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Given any chord, if a second chord is perpendicular to that chord and also passes through the center of the circle, then the first chord must be bisected. This means the first chord has been divided into two equal halves, meaning x must also be 8. | 677.169 | 1 |
Tag: How to prove 3 points are collinear
Introduction Collinearity is a fundamental concept in geometry that refers to points lying on the same straight line. Proving that three points are collinear requires careful analysis and application of … | 677.169 | 1 |
This article is a summary of a YouTube video "Cross products | Chapter 10, Essence of linear algebra" by 3Blue1Brown
Understanding the Cross Product: From Basics to Linear Transformations
TLDRLearn about the cross product, its geometric representation, and its connection to linear transformations. Explore the standard introduction as well as a deeper understanding of this mathematical concept.
Key insights
🔑The cross product of two vectors is the area of the parallelogram they span.
💡The cross product can be positive or negative depending on the orientation of the vectors.
🌟The right hand rule can be used to determine the direction of the cross product.
Q&A
What is the cross product?
—The cross product of two vectors is the area of the parallelogram they span and is perpendicular to both vectors.
How do you determine the direction of the cross product?
—The right hand rule is used, where you point your forefinger in the direction of the first vector, middle finger in the direction of the second vector, and your thumb points in the direction of the cross product.
Can the cross product be positive and negative?
—Yes, the cross product can be positive or negative depending on the orientation of the vectors.
What is the formula for the cross product?
—The cross product formula involves the 3D determinant, where the first column is the basis vectors and the other two columns are the coordinates of the vectors being crossed.
Timestamped Summary
00:09The video introduces the topic of cross products and emphasizes its connection to linear transformations.
00:42In two dimensions, the cross product of two vectors is the area of the parallelogram they span.
01:10The orientation of vectors impacts the sign of the cross product, where positive indicates the vector on the right and negative indicates the vector on the left.
02:00The determinant can be used to compute the cross product, where the matrix's columns represent the vectors being crossed.
04:37The right hand rule is used to determine the direction of the cross product, pointing the forefinger in the direction of the first vector, middle finger in the direction of the second vector, and the thumb points in the direction of the cross product.
05:42The cross product in three dimensions involves a 3D determinant, with the first column containing the basis vectors and the other two columns containing the coordinates of the vectors being crossed.
07:58The cross product represents a vector perpendicular to the two vectors being crossed, with its magnitude as the area of the parallelogram and its direction determined by the right hand rule.
08:28Understanding the geometric representation and computation of the cross product provides insight into areas, orientation, and linear transformations. | 677.169 | 1 |
Get Answers to all your Questions
Write 'True' or 'False' and justify your answer in each of the following AB is a diameter of a circle and AC is its chord such that angleBAC = 30°. If the tangent at C intersects AB extended at D, then BC = BD.
Write 'True' or 'False' and justify your answer in each of the following
AB is a diameter of a circle and AC is its chord such that BAC = 30°. If the tangent at C intersects AB extended at D, then BC = BD.
Answers (1)
Answer True Solution: First of all we solve the question according to give conditions. If we able to prove it then it will be true otherwise it will be false.
Given :BAC =
Diagram : Construct figure according to given conditions then join BC and OC.
To Prove : BC = BD
Proof :BAC = (Given)
[ angle between chord and tangent id equal to the angle made by chord in alternate segment]
[ Radius and tangent's angle is always ]
In OAC
OA = OC (both are radius of circle) [opposite angles of an isosceles triangle is equal]
In
[ sum of interior angle of a trianglE ]
In BCD we conclude that and [sides which is opposite to equal angles is always equal]
Hence Proved.
Hence the given statement is true. | 677.169 | 1 |
supplementary angles
Supplementary angles are a pair of angles that have a sum of 180 degrees
Supplementary angles are a pair of angles that have a sum of 180 degrees. In other words, when you add the measures of two supplementary angles together, the result will always be equal to 180 degrees.
To better understand this concept, let's consider an example. Let's say we have two angles, angle A and angle B. If angle A measures 60 degrees, and angle B is supplementary to angle A, we can find the measure of angle B by subtracting 60 degrees from 180 degrees (since the sum of the measures of supplementary angles is 180 degrees).
It is important to note that supplementary angles do not have to be adjacent or sharing a common vertex. They can be placed anywhere in relation to each other. For instance, one angle could be inside a triangle, while the other might be outside the triangle.
Supplementary angles are commonly found in various geometric situations, such as in a straight line or in a triangle. In parallel lines, for example, alternate interior angles are supplementary. Furthermore, in a triangle, the sum of the three angles is always 180 degrees, making them supplementary to each other.
Knowing that two angles are supplementary can be helpful in solving problems involving angles, as it provides a relationship between their measures | 677.169 | 1 |
What is meant by orthogonality?
1a : intersecting or lying at right angles In orthogonal cutting, the cutting edge is perpendicular to the direction of tool travel. b : having perpendicular slopes or tangents at the point of intersection orthogonal curves.
What is orthogonality in Java?
Orthogonality means that features can be used in any combination, that the combinations all make sense, and that the meaning of a given feature is consistent, regardless of the other features with which it is combined.
Does C++ have orthogonality?
C/C++ is not orthogonal because arithmetic operators do not consistently work on pointers.
What is orthogonality in signals and systems?
Any two signals say 500Hz and 1000Hz (On a constraint that both frequencies are multiple of its fundamental here lets say 100Hz) ,when both are mixed the resultant wave obtained is said to be orthogonal. Meaning: Orthogonal means having exactly 90 degree shift between those 2 signals.
Why is Python orthogonal?
Orthogonal features make the language very expressive and it evident in the succinct expressions used in libraries like Pandas and Numpy. Python is one of the rare languages designed from the outset keeping in mind the beauty and elegance of programs that can be written in the language.
Why orthogonality is important in communication?
Orthogonal signals are used extensively in communications because they can be received and demodulated as separate data streams with very little interference between the orthogonal signals.
What is meant by orthogonality of signals code?
Truly orthogonal codes: Two codes are said to be orthogonal if when they are multiplied together the result is added over a period of time they sum to zero. For example a codes 1 -1 -1 1 and 1 -1 1 -1 when multiplied together give 1 1 -1 -1 which gives the sum zero.
What is orthogonality in communication?
Orthogonality means both signal is having phase difference of 90 degree. Hence, it will not interfere each other. Just like CDMA, all the channels are orthogonal and hence we can use same frequency allocation for all users but signals are decoded based on PN sequence which is used for spreading the signal. | 677.169 | 1 |
90 Degree Mitered Corner Calculator
This calculator is used to calculate the dimensions needed when cutting 90 degree angled corners of a material or structure.
This calculator is used to calculate the dimensions needed when cutting 90 degree angled corners of a material or structure. It is a practical tool to save material and get the right cutting angles for corner joints. It helps you get the right measurements when making your calculations, so you can achieve more efficient and accurate results in your work.
When using the online 90 degree mitered corner calculator: you can calculate by entering trace width and dielectric thickness.
How to Calculate a 90 Degree Mitered Corner?
To calculate a 90 degree mitered corner, you can follow the steps below:
Material Selection: First, determine the material to be used in the corner joint. This material can usually be wood, metal or plastic.
Determine the Cutting Angle: Determine the cutting angle of the material to be used in the corner joint. Generally, 45-degree angle cuts are most common, which creates a 90-degree corner in total.
Cutting Direction: Decide in which direction the material will be cut. Typically, the cut is made towards the inside of the corner at the joint.
Determination of Dimensions: Measure the length and width of the corner where the two pieces will meet. These measurements will be used to determine the correct dimensions required for cutting and joining the materials.
Cutting Process: Cut the material according to the determined angle and dimensions. Use a suitable cutting tool for this process, usually a saw or cutting machine is preferred.
Joining: Join the cut pieces together to form a 90 degree mitered corner. Use a suitable adhesive or jointing method for joining.
By following these steps, you can accurately calculate and create a 90 degree mitered corner.
What is a 90 Degree Mitered Corner?
A 90-degree mitered corner is a joint usually used in the production of a building or furniture. This type of corner refers to an angle where two pieces meet perpendicular to each other, creating 90 degrees in total. This joint is usually achieved by cutting and joining materials such as wood, metal or plastic.
Typically, 90 degree mitered corners are used in various pieces of furniture, frame construction, construction projects and other carpentry applications. It is a common type of joint in wooden frames, window frames, door frames and similar structural components. Mitered cuts ensure that the edges of the joining parts meet properly, resulting in an aesthetically pleasing appearance and structural durability.
This type of corner joint is usually achieved by joining two pieces cut at 45 degree angles towards each other. In this way, both pieces form a total angle of 90 degrees. 90-degree mitered corners provide a solid and smooth joint, making projects strong and aesthetically appealing.
Basic Concepts of 90 Degree Mitered Corner
The basic concepts related to 90 degree mitered corner are as follows:
Miter: refers to the angle at which two pieces are joined. A 90 degree miter represents an angle where the joined parts are perpendicular to each other.
Cutting Angle: In mitered cuts, it is important to cut the material at the correct angle. In 90 degree mitered corners, 45 degree angle cuts are usually used.
Cutting Direction: Refers to the direction in which the material will be cut. In mitered corners, cuts are usually made into the mating parts.
Accuracy: The accuracy of mitered cuts ensures that the joining parts are joined properly. Without accurate cuts, there may be gaps or mismatches between the joined parts.
Connection Methods: Various methods can be used to join mitered corners, including gluing, the use of screws or nails, glue or mechanical connections such as pins.
Material Selection: The type of material to be used for mitered corners affects the cutting method and the connection method. Materials such as wood, metal or plastic are commonly used.
These basic concepts are important elements to understand in the process of calculating and creating 90 degree mitered corners. The right understanding and application will help you achieve solid and aesthetically pleasing results.
Uses of 90 Degree Mitered Corner Calculation
90 degree mitered corner calculation finds use in carpentry, construction, furniture making and many other industrial and craft applications. Some of the uses are as follows:
Furniture Making: 90 degree mitered corner calculations are widely used in the production of wooden furniture, especially in frame construction and furniture joints. Joining furniture pieces at the right angles and cuts is important to achieve a solid and aesthetically pleasing result.
Construction: In the construction industry, 90-degree mitered corner calculations are common in the joining of structural elements. For example, such joints are used in frame walls, window and door frames, stair construction and roof construction.
Carpentry: In carpentry work, especially in woodworking, 90 degree mitered corner calculations are widely used. Cutting and joining wooden pieces at right angles to each other is important for producing high-quality furniture, decorative objects and structural elements.
