of the sine and cosine of the original angle. It finds applications in various fields, including trigonometry, calculus, physics, and engineering.
2. Can "2 sin a sin b" be simplified further?
No, "2 sin a sin b" is already in its simplest form. It represents the double angle formula for sine and cannot be further simplified using trigonometric identities.
3. Are there any other double angle formulas in trigonometry?
Yes, apart from the double angle formula for sine, there are double angle formulas for cosine and tangent as well. These formulas provide relationships between the trigonometric functions of a double angle and the trigonometric functions of the original angle.
4. Can "2 sin a sin b" be used in other branches of mathematics?
While "2 sin a sin b" is primarily used in trigonometry and its applications, it may find relevance in other branches of mathematics that involve trigonometric functions, such as calculus, differential equations, and complex analysis.
5. How can I apply "2 sin a sin b" in real-life problem-solving?
By understanding the concept of "2 sin a sin b" and its applications, you can apply it to solve various real-life problems. For instance, you can use it to analyze vibrations in mechanical systems, process signals in digital communication, or even explore the harmonics in musical instruments.
Summary
In conclusion, "2 sin a sin b" is a powerful mathematical expression that finds applications in various fields, including music, signal processing, and mechanical engineering. By understanding the double angle formula for sine and its significance, we can leverage this concept to solve real-life problems and gain valuable insights. Whether it's analyzing harmonics in music, processing signals in digital communication, or predicting vibrations in mechanical systems, "2 sin a sin b" proves to be an indispensable tool in the realm of mathematics and its applications. | 677.169 | 1 |
Barycentric coordinates
The point D has bartcentric coordinates a, b, c with respect to the points A,B,C, where:
* a is the ratio of the red area to the total shaded area
* b is the ratio of the green area to the total shaded area
* c is the ratio of the blue area to the total shaded area
We have D=aA+bB+cC, and a+b+c=1. | 677.169 | 1 |
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Trigonometry.
.
First, where do these half-angle tangent identities come from?
Use the ratio identity for tangent and fill in the half-angle identities for sine and cosine.
You can leave off the ± sign because you won't have to choose which sign to use with the tangent identity — the square-root sign squares out of the equation.
Put the numerator and denominator under the same radical and then simplify the complex fraction.
Multiply the numerator and denominator by the conjugate (same terms, different sign) of the denominator.
Replace the denominator by using the Pythagorean identity and then simplify by putting the radical over the numerator and denominator, individually.
To find the other form of this half-angle tangent identity, change Step 3 by multiplying the numerator and denominator of the fraction by the conjugate of the numerator instead of the denominator | 677.169 | 1 |
Exibir resultados
Calculation of azimuth and solar elevation angle by given the coordinates and time of observation. It's possible to input coordinates manually or by selecting from the directory of cities.
Calculates the distance between two points of the Earth specified geodesic (geographical) coordinates along the shortest path - the great circle (orthodrome). Calculates the initial and final course angles and azimuth at intermediate points between the two given.
This online calculator converts the value of an angle given in the degrees-minutes-seconds to degrees, expressed in decimal fraction and back from decimal fraction to degrees-minutes-seconds.
Here you can find the set of calculators related to circular segment: segment area calculator, arc length calculator, chord length calculator, height and perimeter of circular segment by radius and angle calculator. | 677.169 | 1 |
Examiners report
Many candidates gave a correct vector equation for the line.
a.
A common error was to misplace the initial position and direction vectors. Those who set the scalar product of the direction vectors to zero typically solved for k successfully. Those who substituted \(k = – 2\) earned fewer marks for working backwards in a "show that" question.
b.
Many went on to find the coordinates of point A, however some used the same letter, say p, for each parameter and thus could not solve the system.
c.
Part (d) proved challenging as many candidates did not consider that \(\overrightarrow {{\rm{AB}}} + \overrightarrow {{\rm{BC}}} = \overrightarrow {{\rm{AC}}} \) . Rather, many attempted to find the coordinates of point C, which became a more arduous and error-prone task.
Examiners report
Part (a) was generally done well with candidates employing different correct methods to find the vector \(\overrightarrow {{\rm{BC}}} \) . Some candidates subtracted the given vectors in the wrong order and others simply added them. Calculation errors were seen with some frequency.
a.
Many candidates did not appear to know how to find a unit vector in part (b). Some tried to write down the vector equation of a line, indicating no familiarity with the concept of unit vectors while others gave the vector (1, 1, 1) or wrote the same vector \(\overrightarrow {{\rm{AB}}} \) as a linear combination of i, j and k. A number of candidates correctly found the magnitude but did not continue on to write the unit vector.
b.
Candidates were generally successful in showing that the vectors in part (c) were perpendicular. Many used the efficient approach of showing that the scalar product equalled zero, while others worked a little harder than necessary and used the cosine rule to find the angle between the two vectors.
Examiners report
The majority of candidates were successful on part (a), finding vectors between two points and using the scalar product to show two vectors to be perpendicular.
a(i), (ii) and (iii).
Although a large number of candidates answered part (b) correctly, there were many who had trouble with the vector equation of a line. Most notably, there were those who confused the position vector with the direction vector, and those who wrote their equation in an incorrect form.
b(i) and (ii).
In part (c), most candidates seemed to know what was required, though there were many who made algebraic errors when solving for the parameters. A few candidates worked backward, using \(k = 1\) , which is not allowed on a "show that" question.
c.
In part (d), candidates attempted many different geometric and vector methods to find the area of the triangle. As the question said "hence", it was required that candidates should use answers from their previous working – i.e. \({\rm{AC}} \bot {\rm{BD}}\) and \({\text{P}}(3{\text{, }}1)\) . Some geometric approaches, while leading to the correct answer, did not use "hence" or lacked the required justification.
Examiners report
Many candidates answered (a) correctly, although some reversed the vectors when finding \(\overrightarrow {{\rm{BC}}} \) , while others miscopied the vectors from the question paper.
a(i) and (ii).
Students had no difficulty finding the scalar product and magnitudes of the vectors used in finding the cosine. However, few recognized that \(\overrightarrow {{\rm{BA}}} \) is the vector to apply in the formula to find the cosine value. Most used \(\overrightarrow {{\rm{AB}}} \) to obtain a positive cosine, which neglects that the angle is obtuse and thus has a negative cosine. Surprisingly few students could then take a value for cosine and use it to find a value for sine. Most left (bii) blank entirely.
b(i) and (ii).
Part (c) proved accessible for many candidates. Some created an expression for \(|\overrightarrow {{\rm{CD}}} |\) and then substituted the given \(p = 3\) to obtain \(\sqrt {50} \) , which does not satisfy the "show that" instruction. Many students recognized that the scalar product must be zero for vectors to be perpendicular, and most provided the supporting calculations.
Examiners report
In part (a), nearly all the candidates correctly found the vector PQ, and the majority went onto find the correct vector equation of the line. There are still many candidates who do not write this equation in the correct form, using "r = ", and these candidates were penalized one mark.
a(i) and (ii).
In part (b), the majority of candidates knew to set the scalar product equal to zero for the perpendicular vectors, and were able to find the correct value of p.
b(i) and (ii).
A good number of candidates used the correct method to find the intersection of the two lines, though some algebraic and arithmetic errors kept some from finding the correct final answer.
Examiners report
While many candidates can find a vector given two points, few could write down a fully correct vector equation of a line.
a.i.
While many candidates can find a vector given two points, few could write down a fully correct vector equation of a line. Most candidates wrote their equation as " \({L_1} = \) ", which misrepresents that the resulting equation must still be a vector.
a.ii.