Decoration and Crafts: 90 degree mitered corner calculations are widely used in home decor, crafts and DIY projects. Accurate corner joints are essential for the assembly of wooden shelves, frame construction, wall panels and the production of various decorative objects.
Industrial Production: In industrial production, 90 degree mitered corner calculations can be used in the processing and joining of materials such as metal or plastic. Such calculations are especially important in the metalworking industry, pipe joints, profile cutting and assembly work.
These are common uses for 90 degree mitered corner calculations, but the list is not exhaustive. These calculations have a wide range of applications in various industries and are important for achieving quality and durable results when done correctly. | 677.169 | 1 |
diagram above, O is the center of the circle. What is the leng
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11 Mar 2015, 06:03
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BookmarksRe: In the diagram above, O is the center of the circle. What is the leng
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11 Mar 2015, 07:05
Answer should be C. triangle ABC is right angled triangle, as angle made by diameter on a triangle is right angle. We need any two sides to get the third side`s length. statement 1 is not sufficient, so is the statement 2. considering 1 + 2 , We can get the diameter of the circle from 2, which is one side of the triangle and from 1 we have another side. So, by Pythagoras theorem, we can get length of AC. Sufficient.
The fact that AB is a diameter guarantees that angle C = 90º. If we had two sides of right triangle ABC, we could find the third using the Pythagorean Theorem.
Statement #1: this gives us only one side of a right triangle: not helpful. This statement, alone and by itself, is not sufficient.
Statement #2: this allows us to solve for the radius and, hence, the diameter, so we can determine side AB. Nevertheless, this gives us only one side of a right triangle: also not helpful. This statement, alone and by itself, is not sufficient.
Combined statements: We get the length of BC from the first statement, and the length of AB from the second. Now, we have two sides of the right triangle, so we can use the Pythagorean Theorem to solve for the third side, AC. Combined, the statements are sufficient.
Re: In the diagram above, O is the center of the circle. What is the leng
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15 Apr 2017, 17:24
23a2012 wrote:In the diagram above, O is the center of the circle. What is the leng
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Updated on: 17 May 2021, 08:35
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Expert Reply
Top Contributor
Bunuel wrote:
Attachment:
cpotg_img4-300x285.png
In the diagram above, O is the center of the circle. What is the length of chord AC?
(1) chord BC = 14 (2) the circle has an area of \(625\pi\)
Kudos for a correct solution.
Target question:What is the length of chord AC?
Given: O is the center of the circle If O is the center of the circle, then AB is the circle's DIAMETER If AB is the DIAMETER, then ∠C = 90°, because ∠C is an inscribed angle containing ("holding") the diameter. So, let's first add this information to the diagram
Statement 1: chord BC = 14 Notice that the length of chord BC has no bearing on the length of chord AC. In fact, here are two diagrams that satisfy statement 1:
In the left-hand diagram, the answer to the target question is chord AC has length 20 In the right-hand diagram, the answer to the target question is chord AC has length 30 Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: the circle has an area of 625π Area of circle = πr² So, we can write: πr² = 625π Divide both sides by π to get: r² = 625 Solve: r = 25 So, the circle's radius = 25, which means the DIAMETER AB has length 50.
This time the length of the diameter has little bearing on the length of chord AC. In fact, here are two diagrams that satisfy statement 2:
In the left-hand diagram, the answer to the target question is chord AC has length 30 In the right-hand diagram, the answer to the target question is chord AC has length 40 Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined When we combine the two statements, we see that we know the lengths of two sides of a RIGHT triangle
So, we COULD apply the Pythagorean Theorem to write: 14² + x² = 50², And we COULD solve the equation to get x = 48. However, performing all of those calculations would be a waste of the time, since we need only show that we COULD answer the target question with certainty. Since we COULD answer the target question with certainty, the combined statements are SUFFICIENT
Re: In the diagram above, O is the center of the circle. What is the leng
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01 Jul 2023 above, O is the center of the circle. What is the leng [#permalink] | 677.169 | 1 |
congruent triangles snowflake activity answer key
Congruent Triangles Snowflake Worksheet Answers – Triangles are among the most fundamental designs in geometry. Understanding triangles is vital to mastering more advanced geometric concepts. In this blog post this post, we'll go over the various types of triangles that are triangle angles. We will also explain how to calculate the area and perimeter of a triangle, and show the examples for each. Types of Triangles There are three kinds of triangles: equal isosceles, as well … Read more | 677.169 | 1 |
A simple visual way to show that the parametric equation of a circle is a helix in our three-dimensional space.
Parametric equation of a circle f1 and f2.
The helix is defined by the intersection of two mutually perpendicular cylindrical surfaces f1 and f2. | 677.169 | 1 |
An inscribed angle is an angle with its vertex on the circle and whose sides are chords. The intercepted arc is the arc that is inside the inscribed angle and whose endpoints are on the angle.
What does inscribed angle mean in geometry?
In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.
What does an arc mean in geometry?
In general, an arc is any smooth curve joining two points. The length of an arc is known as its arc length. In a graph, a graph arc is an ordered pair of adjacent vertices. In particular, an arc is any portion (other than the entire curve) of the circumference of a circle.
How do you find the arc of an inscribed angle?
Theorem 70: The measure of an inscribed angle in a circle equals half the measure of its intercepted arc. The following two theorems directly follow from Theorem 70. Theorem 71: If two inscribed angles of a circle intercept the same arc or arcs of equal measure, then the inscribed angles have equal measure.
What is central angle and inscribed angle?
The degree measure of a central angle is equal to the degree measure of its intercepted arc. For the circle at right with center C, ∠ACB is a central angle. An INSCRIBED ANGLE is an angle with its vertex on the circle. and whose sides intersect the circle.
What is the relationship between arc and inscribed angles of a circle?
By the inscribed angle theorem, the measure of an inscribed angle is half the measure of the intercepted arc.
What is the meaning of arc in circle?
The arc of a circle is defined as the part or segment of the circumference of a circle. A straight line that could be drawn by connecting the two ends of the arc is known as a chord of a circle. If the length of an arc is exactly half of the circle, it is known as a semicircular arc.
What does two arcs mean in geometry?
In mathematics, unless otherwise stated, an arc usually refers to a portion of a circle. Types of arcs. A chord, a central angle or an inscribed angle may divide a circle into two arcs. The smaller of the two arcs is called the minor arc. The larger of the two arcs is called the major arc. | 677.169 | 1 |
rod fixed to a wall, which can be pulled by a chain by applying a force at one of its ends. The position of the rod is defined using a three-dimensional coordinate system.
The angle theta between the force vector and the rod, and the projection of force along the rod needs to be determined.
First, the position vectors for the two ends of the rod are defined. Then the position vector along the rod is determined.
The next step determines the magnitude of the position vector rAB and the force vector.
Now, the dot product of the position vector with the force vector is determined by multiplying the components of the two vectors. Angle theta is then estimated as the inverse cosine function of the ratio of the dot product and the product of magnitudes of the two vectors.
The projection of the force along the rod can be determined as the product of the magnitude of force and the cosine of theta.
2.16: Dot Product: Problem Solving
The dot product is a powerful tool in problem-solving involving vectors, given that the dot product of two vectors is the product of their magnitudes and the cosine of the angle between them measured anti-clockwise. Solving problems involving the dot product requires understanding its properties and developing a step-by-step process to solve them. Here are the main steps to follow when solving any general problem involving the dot product:
Identify the problem: Start by reading the problem and identifying the question that needs to be answered. This will enable you to determine the purpose and direction for solving the problem.
Define the vectors: List the given vectors and represent them in the Cartesian or component form.
Decide which operation to use: The dot product is appropriate when the problem involves finding the angle between two vectors, calculating the component of a vector along a given direction, testing orthogonality, or finding the projection of one vector onto another vector. Ensure that the problem requires the use of the dot product before proceeding.
Calculate the dot product: Multiply the corresponding components of the two vectors and sum their products. This gives the value of their dot product.
Verify the solution: Check your solution to ensure that it satisfies the given conditions in the problem. Be sure to round off the answer appropriately and include the correct units where necessary.
The angle between two vectors can be obtained from the inverse cosine of the dot product of the two vectors divided by the product of the magnitudes of the two vectors. The dot product can also be employed to find the component of a vector along a given direction by projecting it onto a unit vector in the desired direction. This technique is particularly useful for decomposing complex vector problems into simpler components. Additionally, the dot product can be used to test orthogonality between two vectors. If their dot product is zero, the vectors are orthogonal, meaning they are perpendicular to each other. Lastly, the projection of one vector onto another can be found using the dot product by multiplying the magnitude of the first vector by the cosine of the angle between the two vectors. | 677.169 | 1 |
Plane and Spherical Trigonometry, Surveying and Tables
From inside the book
Results 1-5 of 12
Page 194 ... Chain is generally employed in measuring land . It is 4 rods , or 66 feet , in length , and is divided into 100 links . Hence , links may be written as hundredths of a chain . The Engineer's Chain is employed in surveying railroads ...
Page 196 ... chain may be determined . If the slope is considerable , half a chain or less may be used . To construct a perpendicular with a chain : 1. When the point through which the perpendicular is to pass is in the line : Let AB ( Fig . 1 ) ...
Page 202 ... chain , pins , and other instruments that would affect the direction of the needle , are removed from the vicinity of the compass . Even after the greatest care in this respect is exercised , the direction of the needle is often ...
Page 215 ... chains . 3. Required the area of a triangular field whose base is 12.60 chains , and altitude 6.40 . 4. Required the area of a triangular field which has two sides . 4.50 and 3.70 chains , respectively , and the included angle 60 | 677.169 | 1 |
The first six books of the Elements of Euclid, with numerous exercises
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Ergebnisse 1-5 von 6
Seite 34 ... twice the line joining the vertex and the middle of the base . 21. Each angle at the base of an isosceles triangle ... rectangle are equal to one another . 25. If any number of parallelograms be inscribed in a given parallelo- gram ...
Seite 37 ... twice the rectangle contained by the parts . LET the straight line ab be divided into any two parts in c ; the square of a b is equal to the squares of a c , cb , and to twice the rectangle con- tained by a c , cb . a مع с b k f e ...
Seite 39 ... twice the rectangle contained by the whole and that part , together with the square of the other part . LET the straight line ab be divided into any two parts in the point c ; the squares of a b , b c are equal to twice the rectangle ...
Seite 43 ... rectangle cf , fa , together with the square of a e , is equal to the squares of ba , ae : take away the square of ... twice the rectangle contained by the side upon which , when produced , the perpendicular falls , and the ...
Seite 44 Euclides. PROPOSITION XIII . - THEOREM . In every triangle , the square of the side subtending any of the acute angles is less than the squares of the sides containing that angle , by twice the rectangle contained by either of these ...
Beliebte Passagen
Seite 2 - A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another : XVI.
SeiteSeite | 677.169 | 1 |
Lets say that I have a line with one end fixed to the center of a sphere, and the other end can freely rotate. If I were to rotate the line around the x and y axes, what would the coordinates be for the freely-rotating end?
Here's what I need this for:
I have a rectangle, and need to find the coordinates of each vertex when the rectangle is rotated (the rectangle will be rotated from its center at (0,0,0), and can have any width and height). I only need to rotate the rectangle around the x,y (or any pair really, but x,y are just a nice looking pair), because rotating the unincluded axis will be like rolling a pencil back and forth in your thumb.
I have found some helpful resources for questions similar to mine, but they don't solve my problem. For example, this here I've tried to understand, but I have minimal experience with matrices, and this doesn't exactly solve my problem.
1 Answer
1
You say we have a line with one end fixed at the origin; say that the point on this line which is distance 1 from the origin is $v =(a,b,c).$ When I rotate $v$ around the $z$ axis, I won't change it's $z$ coordinate--this will still be $c.$ If I rotate by an angle of $\theta,$ the output vector will be $(cos(\theta)a - sin(\theta)b, sin(\theta)a + cos(\theta)b, c).$
$\begingroup$If I were to then rotate the line along the x after rotating the z as you have shown, would I just input the coordinates from above into: y' = ycos q - zsin q z' = ysin q + zcos q and the x remains the same, would that work?$\endgroup$
$\begingroup$If you would like to first rotate $(a,b,c)$ by an angle of $\theta$ around the $z$ axis and then rotate the result by an angle of $\alpha$ around the $x$ axis, the output would be $(a cos(\theta) - b sin(\theta), a sin(\theta) cos(\alpha) + b cos(\theta) cos(\alpha) - c sin(\alpha), a sin(\theta) sin(\alpha) + b cos(\theta) sin(\alpha) + c cos(\alpha).$$\endgroup$
$\begingroup$also are the rotations "added" onto the line (like a rotation relative to the line), or does it rotate the line to an exact point as if the line could be anywhere, and always rotate to the same place each time?$\endgroup$ | 677.169 | 1 |
Finding the Angle Between Vectors a and b
In summary, the conversation discusses finding the angle between two perpendicular vectors, a and b, with lengths of 2 and 1 respectively. The dot product method is used to find the angle, with the final equation being arccos[(a.b)/(|a||b|)].
Mar 1, 2012
#1
Jane K
3
0
1. Homework Statement
The vector a=2 and vector b=1. The vectors a+5b and 2a-3b are perpendicular. Determine the angle between a and b .
Homework Equations
The dot product a•b=lallblcosθ
The Attempt at a Solution
I've tried a few things but none of it really makes sense. I'm worried that maybe this question doesn't call for the dot product method but I've become fixed on it.
what SammyS means is that the dot product is bilinear, it is linear in each variable:
if a,b,c are vectors, and r is a scalar:
a.(b+c) = a.b + a.c
(a+b).c = a.c + b.c
a.(rb) = r(a.b)
(ra).b = r(a.b)
also, a.b = b.a (the dot product is symmetric).
thus (a+5b).(2a-3b) = 2(a.a) + 5(b.a) - 3(a.b) - 15(b.b)
a.a = |a|2, for any vector a.
you are given |a| and |b|, and you are given that the dot product (a+5b).(2a-3b) = 0.
if you can deduce what a.b is, then you can figure out the angle:
θ = arccos[(a.b)/(|a||b|)]
Mar 2, 2012
#7
Jane K
3
0
Thank-you everyone!
This really helped:)
Related to Finding the Angle Between Vectors a and b
1. What is the formula for finding the angle between two vectors a and b?
The formula for finding the angle between two vectors a and b is given by: cosθ = (a • b) / (|a| * |b|), where θ is the angle between the two vectors and (a • b) represents the dot product of the two vectors.
2. How do you find the dot product of two vectors?
To find the dot product of two vectors, you multiply the corresponding components of the two vectors and then add them together. For example, if vector a = (a1, a2, a3) and vector b = (b1, b2, b3), then the dot product of a and b is given by: a • b = a1*b1 + a2*b2 + a3*b3.
3. Can the angle between two vectors ever be negative?
No, the angle between two vectors is always positive. This is because the range of the inverse cosine function (used to find the angle) is between 0 and π, which means the resulting angle will always be positive.
4. What is the significance of finding the angle between two vectors?
Finding the angle between two vectors is important in understanding the relationship between the two vectors. It can also be used to determine if the vectors are perpendicular (angle = 90°), parallel (angle = 0° or 180°), or at an acute or obtuse angle to each other.
5. Is there a special way to find the angle between two vectors in 3-dimensional space?
Yes, the formula for finding the angle between two vectors in 3-dimensional space is the same as the formula for 2-dimensional space, except now the dot product is replaced by the cross product. The formula is: cosθ = (a × b) / (|a| * |b|), where θ is the angle between the two vectors and (a × b) represents the cross product of the two vectors. | 677.169 | 1 |
Area of an Oblique Triangle Calculator
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Area of an Oblique Triangle Calculator
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Welcome, geometry enthusiasts! If you thought oblique triangles were just for making your geometry textbook look like a dangly earring, you're in for a surprise! They have an area (yes, really!) and we're going to show you how to calculate it. But brace yourself, it's going to be a trigonometrical roller coaster!
The formula to calculate the area of an oblique triangle is:
Area = 0.5 * a * b * sin(C)
Where a and b are the lengths of two sides and C is the angle between them. | 677.169 | 1 |
Get Answers to all your Questions
1. How would you rewrite Euclid's fifth postulate so that it would be easier to understand?
Answers (1)
Euclid's postulate 5: If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.
Now, in an easy way
Let the line PQ in falls on lines AB and CD such that the sum of the interior angles 1 and 2 is less than 180° on the left side of PQ. Therefore, the lines AB and CD will eventually intersect on the left side of PQ. | 677.169 | 1 |
The elements of plane geometry; or, The first six books of Euclid, ed. by W. Davis 16.
сЕКъДА 8 ... angle BAC is equal ( Hyp . ) to the angle EDF Also , the point C shall coincide with the point F , because AC ( Hyp . ) is equal to DF . But the point B was proved to coincide with the point E. Therefore the base BC shall coincide with ...
сЕКъДА 10 ... angle ACD is equal ( I. 5 ) to the angle ADC . But the angle ACD is greater ( Ax . 9 ) than the angle BCD ... ( EDF ) contained by the two sides equal to them , of the other . For , if the triangle ABC be applied to the triangle DEF ...
сЕКъДА 17 ... angle FAG is equal ( I. 8 ) to the angle DCE . Wherefore at the given point A , in the given straight line AB , the ... ( EDF ) contained by the two sides equal to them , of the other ; the base ( BC ) of that which has the greater ...
сЕКъДА 18 ... angle of the one to the third angle of the other . First , let those sides be equal which are adjacent to the angles ... EDF . For , if AB be not equal to DE one of them must be greater than the other . Let AB be the greater of the ...
дГЛОЖИКч АПОСПэСЛАТА
сЕКъДАсЕКъДА 4 - A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another : XVI.
сЕКъДАсЕКъДА 12 - When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle ; and the straight line which stands on the other is called a perpendicular to it.
сЕКъДАсЕКъДАсЕКъДАсЕКъДАсЕКъДА | 677.169 | 1 |
This question is motivated by an answer I provided to a question here on the arc length of a cycloid.
I noticed that the ratio of the circumference of the generating circle (which is also the horizontal distanced traveled through one cycle) to the arc length of the cycloid is
$$ \frac{2\pi r}{8r} = \frac{\pi}{4} . $$
If we take that generating circle and circumscribe a square around it, the ratio of the areas of the circle to the square is the same:
$$ \frac{\pi r^2}{(2r)^2} = \frac{\pi}{4} . $$
Given a circle, is there some sort of deeper relationship between the cycloid it generates and its circumscribed square? Or is this just a coincidence? I don't have much experience with roulettes, so I'm at a bit of a loss here - would appreciate any insight anybody has on whether there's some deeper geometric relationship here.
$\begingroup$@TedShifrin that was my inital thought too, but was tinkering with cyclogons and thought there might have been something there. Also, this isn't the appropriate venue for this, but I just have to say that my first undergraduate math course used your linear algebra textbook, it completely changed how I thought about both algebra and geometry, and I just wanted to thank you for that!$\endgroup$
1 Answer
1
Yes there is a relationship, based on a couple hidden features of cycloidal loci. And there is also a similar relationship between the four-cusped hypocycloid and a regular hexagon inscribed in the same circle, derivable from thecsame concepts.
Making a clean sweep
You already know that when a circle is rolled along a fixed straight line, a fixed point on the circle generates a cycloid loop whose arc length is four times the diameter of the circle. Since the center of the circle also slides through a length equal to (of course) $\pi$ times the diameter, the arc length of the cycloid loop may also be rendered as $4/\pi$ time the length traversed by the center.
Now suppose the straight line is replaced by a circular arc to make an epicycloidal loop (if the fixed arc is externally tangent to the roller) or a hypocycloidal loop (if internally tangent). The calculus is more complicated now, but we find that the arc length of the epicycloidal/hypocycloidal loop is still $4/\pi$ times the arc length traversed by the center of the roller.
To explain this geometrically, consider the picture below in which a cycloid is generated from point $A$ through $A'$ to $A''$. The roller sweeps through an area, which is equivalent to a rectangle whose dimensions are the length of the center's path and the diameter of the roller. Replacing the fixed line with a circular arc leaves this relationship unchanged; half the area is stretched but the other half is squeezed to balance it. (We assume that in the hypocloidal case the fixed arc has large enough radius so that its center is not in the interior of the roller. This assumption will be obeyed in what is to follow.)
We imagine this area as being filled with crisscrossing cycloidal loops whose points on the roller are distributed over the circumference of the same. Thus the arc length of a loop should be proportional to the swept area divided by the circumference. This quotient contains only the length of the centerline path.
Through this proportionality we may relate the length of the cycloid loop to various epicycloid or hypocycloid loops. For instance, suppose that the rolling circle from the cycloid generation is now held fixed and we make a new roller, internally tangent with half the radius. The fixed point now moves from $B$ through $B'$ to $B''$ where it re-contacts the larger circle.
In this case the center of the roller has moved only half its circumference and the roller itself is only half the circumference of that used for the cycloid. Therefore, the length of this path should be one-fourth the length of the cycloid loop. If the $B$ to $B''$ path corresponds to one side of a circumscribed square, then the cycloid loop will match the whole square. But, can we prove the former claim? And is there something to the hypocycloid loop in my second picture looking like a straight line?