Those who recognized that vector perpendicularity means the scalar product is zero found little difficulty answering part (b). Occasionally a candidate would use the given \(p = 6\) to show the scalar product is zero. However, working backward from the given answer earns no marks in a question that requires candidates to show that this value is achieved.
b.
While many candidates knew to set the lines equal to find an intersection point, a surprising number could not carry the process to correct completion. Some could not solve a simultaneous pair of equations, and for those who did, some did not know what to do with the parameter value. Another common error was to set the vector equations equal using the same parameter, from which the candidates did not recognize a system to solve. Furthermore, it is interesting to note that while only one parameter value is needed to answer the question, most candidates find or attempt to find both, presumably out of habit in the algorithm.
c.
Marks available
6
Reference code
18M.1.sl.TZ1.6
Question
Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon. This is shown in the following diagram. | 677.169 | 1 |
Form : ………………………..……
3472/1Matematik TambahanKertas 1Ogos 20072 Jam
SULIT 3472/1
The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used.
ALGEBRA
1
2 am an = a m + n
3 am an = a m - n
4 (am) n = a nm
5 loga mn = log am + loga n
6 loga = log am - loga n
7 log a mn = n log a m
8 logab =
9 Tn = a + (n-1)d
10 Sn =
11 Tn = ar n-1
12 Sn = , (r 1)
13 , <1
CALCULUS
1 y = uv ,
2 , ,
3
4 Area under a curve
= dx or
= dy
5 Volume generated
= dx or
= dy
3472/1 2007 Hak Cipta SBP [ Lihat sebelahSULIT
2
5 A point dividing a segment of a line
( x,y) =
6 Area of triangle =
1 Distance =
2 Midpoint
(x , y) = ,
3
4
GEOMETRY
SULIT 3472/1
STATISTIC
3472/1 2007 Hak Cipta SBP [ Lihat sebelahSULIT
3
1 Arc length, s = r
2 Area of sector , L =
3 sin 2A + cos 2A = 1
4 sec2A = 1 + tan2A
5 cosec2 A = 1 + cot2 A
6 sin 2A = 2 sinA cosA
7 cos 2A = cos2A – sin2 A = 2 cos2A - 1 = 1 - 2 sin2A
8 tan 2A =
TRIGONOMETRY
9 sin (A B) = sinA cosB cosA sinB
10 cos (A B) = cosA cosB sinA sinB
11 tan (A B) =
12
13 a2 = b2 + c2 - 2bc cosA
14 Area of triangle =
1 =
2 =
3 = =
4 = =
5 m =
6
7
8
9
10 P(A B) = P(A)+P(B)- P(A B)
11 P (X = r) = , p + q = 1
12 Mean µ = np
13
14 z =
Answer all questions.
1. Given set A = {9, 36, 49, 64} and set B = {-8, -6, 3, 4, 6, 7, 8}. The relation from set A to set B is "the square root of ", state (a) the range of the relation,
21. A set of numbers has a median of 5 and a standard deviation of 2. Find the median and the variance for the set of numbers
.[ 3 marks ]
Answer: median = ……………………..
variance =.……………..………
22. A box contains 6 black balls and p white balls. If a ball is taken out randomly from
the box, the probability to get a white ball is . Find the value of p.
[ 3 marks ]
3472/1 2007 Hak Cipta SBP [ Lihat sebelahSULIT
13
3
20
3
21
For examiner's
use only
Answer: p = ……………………..
23. 5% of the thermos flasks produced by a company are defective. If a sample of n thermos flasks is chosen at random, variance of the number of thermos flasks that are defective is 0.2375. Find the value of n. | 677.169 | 1 |
22
Page 9 ... polygons , by more than four straight lines . XXIV . Of three sided figures , an equilateral triangle is that which has three equal sides . An isosceles triangle is that which has only two sides equal . TO THE KING , THIS EDITION OF THE ...
Page 176 ... 176 THE ELEMENTS.
Page 177 ... polygons have to one another , the antecedents being ABE , EBC , ECD , and the consequents FGL , LGH , LHK : and the po- lygon ABCDE has to the polygon FGHKL the duplicate ratio of that which the side AB has to the homologous side FG ...
Page 178 ... polygon g 12. 5. ABCDE to the polygon FGHKL : but the triangle ABE has to the triangle FGL the duplicate ratio of that which the side AB has to the homologous side FG . Therefore also the polygon ABCDE has to the polygon FGHKL the ...
Page 214 ... polygon BCDEF : there- B E D fore all the angles at the bases of the triangles are together po- 32.1 greater than all the angles of the polygon 214 THE ELEMENTS. | 677.169 | 1 |
MathHelp.com
You know that you can take side lengths and find trig ratios, and you know you can find trig ratios (in your calculator) for angles. What is missing is a way to go from the ratios back to the original angles. And that is what inverse trig values are all about.
What are inverse trigonometric ratios?
Inverse trigonometric ratios are inverse functions for sine, cosine, and tangent. While you plug angle measures into sine, cosine, and tangent to find side lengths, you plug side lengths into inverse sine, inverse cosine, and inverse tangent to find angle measures.
Can my calculator do inverse trig ratios?
Any calculator that can work with sines, cosines, and tangents will also be able to work with their inverses, usually as a second function of the regular trig buttons.
If you look at your calculator, you should see, right above the sin, cos, tan buttons, notations along the lines of sin−1, cos−1, and tan−1, or possibly asin, acos, and atan. These are what you'll be using to find angles from ratios.
What does the a in asin, acos, and atan stand for?
The a in asin, acos, and atan stands for "arc"; in this context, the arc is the portion of the circumference of the unit circle that corresponds to the angle measure in question. In other words, in a geometrical sense, the arc-functions tell you the angle that matches a length that is a portion of the unit circle's circumference. As Wikipedia puts it, "in the unit circle, 'the arc whose cosine is x' is the same as 'the angle whose cosine is x', because the length of the arc of the circle in radii is the same as the measurement of the angle in radians."
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...which is a long way of stating what the above graphic displays; namely, that the sine and cosine of the right triangle inside the unit-circle's (green-shaded) sector are given by the lengths of the two legs of that right triangle. So, given the length of one or another of its sides, one can find the measure of the sector's angle. And we find those measures by applying inverse trigonometric functions.
(Note: There are arc-functions and inverse trig functions for the other three trig functions — secant, cosecant, and cotangent — but I've never seen them used.)
The first set of notations, with the "minus one" exponent, lists the inverse sine, the inverse cosine, and the inverse tangent. The second set of notations, with the a before each name, lists the arc-sine, the arc-cosine, and the arc-tangent. These are two notations for exactly the same thing.
On inverse trig functions, what does the minus-one power mean?
Inverse trigonometric functions are, in particular, inverse functions. The minus-one power indicates an inverse function, not a reciprocal. For instance, sin−1() is the inverse of the sine function; the reciprocal of the sine function is the cosecant function, csc(). The cosecant and the inverse sine are not the same thing.
How to use your calculator to confirm how inverse trig functions work
You can see how these inverse-trig functions work: Go grab your calculator, and take the sine of some angle value between zero and ninety degrees. Whatever result you get, do the inverse sine (that is, use the sin−1 button) or the arc-sine (that is, use the asin button) of that value, and you should get the value you started with.
That's what the inverses of trig ratios do: they give you the angle that goes with that right-triangle's trig ratio.
Content Continues Below
For the triangle below, find the measure m of angle α, to the nearest degree.