Spiro-mania!
In my younger days I played with a drawing toy in which you place a pen inside a wheel (the pen being a fixed point in the wheel) and roll it around a wheel, ring or rack that is fixed to the paper, thus producing various patterns. If you choose a wheel with $48$ teeth and roll it in a ring interior having $96$ teeth (a $1:2$ ratio), you get just an oval, which becomes longer and narrower if you place the pen near the outer edge of the wheel. Evidently if the pen could be placed at the very edge of the wheel (physically impossible because the gear-teeth are there), this oval would be elongated and flattened to a full diameter of the ring. Which would match a side of a circumscribed square ... .
To prove this mathematically, consider the figure below for the case where the roller is moved up to 90° from the initial contact.
Because of the $1:2$ circumference ratio the roller has the center $O$ of the fixed circle on its boundary (not in the interior, see above), and also the arc on the roller from $C$ to $B'$ has twice as much angular measure as the arc from $B$ to $C$. Now, central $\angle BOC$ has the same measure as arc $BC$, and inscribed $\angle B'OC$ has half the angular measure as arc $B'C$ -- so the two angles are congruent, and this indeed forces $B'$ to lie on the radius $\overline{OB}$.
For a greater rolling angle, first move the roller $180°$ to recontact the locus at $B''$, then roll back through the supplementary acute angle and apply the above reasoning replacing $B$ with $B''$.
Fitting a square peg around a round hole...
So our hypocycloidal loop, with a $1:2$ ratio of diameters or circum-ferences, indeed matches a diameter of the circle and thus one side of a circumscribed square, and measures one-fourth the length of the corresponding cycloid. Therefore the cycloid loop measures the full perimeter of the square.
...and a hexagonal peg inside
At the beginning of this geo-metrical trip, I mentioned that there is a similar relationship between a four-cusped hypocycloid and a regular hexagon inscribed in the same circle. I present that here in the form of a "homework" problem.
A four-cusped hypocycloid is generated as the fixed-point locus obtained when a roller of diameter $r/2$ is rolled inside a fixed circle of diameter $2r$. Using the fact that the hypocloid can be rendered with the parametric equations $x=r\cos^3\theta,y=r\sin^3\theta$, apply standard calculus techniques to find that the length of the complete hypocycloid matches the perimeter of an inscribed regular hexagon.
Compare the arc length traversed by the center of the roller over all four hypocycloidal loops to the corresponding arc length associated with diameter $\overline {BB''}$ above. Conclude that the length of the hypocycloid matches three diameters of the circle, which can be matched by obvious geometric means with the inscribed regular hexagon. | 677.169 | 1 |
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Bihar Board Class 9 Topics And Chapters
Many students struggle to pass their school-level exams because they need help understanding the topics. Have you ever considered the reasons why students struggle to understand the topics? It can be because of the complexity of the topics, the attention span or other factors coming in the way of comprehending the Bihar Board Class 9 Topics. We have a solution to end this struggle to understand difficult topics.
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9th Bihar Board Topics With Video Solutions
To perform well in the exam, you need a comprehensive understanding of the topics. You must make a preparation plan and divide your time wisely. You should focus more on the topics you are weak in. If you want to learn the topics in a new way, you must go through the 9th Bihar Board topics with video solutions given in the article below.
Basic Terms Food Angles Between Lines, Pairs of Angles, Intersecting Lines and Non Intersecting Lines, Parallel Lines and a Transversal, Lines Parallel to the Same Line, Angle Sum Property of a Triangle
7
Triangles
Congruence of Triangles, Criteria for Congruence of Triangles, Properties of a Triangle, RHS and SSS For Congruence, Inequalities in a Triangle
8
Quadrilaterals
Angle Sum Property of a Quadrilateral, Types of Quadrilateral, Properties of a Parallelogram, the Mid-point Theorem,
9
Area of Parallelogram and Triangles
Understanding Area of Plane Figures, Triangles on the Same Base and Between Same Parallels, Parallelogram on the Same Base and Between Same Parallels
10
Circles
Circle and Its Related Terms, Angle Subtended by a Chord at a Point, Perpendicular From the Centre to a Chord, Circle Through Three Points, Equal Chords and Their Distances From the Centre, Angle Subtended by an Arc of a Circle, Cyclic Quadrilaterals
11
Constructions
Basic Constructions, Construction of Triangles
12
Heron's Formula
Area of Triangle Using Base and Height, Area of Triangle Using Heron's Formula, Area of Quadrilateral Using Heron's Formula
13
Surface Areas and Volumes
Surface Area of a Cuboid and a Cube, Surface Area of a Right Circular Cylinder, Surface Area of a Right Circular Cone, Surface Area of a Sphere,Volume of a Cuboid, Volume of a Cylinder, Volume of a Right Circular Cone, Volume of a Sphere | 677.169 | 1 |
Rastko Vuković
July 2024
Saddle
Question: What is a saddle surface?
Answer: When we distort the flat Euclidean surface along the abscissa so that the points further away from the starting point on both sides are higher, and along the ordinate so that the points farther from the same origin are lower, It can also be the other way around, as seen in the picture on the left.
1. When at the beginning (red dot in the picture), a vertical circle starts the abscissa (from above), and another upright, but perpendicular to the first, starts the ordinate (from below), with radii in the order rx > 0 then ry < 0, then the number κ = 1/rxry we call Gaussian curvature of the surface. When κ = 0 is a surface of flat geometry, it is Euclidean. If κ < 0, which we see in the saddle in the picture, is a surface of negative curvature, when κ > 0, as in the case of a sphere, the surface is of positive curvature. We know negative surfaces as hyperbolic or Lobachevsky geometries, and positive surfaces are usually called Riemannian, spherical, or ellipsoidal.
Gauss was the first to find those curves κx = 1/rx and κy = 1/ry (Theorema Egregium, 1827) to observe how the surface of the curve κ = κ1κ2 ≠ 0 not to straighten without deformation. After all, that is why we cannot accurately represent the geography of the globe on flat maps. These ideas were developed by his student Riemann by extending the tensor calculus and then accepted by Einstein through the general theory of relativity.
If we draw equal lengths from the starting point, red in the picture, with the surface of the saddle and its ends, we define a curvilinear "circle" around it with a radius of a given length. The quotient of the circumference and diameter of such a circle is greater than π, than the ratio of the circumference and diameter of a circle of flat geometry. When we draw a curvilinear "triangle" on the saddle surface, connecting three given points with the shortest lines and placing tangents to those lines at the vertices, the sum of the three angles between the tangents will be less than π radians (180 degrees), which is less than the sum of the angles of a right triangle.
2. For example, in the rectangular Cartesian 3-dim coordinate system Oxyz, when all heights z of the saddle surface with projections onto the plane Oxy fall into points (x, y), we get the equation z = x² - y² of the saddle surface. When x, and in another case y, are constants, we have vertical planes that are hyperbolas in the sections of that saddle surface.
The picture on the right is one such hyperbola. The vertical sections of the planes of the saddle surface, as seen in the previous image on the left, are hyperbolas, and that is why the geometry of saddle surfaces is also called hyperbolic.
If we observe the same vertical sections of the saddle surface in the above image, there will be vertical hyperbolas. They become with arms facing upwards, i.e. convex, or with arms facing downwards, which is concave.
3. In the more general case, we can write this surface as z = f(x, y). This is how we work with mappings f: X × Y → ℝ, which, more generally, can be from the m-dim space X ⊂ ℝm in some n-dim space Y ⊂ ℝn. If that surface is compact, or, let's say complete (contains its boundary points), where:
Formally, this is Von Neumann's minimax theorem, whose consequences and interpretations are very important in game theory (Strategies). They are much deeper as well as more important than the simplicity of its confirmation on the surface of the upper picture on the left: the lowest point of the convex and the highest concave of the steepest hyperbola touch in one, so-called. saddle point, shown in red.
4. For example, a seller has a certain daily turnover and an expected income of some z monetary units. That income changes from time to time, so a larger deviation from expectations is less and less likely, but it's bigger news and more reason to fear when income decreases or to rejoice when it increases. The first is probabilities, and the second is information. We ignore the discreteness of the value of money.
When we project probability values and information onto the Oxz and Oyz planes, after leveling, or aligning the lengths of the abscissa and ordinate with the applicate, we get a saddle-shaped income surface. The minimax theorem will then tell the (obvious) truth that the expected value of income is the highest in probability and the lowest in information (news).
Senses
Question: How do I use the minimax theorem in information perception?
Answer: As a general theorem, minimax, with the correct initial conditions, accurately predicts it. Often, it can give good enough estimates even with approximately correct assumptions, and with chaos theory, we distinguish those rare situations when approximate initial conditions lead to significantly different final results. Otherwise, that discussion belongs to the previous conversation (Saddle), and only because of its length is it special.
In the picture on the left we see vectors with n = 1, 2, 3, ... components:
intensities a and b, which in pairs define (when they do not match) only one plane each. According to "Information of Perception", a developing theory, these vectors can be some "physical states", say object A and subject B, and their scalar product is "mutual coupling", then communication, but also perception information. Therefore, the components of these vectors tell us about the chances of two participants perceiving living or dead nature around them.
Even with the number of possibilities n → ∞, according to Borel–Cantelli lemma, only finally many of them will be relevant, so let them be these. Each subject will perceive the environment in a unique way (Only One). With the equal vector \( \vec{a} \) then the "same" participant \( \vec{b}_k \), in the picture k ∈ {1, 2, 3} and at least of the same intensity, the norms of the vector space \( b = \|\vec{b}_k\| \), does not have to span the same plane.
In microphysics, such a number of components is very small. An electron in an atom has only four quantum numbers, then n = 4. As the building elements are also in larger bodies, that is, their number of components (n) is very large, with maybe several millions or billions, but that is irrelevant because these conclusions are also theoretically interesting. We observe the projection of the state B on A, so the value in the bracket of the scalar product is:
The smaller the angle φk, the cosine of that angle is larger, cos φk ∈ (-1, 1), so the projection is larger, b⋅cos φk, states B to state A, although the capacity, or intensity b, may remain constant. However, even with the unchanging capacity of perception of the object by the subject's senses, or the intensity of a and b, the change in the direction of the vector changes the state of the subject B and its relationship to the environment A, as well as its actions. If these two vectors coincided in direction, then there would be some λ such that b = λa for intensities, as well as all coefficients, βik = λαi, in order of index i = 1, 2, ..., n .
The change of these influences is in the idea of reciprocity of game theory, in the manner of this information theory. When the angle φ between the vectors is smaller, the perception of the object by the subject is greater, but it can be said that the effect of the subject on the object is also greater. In the case of competition (valid only for vitals), the mutual information of perception grows with this higher coupling, and with it, the levels of the game. Vitality is hard work, and not everyone is the same.
The greatest masters of the game manage to stick their B vector almost to the opponent's vector. Then the one who manages to maintain the higher levels of the game wins, and the other becomes prey (Sneaking) in a game that he easily loses. The top league level is described as "sticking." He always wins the second level, i.e., "walking" from the first and "dodging," yielding to the principle of the least effect of still life, and both defeat that lowest level of play, which is close to inanimate physical matter. In old conversations and some texts, I called the players of these leagues in order "evil," the middle "manipulators," and at the bottom "good" (Traits).
Once we have cleared this up, it is easier to define the prerequisites for the minimax theorem. Various applications are possible, and here is an easy example. The larger the deviation of the angle φ from the optimal, here zero value, reduces the information of perception. However, as the uncertainty grows, so does the ignorance of B about the object A, and we can apply the minimax theorem: there is a single "saddle point," the maximum information (space X), and the minimum uncertainty (space Y) that joins them.
Successfully tracking this otherwise game-changing saddle point is game mastery that leads to player victory. Let's note that a good move by the opponent is to escape into uncertainty, unpredictability, or creativity, which is otherwise known in complex conflicts. It agrees with the "probability force" interpretation, which I consider my discovery as well as this theory. However, in the case of a less complex game, when there are not too many moves, or the player has enough time, this example can also be reduced to the classic games of Von Neumann: a move that interferes with another's best move — leads wins.
Aposematism
Question: The ease with which you apply the minimax theorem is interesting?
Answer: As in the case of solving problems in geometry, this "ease" came with mastery through many hours of studying the subject. It is nothing special, and you can see it in various skilled craftsmen, believe me. It is equally tempting for the ignorant to think that he could imitate it, which in this case, in the frequent misstatement of theorems, reinforces doubts about mathematics.
In this, I see part of the difficulties of applying abstract methods in the research of history, psychology, and biology. Such sciences, I hope you can see, are present in my discussions, but in a completely unconventional way. On the other hand, the tendency of many to underestimate these methods makes it easier for me to discover them with different knowledge.
Then, unexpectedly, the interlocutor suggested that I say something about aposematism. I found that aposematism, or aposematic, from the Greek apo (further) and semantic (meaning), is perhaps the most common form of warning sign in some animal species for protection from predators. Contrary to hiding colors, animals have developed the ability to discourage potential predators with the bright colors of their bodies, thus letting them know that they are poisonous food or otherwise dangerous. And here is my story.
On the one hand, it is the ability to have more offspring due to brighter skin colors, greater sexual attractiveness of colored couples, greater reproductive power, and better longevity. It is the space X of the saddle surface (Saddle). On the other hand, predators are less likely to prey on them because of their tastelessness or toxicity. This is the space Y. The third prerequisite of the minimax theorem is "completeness". It is achieved through a long series of generations, when the length of an individual's life becomes negligible, almost infinitesimal, in relation to the duration of the evolution of the species. Those three give the "saddle point" in aposematism.
These two "curvatures" exist and complement each other at a common extreme point, the maximum of the first and the minimum of the second, the greatest reproduction when there is the least danger, both coinciding with the bright colors of the body. Thus, the given species becomes an "isolated point." In mathematical analysis, we call the point z ∈ Z an isolated point when there is an environment around z in which there are no others except z points from Z. In biology, a "point z" would be one such species that has no very close relatives.
Aposematism is more widespread in insects than in vertebrates because, looking at it this way, it fits better with the minimax theorem. Most insects live less than a year because they are cold-blooded and do not survive the winter. On the other hand, the fossil record of insects goes back about 400 million years to the Lower Devonian. Therefore, the ratio of the life span of an individual and a species in insects (it tends to zero) is favorable for "completeness." However, further search for such confirmations is naive for mathematics; its positions are not provable, nor are they disputable empirically.
When there are no assumed spaces X and Y of the saddle surface or completeness conditions, then there is no such focus and isolated saddle point. The species has a "streaked series" of transitional evolutionary forms over a relatively short time interval. Developmental forms of such are more often visible.
Singularity
Question: As you tell me this about aposemia, I wonder if saddle points and isolated points are related terms?
Answer: Yes, they are very related. In mathematics, we distinguish them just enough to allow for different variations of related topics (see image link on the left). Simply put, the way we find saddle surfaces, we find isolated points, but vice versa only with small variations.
Namely, if z is a saddle point from the space Z = X × Y (Saddle), then it belongs to X and there is a certain environment where it has the highest value. The same point in the neighborhood of Y has the smallest value. Such z ∈ Z is a saddle point of space Z, and there are no others like it around it. Otherwise said, no matter how small the interval is, around the saddle point, there are points that have both a smaller and a larger value than it. But, conversely, a singular point is a local minimum, maximum, or inflection, and only in the third case is it a saddle point. It is singular and "phallic" when it carries the smallest value but not the largest, or vice versa, when it carries the largest without the smallest, like the two in the next picture.
I hope we distinguish "point position" from "point value." For example, in the place of this-and-this, there is a hill of the height of that-and-that. The advantage of this theorizing is, for those familiar with mathematics, the incredible ease of applying its theorems wherever we have the appropriate assumptions. The first assumption is the Banach space. It is a complete vector space provided with a norm, also called a metric space, but then again contains the limit values of its sequences.
In order to separate ourselves from the trivial view of space as physical or geometric, we interpret it with the metric itself (Distances II). For example, the Hamming metric represents the "distance" by the amount of dissimilarity between two given "places." It formally mimics distances as well as all other recognized measures, but it can also tell us about the distances between, for example, different biological species during evolution. This resolves only one of the three "saddle point" prerequisites: z ∈ Z.
These are the prerequisites of the minimax theorem. The remaining two are that the subspace X in some neighborhoods of z has the largest value, and the subspace Y in some neighborhoods of z has the smallest value, so Z = X × Y. In the following, we need to "just" find opposite tendencies, and there are unique, singular phenomena such as saddle points. The following are two (actually several) examples.
1. Example. It is known that today, in order to increase freedoms, liberalism advocates various needs for "equal representation." We first see this in gender equality, but working consistently, such a principle should be extended to, for example, seeking in work (life) on land an equal representation of fish and vice versa, an equal presence of terrestrial animals in the sea. It turns out that the right to choose stifles diversity.
The concept of freedom can be brought to absurdity like this because more will not always mean better, as it does in other cases of saddle surfaces. Freedom is like that, but there are also many other phenomena that we find in dualism, subject to the "authority" of two opposing principles: "good" against "evil." These are also situations of quantization of information (Packages), now proven differently. □
2. Example. To last as a species, living things need offspring and survival. The first is in the conditions of a local maximum, the space X, while the second is in the conditions of a local minimum, the space Y. The Minimax theorem states that copies of copies in a long series of generations give individuals individual characteristics. Similarly, the "character" of society, which we call culture, is built. Thus, the problem of building a supranational society will arise, with the relatively small weight of the cultures themselves, as opposed to those then increasingly strong general forces of individuality.
Analogous cellular structures, separating singularities, are also found in non-living nature. Molecules, atoms, and even smaller particle waves satisfy the same prerequisites of the minimax theorem when we set things consistently above. They should reproduce (last) and survive (cling to life), i.e., be "born" as new news from the old, because the old "news" is no longer news and due to the law of preservation of information. □
The subspace of the saddle surface, of course, does not have to be a saddle surface, so by abstraction (separation), we can get everything from something else in addition to what is stated here. The job of interpreting the minimax theorem is more extensive than it might seem at first glance.
Standoff
Question: Why do you think that a balanced game of "reciprocity" (first league) between two opponents leads to a draw, rather than a balanced game from other lower leagues?
Answer: Let's say because of the simulations I've been trying, which then go on and on. Living beings (they are the only ones who play) then get tired and define their territory or cooperate. The side that gives in will not bear the reached level of vitality (which is certainly exhausting) and becomes defeated and subordinate. The ideas of simulation and observation in reality, on the other hand, come from theory.
First of all, not all games are equally complex. For example, the tic-tac-toe in the picture (Tic-Tac-Toe), starting with a "x" always wins. The opponent "o" cannot block all the lines that are then opened and lose, and the third move "xxx" in the bottom row is a winner. There are no draws.
A chess game is "open" because the players don't hide their cards, but it seems to be "closed" because the moves have too many possibilities. It is on the border of easy games and the validity of John von Neumann's minimax theorem because we cannot know for sure the best move of the position and, therefore, we cannot make the move that leaves the opponent with no best move, which is valid for the recipe of not losing the game. Although we don't know all possible chess moves, we can still theorize.
The chess game is special as it is one-dimensional, with the move of the first and then the second mover, then the first and the second, and so on, waiting for the opponent's move, which significantly differs from multidimensional games such as war, politics, or economics, in which multiple moves by both sides take place at the same time. to which the perception information in the previous analysis refers (Senses).
Let's imagine that, as recommended by John von Neumann, we play chess in such a way, move by move, that we do not leave the opponent with his strongest answers. He cannot defeat us. If he also played his best with the remaining possibilities, he wouldn't lose either. So the game goes to a draw.
In the reality of the chess game, we do not know all the possible moves, but we will rely on estimates. The best strategy of "reciprocity" in the case of a player of two truly equal estimators, further proportional, timely, and equally uncertain responses, will limit the best responses of the opponent, and we will have a draw game by the previous logic. However, by playing as in a second league (manipulators), when walking between proportional resistance and, in principle, the least effect (dead nature), anything can happen, and there will be victories and defeats for both sides equally and with equal opponents.
Supreme vitality, let's imagine again, would be knowing all possible moves or all their chances, so by drawing them in such a way that the opponent will not have the best left, he cannot lose the game. If the other player plays the same way, he does not lose, and the game cannot go beyond a draw. Complexities such as interstate, political, market, and war are implied here.
More specifically (Senses), the most vital responses are "saddle points." They are such isolated and unique (Singularity), therefore "unsullied" decisions, for which it is not true that "would and could not" be valid for victory. In that extreme state of vitality, which is the other extreme of dead nature (the obligatory principle of least action), there is the certainty of knowing everything possible. Of course, we consider this extreme only theoretically, and with it, we point to possibly the same conclusion as in the case of John von Neumann's situation.
That's why I think that the even game of "reciprocity," players of the first league, leads to a draw more often than the competition of even competitors of lower leagues. Ultimately, due to the structure of the form, which is similar to the beginning and end of the time of the cosmos, or the smallest and largest physical world — which is progressing towards certainty.