They've given me the length of the side that is opposite the angle α, and they have given me the length of the hypotenuse. From "opposite" and "hypotenuse", I can form the sine ratio:
9/10 = sin(α) = 0.9
Plugging 0.9 into sin−1 in my calculator, I get:
α = 64.15806724...
Checking the exercise text, I am reminded that there is a unit of degrees that I need to account for, along with rounding the numerical result to the nearest whole numbers. So my answer is:
m(α) = 64°
Note: There are other units for measuring angles. I'll just be using degrees here, but you may also need to use radians. If so, you'll need to set your calculator's units appropriately. If you are in degree mode, you'll get degree values, not radians, and vice versa. Make sure you're set up correctly.
Find the measure, in degrees, of angle β, accurate to one decimal place.
They've given me the lengths of the side opposite β (being 8) and the side adjacent to β (being 9).
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Since the tangent is opposite over adjacent, I can form the tangent ratio with what they've given me:
8/9 = tan(β)
I won't use the decimal for 8/9, because that could introduce round-off error. Instead, I'll work with the exact fraction and plug tan−1(8/9) into my calculator directly. The result is β = 41.63353934....
Rounding to one decimal place and remembering to affix units to my answer, I get:
m(β) = 41.6°
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Find the length of side p and the measure of angle m, as shown on the diagram. Give each answer correct to the nearest whole number or degree.
How on earth am I supposed to find the angle-measure m and the length of side p when I only have one number for that triangle? All I have is the hypotenuse! Oh, wait...
I can use the angle and hypotenuse on the left-hand triangle to find the height p for both triangles, and this will give me two numbers for the right-hand triangle. With that, I can find the measure m.
The left-hand triangle has opposite, hypotenuse, and angle, so I'll work with the sine ratio:
p/15 = sin(47°)
p = 15×sin(47°) = 10.97030552...
Now that I know that p = 11 (rounded to the nearest whole number), I can find the measure of angle m:
11/18 = sin(m°)
sin−1(11/18) = m° = 37.66988696...
I'm supposed to round this angle measure to the nearest whole degree, so my answer is:
p = 11
m° = 38°
A 5 meter ladder is leaning against a building, with the base of the ladder being two meters from the side of the building. What angle does the ladder form with the ground? Round your answer to one decimal place.
As usual, I start with a picture. It doesn't need to be "exact" or "to scale"; I just need enough of a picture to be able to keep track of what I'm doing.
With respect to the angle they want me to find (which I've indicated in the drawing above by the arc drawn in the lower left-hand vertex), I have the adjacent side and the hypotenuse, so I'll use the cosine ratio.
2/5 = cos(θ)
cos−1(2/5) = θ = 66.42182152...
This is my measure, in degrees, of the angle that the bottom of the ladder makes with the ground on which it is standing. Remembering to round my answer to one decimal place, and to affix the appropriate units, my answer is that the ladder and the ground form an angle of about:
66.4°
Any time you have two sides of a triangle and need an angle, figure out the trig ratio that uses those two sides, and use the appropriate inverse button on your calculator to find the angle that goes with that ratio. And remember to put the "degree" sign on your answer any time that those are the specified units. | 677.169 | 1 |
obtain the desired result.
Applications of "sin a – sin b"
The formula "sin a – sin b" has various applications in mathematics, physics, and engineering. Let's explore some of its practical uses:
1. Calculating the Difference of Sine Values
The primary application of "sin a – sin b" is to find the difference between the sine values of two angles. This can be useful in various scenarios, such as determining the phase difference between two waves or analyzing the oscillatory behavior of a system.
For example, consider a scenario where two waves are interfering with each other. By calculating the difference between their sine values at a particular point in time, we can determine whether they are in phase (constructive interference) or out of phase (destructive interference).
2. Solving Trigonometric Equations
The formula "sin a – sin b" can also be used to solve trigonometric equations. By rearranging the equation and applying appropriate trigonometric identities, we can simplify the equation and find the values of the unknown angles.
For instance, if we have an equation of the form "sin x – sin y = k," where k is a constant, we can use the formula "sin a – sin b" to rewrite it as "2 * cos((x + y)/2) * sin((x – y)/2) = k." By solving this equation, we can find the values of x and y that satisfy the given condition.
Properties of "sin a – sin b"
The formula "sin a – sin b" possesses several properties that are worth noting. Understanding these properties can help us manipulate the formula and apply it effectively in different situations. Let's explore some of these properties:
1. Symmetry Property
The formula "sin a – sin b" exhibits symmetry with respect to the angles a and b. This means that if we interchange the values of a and b, the result remains the same. Mathematically, it can be expressed as:
sin a – sin b = sin b – sin a
This property is useful when simplifying expressions or solving equations involving "sin a – sin b."
2. Double Angle Identity
The formula "sin a – sin b" can be derived from the double angle identity for sine. The double angle identity states that:
sin 2θ = 2 * sin θ * cos θ
By substituting a = θ + φ and b = θ – φ into the double angle identity, we can obtain the formula "sin a – sin b." This property allows us to relate the difference of sine values to the double angle identity.
Examples and Case Studies
To further illustrate the concept of "sin a – sin b," let's consider a few examples and case studies:
Example 1: Calculating the Difference of Sine Values
Suppose we have two angles a = 30 degrees and b = 45 degrees. To find the difference between their sine values, we can use the formula "sin a – sin b."
Using the formula, we have:
sin a – sin b = 2 * cos((a + b)/2) * sin((a – b)/2)
= 2 * cos((30 + 45)/2) * sin((30 – 45)/2)
= 2 * cos(75/2) * sin(-15/2)
By evaluating the trigonometric functions, we can find the difference of sine values.
Example 2: Solving Trigonometric Equations
Consider the equation "sin x – sin y = 1/2," where x and y are unknown angles. To solve this equation, we can use the formula "sin a – sin b."
By rearranging the equation, we have:
2 * cos((x + y)/2) * sin((x – y)/2) = 1/2
By comparing this equation with the formula "sin a – sin b," we can determine the values of x and y that satisfy the given condition.
Summary
In conclusion, the formula "sin a – sin b" is a powerful tool in trigonometry that allows us to calculate the difference between the sine values of two angles. It has various applications in mathematics, physics, and engineering, such as determining phase differences, solving trigonometric equations, and analyzing oscillatory behavior. By understanding the properties and applications of "sin a – sin b," we can effectively apply it to solve problems and gain valuable insights.
Q&A
1. What is the sine function?
The sine function is a trigonometric function that relates the ratio of | 677.169 | 1 |
Parallel Lines Cut By A Transversal Worksheet With Answers
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Web parallel lines & transversals worksheets. When parallel lines are cut, or intersected by a transversal, lots of interesting geometric. Web parallel lines transversal worksheet. When two parallel traces are intersected by a 3rd one, the angles that occupy the same relative place at each. | 677.169 | 1 |
The nine-point
circle of a triangle is the circle passing through the midpoints of the
sides (1,2,3), the feet of the altitudes (4,5,6)
and the midpoints (7,8,9) between the orthocenter
H and the vertices. | 677.169 | 1 |
What Is First Angle Projection?
Are you curious to know what is first angle projection? You have come to the right place as I am going to tell you everything about first angle projection in a very simple explanation. Without further discussion let's begin to know what is first angle projection?
In the realm of engineering and technical drawings, projection methods play a crucial role in accurately representing three-dimensional objects on a two-dimensional plane. First Angle Projection is one such method that holds significance in engineering drawings. This detailed guide aims to demystify the concept of First Angle Projection, exploring its definition, symbols, differences from Third Angle Projection, and practical applications.
What Is First Angle Projection?
First Angle Projection is a method used in engineering drawings to represent a three-dimensional object on a two-dimensional plane. In this projection system, the object is imagined to be positioned in the first quadrant, with the plane of projection between the observer and the object. It is widely used in Europe, Australia, and many other parts of the world.
What Is First Angle Projection Example?
To illustrate First Angle Projection, consider a simple example of a cube. In First Angle Projection, the views of the cube are created as if it is placed in the first quadrant, with the front view appearing above and the side view to the right. This example showcases how First Angle Projection visually represents the object from the perspective of an observer looking towards the object through the projection plane.
Difference Between First Angle And Third Angle Projection
The primary distinction between First Angle and Third Angle Projection lies in the placement of the object in relation to the projection plane. In First Angle Projection, the object is imagined to be in the first quadrant, while in Third Angle Projection, it is placed in the third quadrant. The arrangement of views and symbols also differs between the two methods.
What Is Third Angle Projection?
Third Angle Projection is another method of representing three-dimensional objects on a two-dimensional plane in engineering drawings. In this projection system, the object is imagined to be in the third quadrant, with the plane of projection between the observer and the object. Third Angle Projection is commonly used in the United States and Canada.
First Angle Projection Symbol
The symbol used to denote First Angle Projection is a capital letter 'A' enclosed in a circle. This symbol is typically placed on the drawing to indicate the projection method being used. Recognizing this symbol is essential for engineers and drafters to interpret engineering drawings accurately.
What Is First Angle Projection Pdf?
Understanding First Angle Projection is crucial for professionals working with engineering drawings. Various resources, including PDF documents and guides, delve into the intricacies of First Angle Projection, providing detailed explanations, examples, and illustrations to enhance comprehension.
Third Angle Projection Symbol
Conversely, the symbol for Third Angle Projection is a capital letter 'T' enclosed in a circle. This symbol is placed on engineering drawings to signify the use of Third Angle Projection. Familiarizing oneself with these symbols is key to avoiding misinterpretations in the field of engineering.
First And Third Angle Projection
The choice between First Angle and Third Angle Projection depends on the regional standards and practices. While First Angle Projection is prevalent in Europe, Australia, and most of the world, Third Angle Projection is commonly used in the United States and Canada. Engineers and drafters must adhere to the standards specified in their respective regions.
First Angle Projection Used In Which Country
First Angle Projection is widely used in countries like those in Europe and Australia. Understanding the projection method adopted in a specific country is crucial for engineers collaborating on international projects.
Conclusion
In conclusion, First Angle Projection is a fundamental method in engineering drawings, providing a systematic way to represent three-dimensional objects on a two-dimensional plane. Its symbol, arrangement of views, and differences from Third Angle Projection contribute to its unique characteristics. Whether you are a student, engineer, or professional in the field, grasping the principles of First Angle Projection enhances your ability to interpret and create accurate engineering drawings. As global standards vary, recognizing the prevalent projection method in a specific region ensures effective communication and collaboration in the dynamic field of engineering.
FAQ
What Is Third Angle And First Angle Projection?
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. Figures 5.13 and 5.14 show the differences using an example of a large ship's gun barrel.
What Is The First Angle Projection In Gd&T?
When the large end of the cone in the section view is closest to the top view, this is known as first angle projection. Traditionally, the first angle projection symbol is drawn with the top view on the left and the side view on the right.
What Is 2nd Angle Projection?
The second angle projection is one of the multiview projections. It is located in the second quadrant. The same with the fourth angle, it is not used in multiview projections. The projection system is composed of the vertical plane (VP) and horizontal plane (HP).
What Is First Angle Projection Wikipedia?
First-angle projection: In this type of projection, the object is imagined to be in the first quadrant. Because the observer normally looks from the right side of the quadrant to obtain the front view, the objects will come in between the observer and the plane of projection.
I Have Covered All The Following Queries And Topics In The Above Article | 677.169 | 1 |
If the two consecutive interior angles of a polygon differ by 5 degrees and the shortest angle is 120 deg.Then detemine the sides of the polygon.
If the two consecutive interior angles of a polygon differ by 5 degrees and the shortest angle is 120 deg.Then detemine the sides of the polygon.
Aparna Tiwari,
13 years ago
Grade:12
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Sukhendra Reddy Rompally B.Tech Mining Machinery Engg, ISM Dhanbad
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13 years ago
Dear student,
the sum of all interior angles for an n-sided polygon=(n-2)180°.you can check it by substuting different values of n.now,the sum of all interior angles of the given polygon is an A.P with first term 120° and common difference 5° ,which is given by n/2 (2*120° + (n-1)5°) = n/2 (235 - 5n) = 180(n-2). Solve the quadratic to get = 9 or n = 16. Hope you understand the solution.if not,feel free to call me on 07209736303. ALL THE BEST. PLZZ APPROVE MY ANSWER IF YOU LIKE IT | 677.169 | 1 |
Triangle Congruence Worksheet PDF
Triangles are unique shapes that have three sides and three angles. Understanding triangle congruence is important in order to solve problems that involve triangles. This worksheet offers a comprehensive study guide on the topic of triangle congruence.
This worksheet covers the different types of congruent triangles including SSS, SAS, ASA, and AAS. It also discusses reflection, rotation, and translations as they apply to congruent triangles. The questions in this worksheet cover a range of difficulty from basic to more advanced. The worksheet also includes a detailed answer key.
This triangle congruence worksheet PDF is a great resource for students and teachers alike. It is appropriate for students from middle school to college. The worksheet can be used in the classroom, given as a homework assignment, or used for self-study. The PDF is easy to download and print, making it a convenient and useful resource.
If you are looking for a comprehensive study guide on triangle congruence, then this worksheet PDF is the perfect resource for you. It will help you master the different types of congruent triangles and how to apply them to solve problems. | 677.169 | 1 |
Year 6 | Reflection Worksheets
These Year 6 reflection worksheets feature a squared grid divided into four quadrants, with two shapes (Shape A & Shape C) positioned in the 4th quadrant. Learners are tasked with reflecting Shape A in the x-axis to create Shape B, subsequently drawing it and recording the coordinates of Shape B's vertices. Following this, learners reflect Shape C in the y-axis to generate Shape D, drawing it and documenting the coordinates of Shape D's vertices. In the final segment, pupils are challenged to reflect both Shape A and C in the 4th quadrant, drawing these shapes accordingly.
This worksheet poses a challenge, requiring children to employ critical thinking skills while mastering the art of reflecting shapes in all four quadrants on a grid.
Our Year 6 geometry: position and direction worksheets are created to match the KS2 national curriculum. They can be combined with your ideas and primary teaching strategies for learning activities, homework, differentiation and lesson plans. | 677.169 | 1 |
Answer: c
Explanation: The rolling circle is known as generating circle, and the straight line on which it rolls is known as the directing or baseline.
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4. The plane curve generated by a point on the circumference of a circle, when it rolls on a fixed straight line without slipping is __________
a) Cycloid
b) Involute
c) Conic
d) Ellipse View Answer
Answer: a
Explanation: When a circle rolls on a straight line, the instantaneous positions of the point of intersection are obtained by periodically moving the center of the circle to different positions parallel to the line, while maintaining the same height of the division's point.
Answer: a
Explanation: When r and R are the radii of the rolling and the generating circle, respectively, the hypocycloid is a plane curve generated by a point on the circumference of a circle, when it rolls without slipping on another circle and inside it. The epicycloid is a plane curve generated by a point on the circumference of a circle when it rolls without slipping on the circumference of another circle.