Discrete
Question: What do you mean by "finite divisibility"?
Answer: The term "finitely divisible" is a description of a discrete set. These are, say, subsets of the natural numbers ℝ = {1, 2, 3,...} which are not bounded from above but are bounded from below. Discrete integers ℤ = {0, ±1, ±2,...}. However, they are also rational numbers.
because they are countable, i.e., we can arrange them in one (infinite) sequence, but then this expression is not happy.
More precisely, the set S ⊂ X is discrete when every point x ∈ S has a neighborhood U such that S ∩ U = {x}. Then the points S of the set are isolated, and we have the singularity of every point of the given (sub)space. In short, a set is discrete if it has a discrete topology if every subset of it is open.
This specification has interesting consequences for perceptual information, one of which concerns its smallest packages. Namely, according to premise 2.5 of Banach spaces, "when the additive mapping is continuous at a point in the space, it is continuous in the entire space," and "additive mapping" is a process in which the sum of the originals is equal to the sum of the copies, which is guaranteed by the law of conservation. This further means that the world information perception is cellular, so to speak for the world of singular phenomena.
On the other hand, the discreteness of the coupled information of the subject and the object does not oblige the individual information to be discrete. It is confusing at first glance, like some statement that the set of rational numbers is "everywhere dense" (no matter how small the interval given, there will be rational numbers). Overall, the term "finite divisibility" is a simplistic usage intended for a kind of discreteness for a set like the natural numbers.
Summation
Question: What kind of "coupled information" are you talking about?
Answer: Linked information is the basic concept of information perceptions, that is, the theory of information that I am working on. As with any new phenomenon, the brain first resists it and then perhaps realizes its possible accuracy, simplicity, and beauty. Therefore, I will underline the explanation of information coupling again, with a few banally known examples.
1. Feeding evil we get evil (+ ‧ - = -); going evil to good (- ‧ + = -) we get evil. Answering with good on goodwill (+ ‧ + = +) returns good to us, and we must return evil with evil (- ‧ - = +); just like that, otherwise, we lose. This is not a matter of choice but an obligation — on the recommendation of the part of this theory that concerns game theory (Reciprocity).
where the strings a = (a1, a2,..., an) and b = (b1, b 2,..., bn) represent the state of subjects (or objects) of perception. The higher this number Q, the greater their mutual perception, and the stronger the coupling of information. In addition, the corresponding components ak and bk, respectively for k = 1, 2,..., n, are related information of communication participants (couplers).
2. The picture on the top right with the attached link explains the work of the force on the road:
W = F⋅r = |F||r|cos∠(F,r) = Fxrx + Fyry + Fzrz,
where F = (Fx, Fy, F z) and r = (rx, ry, rz) force vectors and the direction in which it should act. When the angle, θ = ∠(F,r), between the given vectors, the smaller cosine of the one is larger, -1 ≤ cos θ ≤ 1. Then the larger scalar productW.
When the force and path components are both positive (Fk > 0 and rk > 0), or both negative (Fk < 0, rk < 0), their product is positive (Fkrk > 0), and that sum is a contribution to a larger sum (W). However, when the first of the factors is positive and the second is negative (Fk > 0 and rk < 0), or the first is negative while the second is positive (Fk < 0 and rk > 0), their product becomes negative (Fkrk < 0), and such addition reduces the total sum (W var>).
3. Measurement in quantum mechanics, the interaction of a quantum state, a vector written in Dirac brackets |ψ⟩, in a process to obtain the quantity A, which can be the position of the particle, energy, momentum, and the like, is generally observable. Eigenequation of that process is Â|uk⟩ = λk|uk⟩, where λk eigenvalues that correspond exactly to the eigenstates |uk⟩ of the given state of the measurement system |ψ⟩.
Instead of one outcome, as it would be in classical mechanics, here one of n ∈ ℕ possibilities of observables, indexed k = 1, 2, 3,..., n. Although we do not know the exact outcome, we do know the probabilities of all possible Pr(λk) = |⟨uk|ψ ⟩|². This is the difference between the micro and macro worlds of physics, in which quantum works only with chances; it is called superpositions of states, which in mathematics would be called probability distributions.
4. If we put ak = ⟨uk|ψ⟩ it will be Pr(λk) = |ak|² the probability of k-th observable in a given state |ψ⟩, in a given measurement, is such that:
|ψ⟩ = a1|u1⟩ + a2|u2⟩ + ... + an|un⟩,
where |uk⟩ are the unit vectors of the given measurement settings. In the second setting of the same, we would have a measurement state of the unit vectors |vk⟩, such that:
|φ⟩ = b1|v1⟩ + b2|v2⟩ + ... + bn|vn⟩.
In particular, with two measurement settings, we also find the well-known "spooky action at a distance" (Spooky), from whose later interpretation a new field of quantum mechanics arose, quantum entanglement. The interaction of these measurement device states is the scalar product ⟨φ|ψ⟩, where ⟨φ| = |φ⟩† label adjoint of the given vector (conjugate transposed).
5. Only under equal measurement conditions would it be ⟨φ| = ⟨ψ|, when each of the coefficients bk = ak, so then:
⟨φ|ψ⟩ = ⟨ψ|ψ⟩ = a1*a1 + a2*a2 + ... + an*an = 1,
because this is the sum of the distribution probabilities, and ak*ak = | ak|² = Pr(λk). In other cases of measurement settings, we will have "only" information of perception, then of a local situation.
However, the product of the different states is the sum of the products:
The inequality on the right is known Cauchy–Schwarz which now tells us that the sum of the products is less than or equal to one, which means that ⟨φ|ψ⟩ can formally be some coupling probability of two subjects of a given measurement.
6. When we have several superpositions at once, then the total state can be understood as a combination of many quantum states, just as a macro-body consists of small particles. The sum of vectors is again a vector, a sum of superpositions, or a sum of probability distributions, whose components in an ever-increasing set converge to information (Exponential II). There is also a new quality.
where the first vector is not conjugated, because the imaginary is lost in the macro world, it becomes negligible. However, the interpretation also extends to the coupling of the two participants and their perception of information. The larger this scalar S is, the greater the communication between the two states, and the power of mutual perception increases with it; that is the level of play in the case of vital subjects.
thus explaining the results of (1) multiplying the signs. When the factors in the sum are both "good" (+⋅+) or both "evil" (-⋅-), the sum is positive; otherwise, it is negative.
The higher the sum of S, the higher the level of the game, due to its greater vitality. This is when we return "good" to "good" or "evil" to "evil." Also, the strength of the game increases by joining relatives. On the contrary, different signs (good on evil or evil on good) lower the level of the game, leading to the loss of the game. Separating the components into such (various) ones increases the power of the game. In short, joining together makes us stronger, as does separating the "wheat from the chaff" with consistent answers (Win Lose), and the rest weakens us.
Determinant
Question: You don't give an example with determinants in your answer?
Answer: In the previous answer (Summation) many things can be listed because, with concreteness, there are (correct) theories with countless applications alongside practices. The theory is always "everyone's," while practice is "someone's" (Abstractions). However, I agree that determinants are useful and can be a difficult story, that's for sure. Let's try to understand them.
Commutator is a determinant of order two, area and information, [Ã, B̃] = A xBy - BxAy. It is a form of the Heisenberg uncertainty relations of the position and momentum operators [x̃, p̃], or [t̃, Ẽ] of time with energy, because both are physical actions and are of the order of magnitude of Planck's constant. If the first operator is larger, the second is smaller, and vice versa, so that the commutator is always of the same area. Operators have intensity, norm (2.4.IV), and the mentioned ones are also linear.
The determinant of the third order is the volume V[u, v, w] of three 3-dim vectors:
so the volume is the sum of commutators, the sum of actions, or, let's say, formally, the sum of products, i.e., the information of perception.
The determinant of the fourth order is the 4-dim volume spanned by the vectors from its four columns, and this is how we interpret the general determinantsn -th order in the tensor calculus. In addition to everything previously said, in linear algebra, the determinant is specially treated as a product of the eigenvalues of the operator it represents. And then, again, the determinant of the nth order is:
V = a1b1 + a2b2 + ... + anbn
where for the series (ak) of the components of the subject, we can take any row of the given determinant when the components of the object (bk) are corresponding cofactors.
If two different development lines were identical or proportional, the determinant would be zero. That famous theorem of algebra now says that there is no information (V = 0), there are no such identical subjects (ak = λa'k, index order k = 1, 2,..., n), which can be underlined under the long-known Riesz's theorem (Functional), and now the principled uniqueness of information. Otherwise, each feature of the determinants will give some interpretation of the world of information.
Topology
Question: I'm a little unclear about that abstract concept of "space", can you clarify it for me?
Answer: The simplest "space" in mathematics is "topology". It is also the most abstract. Topologically space is without measurement of distance, as well as without direction. Simply put, it is a set to which the unions and intersections of its subsets belong.
For example, a topology T is a set X whose elements are all sets: ∅, {1}, {2}, {3}, {1,2}, {2,3}, {3,1}, {1,2,3}. An empty set is an element of every set; so is the first of the above, followed by the union of letters or numbers 1, 2, and 3.
We can arrange these elements, or subsets, as in the picture on the right, which is not important for the topology itself, but here it is on a scaffold organized for easier understanding of a special (sub)type of topological spaces, which we call "metric spaces." Given a discrete topology, its subtype is also a discrete metric space. Nevertheless, they are topological spaces equipped with "metrics", i.e., with the expression of distances between elements, then called points, which simulates measurement in physical or geometric space.
In the figure, the distance d(x, y) between the points x and y is simply the number of different elements of those subsets. For example, d(∅, {2}) = 1, d({1}, {1,3}) = 1, but d ({2}, {1,3}) = 3. So, we have: d(x, x) = 0, if x ≠ y then d(x, y) > 0, also d(x, y) = d(y, x) and d (x, y) ≤ d(x, z) + d(z, y). And, for the example in the picture, the axioms of metric spaces apply. It is a metric space, a set of points related to a given distance and subsumed under a given topological space.
A step further in detailing would be the norm of the point ∥x∥ = d(∅, x), then the vector x. In this example, say ∥∅∥ = 0, ∥{3}∥ = 1, ∥{1,2}∥ = 2, and ∥{1,2,3}∥ = 3, and, in general, the norm of a set is the number of elements of that set. Analogously, we define the discrete vector space of this example. As we can see, the more detailed we are, the farther we are from the starting topology, which is the roof over what we carry out.
We can associate probabilities with the points of these spaces at each phase of extraction. Here is an example of such an assignment:
The sum of all of them is 1, so when we look at these points as independent random events, then we have one probability distribution here. The starting set X in this way became a probability space. The mean (Shannon) information of the entire set, distribution X, is:
serves to convert logarithms from one base to another, where logbb = 1.