6. The locus of a point maintaining a constant distance from a fixed point is known as:
a) An ellipse
b) A circle
c) A parabola
d) A straight line View Answer
Answer: b
Explanation: Where the fixed point of a circle is center and distance between centers to the edge of the circle is constant and that is called radius.
7. The path traced by the instantaneous positions of any point in a moving member is known as__________
a) Ellipse
b) Parabola
c) Hyperbola
d) Locus of points View Answer
Answer: d
Explanation: Locus of points is very important to ensure the proper function of the member in any machine or in any applications.
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8. The locus of a point moving between a fixed point and a fixed straight line at constant ratio is __________
a) a conic
b) an involute
c) a cycloid
d) a polygon View Answer
Answer: a
Explanation: The conic sections or the conics, as they are simply called, are the curves obtained by intersecting or cutting a right circular cone with a section, or by cutting a plane located in a different orientation.
9. The curve traced by the locus of a point moving in a plane such that the ratio of its distance from the fixed point and a fixed straight line is equal to one is known as __________
a) Ellipse
b) Parabola
c) Circle
d) Hyperbola View Answer
Answer: b
Explanation: When the cutting plane is inclined to the base and the axis becomes parallel to one of the end generators, thereby cutting a few generators of the cone as indicated by c-c, the conic section obtained is known as a parabola | 677.169 | 1 |
41
Página 17 ... divided into four equal parts , into eight , into sixteen , and so on . PROP . X. PROB . To bisect a given finite straight line , that is , to divide it into two equal parts . Let AB be the given straight line ; it is required to divide ...
Página 30 ... divided into as many trian- gles as the figure has sides , by drawing straight lines from a point F within the figure to each of its angles . And , by the preceding proposition , all the angles of these triangles are equal to twice as ...
Página 32 ... divided into four triangles , which contain eight right angles ; that is , as many times two right an- gles as there are units in the number of sides diminished by two . But to avoid all ambiguity , we shall henceforth limit our ...
Página 49 50 ... divided into any two parts , the square of the whole line is equal to the squares of the two parts , together with twice the rectangle con- tained by the parts . Let the straight line AB be divided into any two parts in C ; the square | 677.169 | 1 |
in AC (alternating current) circuits, the formula helps calculate the phase difference between voltage and current waveforms. This knowledge is crucial for designing efficient power transmission systems and optimizing electrical networks.
3. Robotics and Computer Graphics
In the field of robotics and computer graphics, the formula "sin a – sin b" is utilized for calculating the rotation between two objects or coordinate systems. By determining the difference in orientation between two entities, it becomes possible to perform transformations, such as rotating or aligning objects in a virtual environment.
For example, in computer animation, the formula can be used to animate characters or objects smoothly. By calculating the difference in orientation between two frames, the animation software can interpolate the intermediate frames, creating realistic and fluid motion.
Real-World Examples
Let's explore a few real-world examples where the formula "sin a – sin b" is applied:
Example 1: Sound Localization
In the field of audio engineering, sound localization is the process of determining the direction from which a sound originates. By using an array of microphones, the phase difference between the sound waves arriving at each microphone can be calculated using the formula "sin a – sin b." This information helps in accurately localizing the sound source, enabling applications such as virtual reality audio, hearing aids, and surveillance systems.
Example 2: Robotics Path Planning
In robotics, path planning involves finding an optimal path for a robot to navigate from one point to another. The formula "sin a – sin b" can be used to calculate the orientation difference between the robot's current position and the desired target position. By determining the angular difference, the robot can adjust its orientation and move efficiently towards the target, avoiding obstacles and optimizing its path.
Q&A
Q1: How is "sin a – sin b" different from "sin(a – b)"?
A1: TheQ2: Can the formula "sin a – sin b" be used for any angle values?
A2: Yes, the formula "sin a – sin b" can be used for any angle values. However, it is important to ensure that the angles are measured in the same unit (radians or degrees) to obtain accurate results.
Q3: Are there any other trigonometric identities related to "sin a – sin b"?
A3: Yes, there are several trigonometric identities related to "sin a – sin b," such as the sum of two sines, the product of two sines, and the double angle formula for sine. These identities provide additional tools for solving trigonometric equations and simplifying complex expressions.
Q4: Can the formula "sin a – sin b" be extended to other trigonometric functions?
A4: Yes, similar formulas can be derived for other trigonometric functions like cosine and tangent. These formulas involve the corresponding trigonometric identities and can be used to calculate the differences between the values of these functions for different angles.
Q5: How can I apply the formula "sin a – sin b" in my own calculations?
A5: To apply the formula "sin a – sin b," you need to know the values of angles a and b. Once you have these values, substitute them into the formula and evaluate the expression. Make sure to use the correct unit (radians or degrees) for the angles to obtain accurate results.
Summary
In conclusion, the formula "sin a – sin b" is a powerful tool in trigonometry that allows us to calculate the difference between the sine values of two angles. It finds applications in various fields, including wave interference, electrical engineering, and robotics. By understanding the concept and applications | 677.169 | 1 |
The Pythagoras theorem that is applicable to right-angle triangles is an important component of trigonometry, and according to it area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two sides that meet at the right angle.
Easy access to various number crunching devices makes us indifferent to the very fact that calculating the square of a number is not all that easy, and more so finding the square root of a number is really a herculean task as there exists no simple formula to calculate the square root.
You all would be surprised to know that an ancient Tamilmathematician and poet Pothayanar, who lived around 800 BC, had propounded the following quatrain depicting the method of finding the length of the hypotenuse of a right-angle triangle using the length of the sides, and simple fractions, without getting into intricacies of calculating square or square root:
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Comments
This formula is simply false, rubbish, or incorrect. It does not, in any way, guarantee the hypotenuse of a right triangle, and should never be used in general situations. For example, 21 and 20 gives 28.375, but Pythagorean theorem gives 29. The article was very incorrect and contained many mistakes.
The formula 7/8*A+B/2 should be viewed in light of the situations where it would be used and not a general mathematical result. It would have been used on fields or construction sites, where the horizontal distance (H) and vertical distance(V) could be actually measured (in some units) and the hypotenuse (H) was required. The construction worker, even if the Pythagorus theorem or equivalent were then known, would have hard time calculating the square-root. For the usual kind of values obtained in construction sites the Bodhayana formula is good enough, with an error of a couple of percentage of the values of H and V involved. It requires division by 8 (simple to measure out in the field) and by 2. It is a site workers formula.
Pothayanar's rule is *not* an alternative to the Right Triangle Theorem commonly referenced as Pythagoras' Theorem. Showing that it works for cases where the short sides of a right triangle are in specific ratios of 4:3 or 12:5 does not constitute a proof for all right triangles. Hence, Pothayanar does not hold for all right triangles unlike the Right Triangle Theroem.
It is simply rubbish. In all examples, switch a side with b and you will get all wrong answers.
The most commonly stated example is a triangle where a is 3 and b is 4 or vice versa, the answer is alway 5. Try the formula of Pothayanar, it won't work. | 677.169 | 1 |
The Elements of Euclid, the parts read in the University of Cambridge [book ...
1. DESCRIBE a circle of given radius, which shall pass through two given points.
2. If with the vertex of an isosceles triangle, as centre, a circle be described cutting the base or base produced, the parts of it intercepted between the circle and the extremities of the base will be equal.
3. If two circles cut each other, any two parallel lines drawn through the points of section to cut the circles are equal.
4. If two circles cut each other, draw through one of the points of section a line which shall be terminated in the circumferences and be bisected in that point.
5. A chord PAQ cuts the diameter of a circle in A in an angle which is half a right angle: shew that the squares of AP and AQ are together double of the square of the radius.
6. If two chords intersect in a circle, the difference of their squares is equal to the difference of the squares of the difference of the segments.
7. Two parallel chords in a circle are respectively six and eight inches in length, and are one inch apart: how many inches in length is the diameter ?
8. Draw a line cutting two concentric circles, so that the part of it intercepted by the circumference of the greater may be double the part intercepted by that of the less.
9. If two circles cut each other, the greatest line that can be drawn through the point of intersection is that which is parallel to the line joining their centres.
10. Describe three equal circles touching one another, and also another which shall touch them all three.
11. How many equal circles can be described around another circle of the same magnitude, touching it and one another?
12. Describe a circle which shall pass through a given point, and touch a given circle in a given point, the two points not being in a tangent to the given circle.
13. Describe a circle which shall touch a given circle in a given point, and also touch a given straight line.
14. If from any point without a circle two lines be drawn making equal angles with the line through the centre from that point, they will cut off equal segments from the circle.
15. Through a given point within a circle draw the least possible chord.
16. Of all lines which touch the interior and are bounded by the exterior of two circles which touch internally, the greatest is that which is parallel to the common tangent.
17. Shew that the two tangents to a circle drawn from the same point without it are equal to one another; and hence prove that the sums of the opposite sides of any quadrilateral described about a circle are equal, and the angles subtended at the centre of the circle by any two opposite sides together equal to two right angles.
18. If any line be drawn touching a circle, the part of it intercepted between the tangents at the extremities of any diameter subtends at the centre a right angle.
19. In the diameter of a circle produced determine a point from which a tangent drawn to the circle shall be equal to the diameter.
20. Describe a circle which shall pass through a given point, have a given radius, and touch a given line.
21. Describe a circle whose centre shall be in the perpendicular of a given right-angled triangle, and which shall pass through the right angle and touch the hypo
thenuse.
22. A is any point in the diameter (or diameter produced) of a circle, whose centre is O, OB a radius perpendicular to the diameter: if AB cut the circle in P, and the tangent at P cut AO in C, shew that AC=CP.
23. A common tangent is drawn to two circles which touch externally if a circle be described on that part of it which lies between the points of contact, as diameter, it will pass through the point of contact of the two circles, and be touched by the line joining their centres.
24. Describe a circle with given radius and its centre in a given line, which shall touch another given line.
25. Describe a circle, which shall touch a given line in a given point, and also touch a given circle.
26. Draw a common tangent to two circles, when the points of contact are (i) on the same side, and (ii) on opposite sides, of the line joining their centres.
27. Draw a line which shall touch a given circle, and make with a given line a given angle.
28. Describe two circles of given radii, which shall touch each other, and the same given line on the same side of it.
29. If two circles touch each other, and parallel diameters be drawn, then lines which join the extremities of these diameters will pass through the point of
contact.
30. The line, drawn from the vertex of an equilateral triangle to meet the circumscribing circle in any point, is equal to the sum or difference of the two lines drawn from the extremities of the base to that point, according as it does or does not cut the base.
31. AB, AC are any two chords of a circle, D, E, the bisections of the arcs AB, AC: let DE cut AB, AC in F, G, and shew that AF-AG.
32. If the opposite angles of a quadrilateral be together equal to two right angles, shew that a circle may be described about it, and find its centre and radius.
33. ABCD is a parallelogram; draw CE perpendicular to the diagonal BD, and shew that perpendiculars upon AB, AD at the points B, D, will intersect in CE.
34. The circles described on the three sides of a triangle, so as to pass through the points of intersection of the perpendiculars upon them from the opposite angles, are equal to each other.
35. Two circles intersect in A, B, the centre of one being in the circumference of the other: draw any chord ACD cutting them both, and shew that CB= CD.
36. If from any two points in the circumference of a circle there be drawn two lines to a point in any tangent to the circle, they will make the greatest angle when drawn to the point of contact.
37. Given three points in a circle: shew how we may find any number of other points, without knowing the position of the centre.
38. If through the angles of a quadrilateral, lines bisecting them be drawn, the points in which each line intersects the adjacent ones will all lie in the circumference of a circle.
39. ABC is a semicircle, ADC a quadrant, upon the same line AC and on the same side of it; from any point B in the semicircle draw BA, BDC, and shew that BA and BD are equal, and that the longer only of the lines AB, AC, can cut the circle ADC.
40. If any chord of a circle be bisected by another and produced to meet the tangents at the extremities of the bisecting line, the parts intercepted between the tangents and the circumferences are equal.
M
41. From the extremities A, C of a given circular arc, equal arcs AB, CD are measured in opposite directions: shew that the chords AC, BD are parallel.
42. The arcs intercepted between any two parallel chords of a circle are equal; and if any two chords of a circle intersect each other, the sum of the arcs intercepted by them is equal to the sum of the arcs intercepted by diameters parallel to them.
43. A, B, C, A', B', C' are points in the circumference of a circle: if AB, AC be respectively parallel to A'B', A'C', shew that BC' is parallel to B'C.
44. If two equal circles cut each other, and from either point of intersection a circle be described cutting them, the point where this circle cuts them and the other point of intersection of the equal circles are in the same straight line.
45. If two equal circles cut each other and from either point of intersection a line be drawn cutting the circumferences, the part of it between them will be bisected by the circle, whose diameter is the common chord of the equal circles.
46. If two circles cut each other, and any two points be taken in the circumference of one through which lines are drawn from the points of intersection cutting that of the other, the lines, joining the points of section with the latter circle of those drawn through the same point, will be equal.
47. A, B are given points: if AC, BC are drawn, making a given angle with each other, shew that the line bisecting that angle passes through a fixed point.
48. If perpendiculars be dropped from the extremities of any diameter upon any chord of a circle, the parts of the chord, intercepted between them and the circle, will be equal, and the less perpendicular shall be equal to the segment of the greater contained between the chord and circumference. | 677.169 | 1 |
...segments of one of them is equal to the rectangle contained by the segments of the other 8 marks. 7. If a straight line be divided into two equal, and...the squares on the two unequal parts are together double of the squares on half the line and the line between the points of section. Prove this proposition...
....: 4 sq. on AC+2 sq. on DC=2 sq. on ^C + sqs on AD, DB; or 2 sqs on AC, DC = sqs on AD, DB. 153, 3. If a straight line be divided into two equal and also...unequal parts, the squares on the two unequal parts are equal to twice the rectangle contained by the two unequal parts together with four times the square...
...described upon the other two sides of it, the angle contained by these two sides is a right angle. 5. If a straight line be divided into two equal, and also into two unequal parts, the squares of the two unequal parts are together double of the square of half the line, and of the square of the...
...lines are together equal to the squares of the hypothenuse and of the line drawn parallel to it. 3. If a straight line be divided into two equal, and also into two unequal parts, the squares of the unequal parts are together double of the square of half the line and of the square of the line...
...angles ; and the three interior angles of every triangle are together equal to two right angles. 11. If a straight line be divided into two equal, and also into two unequal parts ; the squares of the two unequal parts are together double of the square of half the line, and of the square of the...
...point outside the straight line BAG such that'AB2+ PC2=AC° + PB2, PA will be at right angles to BC. 8. If a straight line be divided into two equal and also into two unequal parts, the sum of the squares on the two unequal parts is -equal to twice the sum of the squares on the part between... | 677.169 | 1 |
Euler's rotation theorem tells us that any rotation in 3D can be
described by 3 angles. Let's call the 3 angles the Euler angle vector
and call the angles in the vector \(alpha\), \(beta\) and
\(gamma\). The vector is [ \(alpha\),
\(beta\). \(gamma\) ] and, in this description, the order of the
parameters specifies the order in which the rotations occur (so the
rotation corresponding to \(alpha\) is applied first).