Step by step, the initial very broad notion of topological space takes on more and more concrete forms and becomes less and less a theory and more and more something special. As we reach the end with detailing, the model becomes so practical that it is unique that it is reduced to physical reality. It is noticeable that there are many ways of concretizing or reducing endlessly applicable theories to ultimate reality. This follows not only from the very logic of space reduction but also from the multiplicity of the ultimate goal.
The multiplicity is in the selection of initial, say "numbers" 1, 2,..., n → ∞, which can be any objects or concepts, with the assignment of a "distance", which dictates the definition of a "norm", but, again, with the freedom to choose the probabilities p1, p2,..., pn elements, while their unions do not even have to be distributions. The resulting "space" can be the physical space-time of an event or a piece of change, and not just a concrete state in the process.
The distances of such "parts of change" are getting bigger as the jump from one to the other is less likely, so this mathematical formalism, which has been developed for centuries, fits quite well into the new (my) theory of information. And I mentioned some other properties of topologies in the metric attachment (2.11. Statement). The Dimension of the point and the finite set of points is zero.
Equilibrium II
Question: What is the "equilibrium point" of the payoff matrix?
Answer: A zero-sum matrix game results in a saddle point if the smallest values are chosen from the row and the largest from the column. In the picture on the left are the smallest species values (down, right): (1,4):3, (2,4):2, (3,4):6, (4,4):8, and (5,1):1. And the highest value is 8 in column 4. I will explain why game developers are crazy about this and how it is achieved later.
1. Here is one of the famous easier examples. In the zero-sum game, in the next picture on the right, Ross has to choose one of three moves, type A, B, or C, and Colin one of two, column A or B. They did not know what the opponent played until the payoff. For example, if Rosa chose row A and Colin chose column A, Rosa gets 2 points and Colin loses them (he "gets" -2). However, if the moves are (A, B), Colin Rossi takes 3 points. Knowing the payoff matrix, Rosa will not choose B because she gains nothing or loses 2. Colin, on the other hand, can lose 10 points if he chooses B.
Let's note that for the matrix of the zero-sum game, it is enough to know only the first payoff numbers, these red ones, Rosе. So the previous one, from the left, is a 5×5 matrix, where the first player always wins, but with different amounts.
By choosing the fourth row, his gain is maximum (8 points), because a cunning opponent will always choose the fourth column. The opponent chooses from the columns of the highest value to decide on the suit with the smallest of them: (3,1):18, (3,2):24, (5,3):22, (4,4): 8 and (5,5):15, to reduce the other's profit. The point (4,4) of the given payoff matrix is therefore "equilibrium," because the amount 8 is common to both players, for the rows (min-max) and columns (max-min).
2. Not every matrix has this equilibrium (the saddle point). Let us write the types (8, 1, 9), (7, 2, 6), and (3, 4, 5) one below the other in the form of a matrix of type 3×2 without a saddle point. The probability that a randomly generated matrix of type m×n has a saddle point is m!n!/(m + n - 1)!. Thus, a randomly formed square matrix of the second order (type 2×2) with probability 2/3 will have an equilibrium point and a square matrix of the third order with probability 0.3.
3. With the link of the first image on the left, you will see a Java program for finding the saddle point of a given matrix. The following code works similarly in Python 3:
Theme:
Von Neumann, the founder of game theory (1928), is quoted as saying, "As far as I can see, there could be no theory of games... out of the Minimax Theorem... I thought there was nothing worth publishing until that theorem was proved." This text was completed in May 2024. | 677.169 | 1 |
Hint: First we have to know the order of rotation is the number of times the figure coincides with itself as it rotates through \[{360^o}\]. We have to know the angle of rotation of a line segment. Then using the definition of order of rotation find the order of rotation of a line segment.
Complete step by step solution: Rotation is the process or act of turning or circling around something. An example of rotation is the earth's orbit around the sun. One rotation around a circle is equal to \[{360^o}\]. An angle measured in degrees should always include the degree symbol \[^o\] or the word "degrees" after the number. For example, \[{180^o} = 180\] degrees. A complete rotation about the centre point is equal to \[{360^o}\] or \[2\pi \] radians. The rule for a rotation by \[{180^o}\] about the origin is \[\left( {x,y} \right) \to \left( { - x, - y} \right)\] . In geometry, a line segment is a part of a line that is bounded by two distinct endpoints, and contains every point on the line that is between its endpoints. The angle of rotational symmetry or angle of rotation is the smallest angle for which the figure can be rotated to coincide with itself. Example: The angle of rotation is \[{60^o}\] and the order of the rotational symmetry is \[6\]. The angle of rotation of a line segment is \[{180^o}\]. So, order of rotation of a line segment is \[\dfrac{{{{360}^o}}}{{{{180}^o}}} = 2\]. Hence, the Option (D) is correct. So, the correct answer is "Option D".
Note: Note that the order of rotational symmetry of a circle is how many times a circle fits onto itself during a full rotation of \[360\] degrees. A circle has an infinite 'order of rotational symmetry'. In simplistic terms, a circle will always fit into its original outline, regardless of how many times it is rotated. | 677.169 | 1 |
geometry triangle congruence proofs cpctc worksheet answersGeometry Triangle Proofs Worksheet Answers – Triangles are one of the most basic shapes found in geometry. Understanding triangles is crucial to developing more advanced geometric ideas. In this blog we will explore the various kinds of triangles triangular angles, the best way to determine the perimeter and area of a triangle, and present details of the various. Types of Triangles There are three types of triangles: equilateral isoscelesand scalene. Equilateral triangles have three equally sides and threeTriangle Congruence Cpctc Worksheet Answers – Triangles are among the most fundamental shapes of geometry. Understanding triangles is vital to learning more advanced geometric concepts. In this blog this post, we'll go over the various kinds of triangles triangular angles, the best way to determine the length and width of a triangle, and offer illustrations of all. Types of Triangles There are three types to triangles: the equilateral, isoscelesand scalene. Equilateral triangles have three equally sides as well … Read more | 677.169 | 1 |
What you'll learn
2. Whether the Heron's formula can always be used to find the area of right angled triangle?
3. How can we use the Heron's formula to calculate the area of Scalene Triangle?
4. How Heron's formula is used to find the area of Equilateral and Isosceles triangles?
Requirements
Basic understanding of algebra, geometry, and trigonometry.
Description
1. Who is given the credit of Heron's formula?
2. In questions, where height and base is given, we can find the area of triangle using a simple formula. What is that formula?
3. In cases when the height of triangle is difficult to calculate or is not given and all the three sides of triangle are given, which formula can be used to find the area of triangle?
4. What is Heron's formula?
5. What is the meaning of semi-perimeter? How it is arrived at?
6. What is the meaning of perimeter? How it is calculated?
7. Write the names of some most common triangles.
8. What is the meaning of right-angled triangle?
9. In a right angled triangle, what is the name given to the side which is opposite to the right angle?
10. In a right angled triangle, one side is longest. What is the name given to other two sides of the triangle?
11. Whether we can use Heron's formula to find the area of right angled triangle?
12. In case of right angled triangle where only base and height are given, can we use Heron's formula to find the area of triangle?
13. What is the meaning of scalene triangle?
14. Can we use Heron's formula to calculate the area of scalene triangle? Whether the length of two sides of scalene triangle will serve the purpose to find the area of triangle with the help of Heron's formula?
15. What is meant by Equilateral Triangle?
16. If in an equilateral triangle, only the length of one side is given. Can we still use the Heron's formula to calculate the area of triangle?
17. What is the meaning of Isosceles Triangle?
18. How many equal sides are there in an isosceles triangle?
19. What is the name given to the unequal side of isosceles triangle?
20. Can we use the Heron's formula to calculate the area of isosceles triangle if the length of equal and unequal sides is given?
Who this course is for:
The students who are interested to learn about Heron's formula | 677.169 | 1 |
What are the symmetrical figures?
Something is symmetrical when it is the same on both sides. A shape has symmetry if a central dividing line (a mirror line) can be drawn on it, to show that both sides of the shape are exactly the same.
How do you know if a shape is symmetrical?
Symmetry. A 2D shape is symmetrical if a line can be drawn through it and either side is a reflection of the other. The line is called a line of symmetry. This is sometimes called a 'mirror line' or 'mirror symmetry', because if you put a mirror on the line, the reflection would show the whole shape.
What do all figures with line symmetry have?
Here, for instance, if we fold the figure cut out exactly at the center vertically, its halves will be congruent. The line of the fold is the line of symmetry. Figures or shapes that have exact resemblance to its other part, when divided into two or more equal parts are called symmetrical.
What is the purpose of symmetry?
Symmetry is a fundamental part of geometry, nature, and shapes. It creates patterns that help us organize our world conceptually. We see symmetry every day but often don't realize it. People use concepts of symmetry, including translations, rotations, reflections, and tessellations as part of their careers.
Which shape has 2 lines of symmetry?
Rectangle
Rectangle. A rectangle has two lines of symmetry. It has rotational symmetry of order two.
What are the 4 types of symmetry?
The four main types of this symmetry are translation, rotation, reflection, and glide reflection.
Which shape has only one line of symmetry?
trapeziums
A give example?
Symmetry is an attribute where something is the same on both sides of an axis. An example of symmetry is a circle that is the same on both sides if you fold it along its diameter. Similarity of form or arrangement on either side of a dividing line or plane.
Which triangle has 2 lines of symmetry?
isosceles triangle
Any isosceles triangle has line symmetry. The dashed lines represent lines of symmetry, and each shape is said to be symmetrical about this line. A shape can have more than one line of symmetry. Thus a rectangle has two lines of symmetry, an equilateral triangle has three lines of symmetry, and a square has four.
What shape has only one line of symmetry?
Trapezium. A in math example?
In Mathematics, symmetry means that one shape is identical to the other shape when it is moved, rotated, or flipped….Infinite Lines of Symmetry.
Number of lines of symmetry
Examples of figures
Exactly one line of symmetry
Isosceles triangle
Exactly two lines of symmetry
Rectangle
What is natural symmetry?
The two main types of symmetry are reflective and rotational. Reflective, or line, symmetry means that one half of an image is the mirror image of the other half (think of a butterfly's wings). Human beings, insects, and mammals all show bilateral symmetry. Man is naturally attracted to symmetry.
Which is the best definition of a symmetrical figure?
1 Answers. A symmetrical figure is one that has a point, line, or plane of symmetry. The figure looks the same whether you go in one direction from the point/line/plane of symmetry, or whether you go in the opposite direction.