In order to specify the meaning of an Euler angle vector we need to
specify the axes around which each of the rotations corresponding to
\(alpha\), \(beta\) and \(gamma\) will occur.
There are therefore three axes for the rotations \(alpha\),
\(beta\) and \(gamma\); let's call them \(i\)\(j\),
\(k\).
Let us express the rotation \(alpha\) around axis i as a 3 by 3
rotation matrix A. Similarly \(beta\) around j becomes 3 x 3
matrix B and \(gamma\) around k becomes matrix G. Then the
whole rotation expressed by the Euler angle vector [ \(alpha\),
\(beta\). \(gamma\) ], R is given by:
axes i, j, k are the z, y, and x axes respectively. Thus
an Euler angle vector [ \(alpha\), \(beta\). \(gamma\) ]
in our convention implies a \(alpha\) radian rotation around the
z axis, followed by a \(beta\) rotation around the y axis,
followed by a \(gamma\) rotation around the x axis.
the rotation matrix applies on the left, to column vectors on the
right, so if R is the rotation matrix, and v is a 3 x N matrix
with N column vectors, the transformed vector set vdash is given by
vdash=np.dot(R,v).
extrinsic rotations - the axes are fixed, and do not move with the
rotations.
a right-handed coordinate system
The convention of rotation around z, followed by rotation around
y, followed by rotation around x, is known (confusingly) as
"xyz", pitch-roll-yaw, Cardan angles, or Tait-Bryan angles.
It's possible to reduce the amount of calculation a little, by
combining parts of the angle_axis2mat and mat2euler
functions, but the reduction in computation is small, and the code
repetition is large.
The direction of rotation is given by the right-hand rule (orient
the thumb of the right hand along the axis around which the rotation
occurs, with the end of the thumb at the positive end of the axis;
curl your fingers; the direction your fingers curl is the direction
of rotation). Therefore, the rotations are counterclockwise if
looking along the axis of rotation from positive to negative.
The cy fix for numerical instability below is from: Graphics
Gems IV, Paul Heckbert (editor), Academic Press, 1994, ISBN:
0123361559. Specifically it comes from EulerAngles.c by Ken
Shoemake, and deals with the case where cos(y) is close to zero: | 677.169 | 1 |
ASA vs. AAS: Know the Difference
ASA (Angle-Side-Angle) and AAS (Angle-Angle-Side) are both methods to prove triangle congruence: ASA requires two angles and the included side, AAS two angles and any side.
Key Differences
ASA (Angle-Side-Angle) is a criterion for proving the congruence of two triangles by comparing two angles and the side between them. In contrast, AAS (Angle-Angle-Side) proves congruence using two angles and any side, not necessarily the one between the angles | 677.169 | 1 |
What is a 3-4-5 Triangle?
It's no secret that flooring installers and contractors need to be comfortable with math in order to do a good job. Many of them have discovered some extremely useful formulas to implement to make sure they are accurate in what they are doing. When they are trying to determine if an angle is a right angle, they can turn to this specialized rule.
The 3-4-5 triangle can be used to show whether or not a corner of a room is indeed 90 degrees. First, the installer needs to measure along one of the walls of the corner for about 3 feet. Then, they should measure along the other wall for 4 feet. If the diagonal between these points is 5 feet, then the corner must be a square angle. Rather than depend on guesswork or estimations, the 3-4-5 triangle will provide excellent confirmation that they are indeed working with proper angles.
Remember that the 3-4-5 triangle method can also be expanded by using multiples, like 6-8-10 and so forth. There's no easier method to determine if flooring installers are indeed working with right angles | 677.169 | 1 |
Draw a vertical diameter and a horizontal diameter with the ruler. A diameter is a straight line connecting two opposite points on a circle that passes through the centre point. This will create a cross shape and will mark four corners of the regular octagon.
How do you cut the perfect octagon out of paper?
Steps
Start with a square piece of paper.
Fold the square in half along both diagonals to find the middle of the square. Open.
Fold all corners so they meet in the middle. Open.
Fold an angle bisector of the angle formed by the creases you just made and the edges of the paper.
These creases form a regular octagon.
What is the angle for an octagon?
An octagon has 8 exterior angles. The sum of the exterior angles of an octagon is 360°. Regular octagon: An octagon in which all sides and angles are equal. Each angle of a regular octagon is 135°.
How do you draw a regular octagon?
Regular Octagon. Draw a large circle at the center of the page. Draw a straight horizontal line splitting the circle to equal parts. Draw a straight vertical line (also splitting the circle to equal parts) of which its center will intersect with that of the horizontal line.
What angle makes an octagon?
Set your miter saw at 22.5 degrees. This is the angle you need to cut at to make an octagon shape.
How long is each side of an octagon?
A true octagon has eight equal sides. The length and width should be the same….
A
B
48″
19.875″
49″
20.25″
50″
20.75″
51″
21.125″
Are all sides equal in an octagon?
Properties of Octagon These have eight sides and eight angles. All the sides and all the angles are equal, respectively. There are a total of 20 diagonals in a regular octagon. Sum of all the exterior angles of the octagon is 360°, and each angle is 45°(45×8=360). | 677.169 | 1 |
geometryCrossRatiocompute the cross ratio (or anharmonic ratio, or double ratio) of four points
Calling SequenceParametersDescriptionExamples
Calling SequenceCrossRatio(A, B, C, F)Parameters
A, B, C, F-four points
DescriptionGiven four points A, B, C, F, the routine CrossRatio(A, B, C, F) computes the cross ratio of A, B, C, F taken in this order.If A, B, C, F are four distinct points on an ordinary line, the cross ratio of A, B, C, F taken in this order is defined as:LGIzYpLUklbXJvd0dGJDYlLUkjbWlHRiQ2JlEwU2Vuc2VkTWFnbml0dWRlYuNiUtRjE2JlEiQUYnRjRGN0Y6LUY+Ni5RKCZjb21tYTtGJ0Y3RkFGQy9GRkY2RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUYxNiZRIkNGJ0Y0RjdGOkY3RkEtRi42JUYwRj0tRlc2JS1GLjYlRl5vRmhuLUYxNiZRIkJGJ0Y0RjdGOkY3RkEv9wLyUpYmV2ZWxsZWRHRjkvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1GIzYpLUYuNiVGMEY9LUZXNiUtRi42JUZlbkZobi1GMTYmUSJGRidGNEY3RjpGN0ZBLUYuNiVGMEY9LUZXNiUtRi42JUZfcUZobkZnb0Y3RkFGam9GXXBGYHBGYnBGZHBGam9GXXBGYHBGYnA=The command with(geometry,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 | 677.169 | 1 |
Lesson 4 – GEOMETRY OF SPACE (part2)
3.3 EXAMPLES OF SURFACES IN TECHNICAL PRACTICE
In our studies we have made our drawing in the plane of our drawing paper, using a ruler and a compass. In space we cannot work with these tools, so we have to make our spatial geometry constructions in our heads, in our minds – and then we can record them on a drawing paper using the rules of descriptive geometry.
Firstly we have to define the rules of spatial drawings making. We need to introduce some new concepts.
3.3.1 DRAWING PLANES
A PLANE is considered to be constructed if the spatial elements which clearly define the position of the plane in space have been found.
We can draw a plane:
• with three points
• on a straight line and an external point
• an intersecting pair of straight lines
• a pair of parallel lines
In the case of two intersecting planes are given, their intersection lines are given thus we can determine the intersection of two planes in space. Once we have defined a plane in space, we can do all the drawings in this plane what we do in descriptive geometry.
3.3.2 DRAWING OBJECTS WITH FLAT LINE SHAPES
The FLAT LINE SHAPES are prsim, cube, pyramid. The prism or the cube can laying on their any planes but the pyramid just can only stand on its base plate. The standing object has a vertical axis and its ending plane is horizontal. The vertical edges look like point on the base plane.
29. Figure Objects build from flat surfaces
Various engineering components can be designed using the geometric forms of cubes, rectangular prisms (or cuboids), and pyramids(See Figure 29):
Cubes: Being equal in all dimensions, cubes can be used in certain modular designs or components that require symmetry. Examples include:
Square shaft ends
Structural blocks for assemblies or constructions.
Calibration blocks in metrology.
Certain types of weights or counterbalances.
Rectangular Prisms (Cuboids): With their three distinct dimensions, these are one of the most common shapes in mechanical design. Components or structures derived from this shape include:
Casings or enclosures for electronics or machinery.
Shipping containers or storage boxes.
Battery cells in certain configurations.
Base structures for platforms or machinery.
Pyramids: In buildings and building services engineering, pyramid or conical shapes can be observed in various applications due to their distinct structural and functional advantages:
Roofs: Pyramid-shaped roofs, often seen in traditional or historic architecture, can aid in water runoff, offer a distinctive aesthetic, and provide additional space beneath.
Ventilation and Exhaust: Conical ducts or vents, used in HVAC systems, help in regulating and directing airflow. Their tapering design can increase or decrease the velocity of air or other fluids, depending on the application.
Acoustics: In some spaces, conical structures or surfaces can be implemented to manage sound reflection or absorption, optimizing the acoustic properties of a room.
3.3.3 DRAWING OBJECTS WITH SOLIDS OF ROTATION
The SOLIDS OF ROTATION are cylinder, cone, sphere and the circumferential ring. Combinations of these can produce many of the shapes used in engineering practice. Examples include the junction of tubes of different diameters, cylindrical and conical sheet metal parts or a solid shafts with cylindrical bores.
Various components can be designed with cylindrical, conical, and spherical surfaces.
Cylindrical surfaces are commonly used in parts such as shafts, rods, barrels, rollers, and sleeves. They provide a uniform cross-section and are typically utilized for rotating elements or parts that need to slide within other components.
Conical surfaces, having a tapering shape, can be found in parts such as nozzle tips, tapered bearings, and certain types of gears. Their design often serves specific functions, like guiding or centering components, or enabling tighter fits between parts. The tapering design offers unique advantages in various components. Some applications include:
Certain types of nozzle or vent designs, directing flow in a particular direction.
Spikes or pointed feet for machinery or equipment to ensure grip or stability on a surface.
Spherical surfaces are seen in components like ball bearings, ball joints, and certain types of lens. They provide omnidirectional movement or rotation, making them ideal for parts that require unrestricted movement in multiple directions.
Each of these geometric designs offers unique advantages depending on the application and the specific requirements of the component in the larger mechanical system. The Figure 30 below shows a shaft designed as a combination of cylinders and cones.
30. Figure Shaft made up of cylinders and cones
3.3.4 COMPLEX SHAPES OF PARTS
In mechanical engineering, most surfaces are born from a combination of flat planes and revolution bodies(Figure 31,32). The design and development of components and structures often involve a harmonious blend of flat planes and bodies of revolution. While flat planes provide stability, ease of manufacturing, and simplicity in assembly, bodies of revolution, generated by rotating a curve about an axis, offer smooth transitions, efficient flow dynamics, and uniform stress distribution. Cylinders, cones, and spheres, all examples of revolution bodies, are frequently combined with flat surfaces to create parts like shafts with flanges, conical nozzles with mounting bases, or spherical tanks with planar supports.
The melding of these two fundamental geometries allows engineers to tailor designs to specific functional requirements while optimizing manufacturing processes. This synergy between flat planes and curved surfaces is a testament to the adaptive and innovative nature of mechanical engineering, ensuring both functional efficiency and design elegance.
31. Figure Complex parts
In the differential mechnaism shown in Figure 32, the components formed using complex surfaces can be clearly observed. When looking at the axes of the components, parallel, intersecting or skew lines(axes of the parts) are clearly visible. The axes of the parts are marked with a dotted line. | 677.169 | 1 |
...PROP. XXI. THEOR. See N. IF B, C, the ends of the side BC of the triangle...
...of a triangle there N. "side. Prop. XXI. Theor. If, from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but (hall contain a greater angle. Prop. XXII. Prob. To make a triangle of •which the sides shall be...
...ends.of the side of a triangle, there be drawn two straight lines to a point within the trir angle, these shall be less than the other two sides of the triangle, but shall contain a greater angle. Let the two straight lines BD, CD be drawn from B, C, the ends of the side BC of the triangle...
...side of a triangle, therePropoíiíion. XX/. Theoren. If В, С, the ends of the side BC of the triangle...
...side of a triangle, there be drawn two straight lines to a point within the triangle, these two lines shall be less than the other two sides of the triangle, but shall eontain a greater angle. L'el the two straight lines BD, CD be drawn from B, C, the em's of the side...
...from the ends of the side of a triangle, there be drawn See N. two straight lines to a point 1cithin the triangle, these shall be less than the other two...sides of the triangle, but shall contain a greater angle. Let ABC be a triangle, and from the points B, C, c 13. 1. 1 5. 1. t 9 Ax. 19. 1. the ends of...
...QED PROP. XII. THEOR. If from the ends of the side of a triangle there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle. Let the two straight lines BD, CD be drawn from B, C, the ends of the side BC of the triangle ABC,...
...side of a triangle, there be drawn two straight lines to a point within the triangle, these two lines shall be less than the other two sides of the triangle, but shall contain a greater angle. XXIV. If two triangles have two sides of the one equal to two sides of the other, each to each,... | 677.169 | 1 |
Relation between affine parameters
In summary, Affine parameters are variables used to describe the geometric relationship between two objects in a specific space. They are important in fields such as physics, computer graphics, and image processing as they allow us to define and analyze transformations. Affine parameters are closely related to affine transformations, which describe how the parameters of an object change during a transformation. They can only be used to describe linear transformations, and they are unique to each specific transformation and coordinate system.
Nov 10, 2011
#1
Identity
152
0
In Euclidean three-space with coordinates [itex](x,y,z)[/itex] and line element
[tex]ds^2 = dx^2+dy^2-dz^2[/tex]
It is easy to show using the geodesic equation that:
[itex]x = lu+l'[/itex], [itex]y=mu+m'[/itex], [itex]z=nu+n'[/itex]
where [itex]u[/itex] is an affine parameter.
However, is it possible to find a relation between [itex]l,m,n[/itex]?
Shouldn't l,m, and n be analogous to a four-velocity in Minowskian space, and hence have a constant norm?
What are affine parameters?
Affine parameters are variables used to describe the geometric relationship between two objects in a specific space. They are usually used in mathematics and physics to define transformations and coordinate systems.
What is the importance of affine parameters in science?
Affine parameters are important in science because they help us understand the relationship between two objects in a particular space. They allow us to define and analyze transformations, which are crucial in fields such as physics, computer graphics, and image processing.
How are affine parameters related to affine transformations?
Affine parameters and affine transformations are closely related. Affine transformations are mathematical operations that map one set of affine parameters to another. In other words, they describe how the affine parameters of an object change when it undergoes a transformation. | 677.169 | 1 |
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