How to make a symmetric shape or pattern?
Making symmetric figures and patterns – Example 1 Step 1: Take a piece of paper and fold it. Step 2: Cut out any figure which starts and ends on the fold line such as the one shown below. Step 3: Unfold the figure. You have a symmetric figure where the fold line is the line of symmetry.
What is the dotted line in a symmetric figure called?
The dotted line in each of the symmetric figures above that divides the figure into two equal halves is called the line of symmetry. In the figures above, there are no lines of symmetry that divide each of the figures into two equal halves.
How many lines of symmetry does a square have?
has 3 Lines of Symmetry A Square (4 sides) has 4 Lines of Symmetry A Regular Pentagon (5 sides | 677.169 | 1 |
Trigonometry Word Problems
Trigonometry word problems involve applying the principles of trigonometric ratios—sine, cosine, and tangent—to solve real-world scenarios such as determining heights, distances, and angles. Mastery of these problems requires a thorough understanding of right-angle triangles and the Pythagorean theorem. By practising, students can enhance their problem-solving skills and effectively use trigonometry in practical situations.
Introduction to Trigonometry Word Problems
Trigonometry word problems apply the principles of trigonometry to real-life scenarios. Understanding these problems helps you see the practical uses of mathematics, from measuring heights to navigating maps.
The Importance of Trigonometry Word Problems
Understanding trigonometry word problems allows you to apply trigonometric concepts to practical situations. This not only enhances your problem-solving skills but also shows how maths is used in various fields such as engineering, physics, and navigation.
Trigonometry: A branch of mathematics that studies relationships between side lengths and angles of triangles.
Key Trigonometric Functions
In trigonometry, three main functions are used to solve problems involving right-angled triangles Given a right-angled triangle where the opposite side is 3 units, the adjacent side is 4 units, and the hypotenuse is 5 units:
sin(θ) = 3/5
cos(θ) = 4/5
tan(θ) = 3/4
Steps to Solving Trigonometry Word Problems
To tackle trigonometry word problems effectively, follow these steps:
Understand the problem: Read the problem statement carefully to determine what is being asked.
Identify the right triangle: Sketch the scenario and highlight the right-angled triangle.
Choose the correct function: Decide whether to use sine, cosine, or tangent based on the given information.
Set up the equation: Write the equation using the chosen trigonometric function and solve for the unknown.
Check your answer: Confirm if the solution is reasonable and correctly answers the problem.
Sometimes, trigonometry word problems may involve more complex scenarios, such as non-right triangles or 3D contexts. In such cases, you might need to use advanced trigonometric principles like the law of sines or the law of cosines. These laws generalise the basic trigonometric functions and can handle various triangle types.
Example Problem
Example: A tree casts a shadow of 10 metres, and the angle of elevation of the sun is 30 degrees. Find the height of the tree.Solution:
Let the height of the tree be h.
tan(30) = h / 10
Since tan(30) = 1/√3, then 1/√3 = h / 10.
h = 10 / √3 ≈ 5.77 metres.
Common Mistakes to Avoid
When solving trigonometry word problems, some common mistakes include:
Not identifying the right function: Ensure you choose the correct trigonometric function (sine, cosine, or tangent) based on the given information.
Forgetting to use the correct units: Always keep track of the units and convert them if necessary.
Not checking your work: Always recheck your calculations and ensure your answer makes sense in the context of the problem.
Not using a calculator properly: Ensure your calculator is set to the correct mode (degrees or radians) based on the angle measures used in the problem.
Trigonometry Word Problems Examples
Simple Trigonometry Word Problems Examples
Simple trigonometry word problems often involve right-angled triangles and basic trigonometric functions like sine, cosine, and tangent. Let's take a look at an example.
Example: A ladder is leaning against a wall, forming a right-angled triangle with the ground. If the ladder is 5 metres long and the angle between the ladder and the ground is 60 degrees, find the distance from the base of the ladder to the wall.Solution:
Let the distance from the base of the ladder to the wall be x.
Using the cosine function, cos(60) = x / 5
Since cos(60) = 1/2, then 1/2 = x / 5.
So, x = 5 / 2 = 2.5 metres.
Always start by identifying what is given and what you need to find. This will help you choose the right trigonometric function.
Intermediate Trigonometry Word Problems Examples
Intermediate problems may require you to use multiple trigonometric functions and incorporate more complex geometric shapes. These problems will often involve finding unknown angles or distances.
Example: You are looking at the top of a building from a distance of 50 metres. If the angle of elevation is 45 degrees, find the height of the building.Solution:
Let the height of the building be h.
Using the tangent function, tan(45) = h / 50
Since tan(45) = 1, then 1 = h / 50.
So, h = 50 metres.
For more complex shapes, you might need to break the problem into smaller right-angled triangles. This approach helps simplify calculations and ensures that you use the correct trigonometric function.
Advanced Trigonometry Word Problems Examples
Advanced problems often involve non-right triangles or three-dimensional geometries. These problems typically require the use of the law of sines or the law of cosines to find unknown measurements.
Example: In triangle ABC, you know side a = 8, side b = 6, and angle C = 60 degrees. Find side c.Solution:
Using the Law of Cosines, \(c^2 = a^2 + b^2 - 2ab \cos C\)
\(c^2 = 8^2 + 6^2 - 2 \cdot 8 \cdot 6 \cdot \cos(60)\)
\(c^2 = 64 + 36 - 96 \cdot (1/2) \)
\(c^2 = 64 + 36 - 48 \)
\(c^2 = 52 \)
\(c = \sqrt{52} \)
\(c \approx 7.21 \) units
Applications of Trigonometry Word Problems
Trigonometry is not just an abstract mathematical concept; it has numerous practical applications in everyday life, science, and engineering. Understanding how to solve trigonometry word problems can help you navigate through various real-world scenarios effectively.
Everyday Applications of Trigonometry Word Problems
Trigonometry is applied in many everyday activities, whether you realise it or not. Here are a few areas where trigonometric calculations are commonly used:
Navigating Maps: You can use trigonometry to calculate distances and angles between various points on a map.
Measuring Heights: Determining the height of a tree, building, or any other tall object without climbing it can be done using trigonometric functions.
Photography: Calculating the ideal angle and distance to take a picture often involves trigonometry.
Example: You want to measure the height of a flagpole. Standing 20 metres away, you measure the angle of elevation to the top of the pole to be 30 degrees. Find the height of the flagpole.Solution:
Applications of Trigonometry Word Problems in Science
In science, trigonometry plays a crucial role in fields such as physics and astronomy. By understanding trigonometric word problems, scientists can make accurate measurements and predictions. Here are some specific applications:
Example: Astronomers determined the angle between Earth and a distant star is 0.5 degrees, and the distance from Earth to the star is 4 light-years. Find the distance between two observation points on Earth if the angle at the distant star is measured to be 0.001 degrees.Solution:
In more complex problems, astronomers and physicists may use spherical trigonometry, which is an extension of trigonometry that deals with spheres rather than planes. This is particularly useful for calculations involving planetary orbits and celestial navigation, where the curvature of space becomes significant.
Applications of Trigonometry Word Problems in Engineering
Engineering makes extensive use of trigonometry, especially in areas such as civil, mechanical and electrical engineering. By applying trigonometric word problems, engineers can design and build structures, machinery, and systems efficiently.
Law of Cosines: \( c^2 = a^2 + b^2 - 2ab \cos C \)
Structural Engineering: Calculating the forces in various parts of a structure, such as beams and trusses, often involves trigonometric principles.
Understanding the law of cosines is crucial for solving problems involving non-right triangles.
Trigonometric Ratios in Word Problems
Trigonometric ratios are pivotal in solving various mathematical problems. They allow you to find unknown angles or distances in different types of triangles. Applying these ratios in word problems helps you understand their real-world functionality.
Using Sine, Cosine, and Tangent in Word Problems
Sine, cosine, and tangent are the primary trigonometric functions used in solving right triangle problems. They help you relate the angles to the lengths of the sides in a triangle. Here's a quick refresher on these functions You are standing 30 metres away from a tree and the angle of elevation to the top of the tree is 45 degrees. Find the height of the tree.Solution:
Using the tangent function: \( \tan(45) = \frac{\text{height}}{30} \)
Since \( \tan(45) = 1 \), then \( 1 = \frac{\text{height}}{30} \)
So, the height is 30 metres.
Always ensure your calculator is in the correct mode (degrees or radians) based on the given angle.
Right Triangle Trigonometry Word Problems
Right triangle trigonometry involves solving problems where one angle is 90 degrees. The three main trigonometric ratios — sine, cosine, and tangent — are especially useful in these problems.
Identify the right angle: Determine which angle is 90 degrees to simplify your calculations.
Choose the appropriate ratio: Decide whether to use sine, cosine, or tangent based on the sides and angles you are dealing with.
Example: A 20-foot ladder is leaning against a wall, making a 60-degree angle with the ground. Find how high up the wall the ladder reaches.Solution:
In more complex problems involving right triangles, it might be helpful to use the Pythagorean theorem in conjunction with trigonometric ratios. The Pythagorean theorem states that for a right triangle, the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\) and \(b\)): \( c^2 = a^2 + b^2 \). This theorem can be particularly useful when you have two sides and need to find the third.
Trigonometric Ratios in Non-right Triangle Word Problems
When dealing with non-right triangles, trigonometric ratios can still be utilised, but you may need to employ the law of sines or the law of cosines. These laws help solve for unknown sides or angles.The law of sines states: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] While the law of cosines is: \[ c^2 = a^2 + b^2 - 2ab \cos C \]
Example: In triangle ABC, you know that angle A = 30 degrees, angle B = 45 degrees, and side a = 10 units. Find the length of side b.Solution:
Frequently Asked Questions about Trigonometry Word Problems
How can I set up a trigonometry word problem involving angles of elevation and depression?
Identify the observer's position, the object, and the horizontal ground line. Use the given angle of elevation or depression to form a right-angled triangle. Label known sides or angles. Apply trigonometric ratios (sine, cosine, tangent) to solve for unknown measurements.
What are common mistakes to avoid in trigonometry word problems?
Common mistakes include misidentifying angles, confusing trigonometric ratios (sine, cosine, tangent), using the wrong units, and not considering which side is opposite, adjacent, or the hypotenuse. Always double-check calculations and ensure you understand the context of the problem.
What strategies can I use to solve trigonometry word problems more effectively?
Identify and label the known and unknown values. Draw a diagram to visualise the problem. Use the appropriate trigonometric functions and identities. Double-check calculations and ensure all angles are in the correct units.
How can I apply the Pythagorean theorem to trigonometry word problems?
The Pythagorean theorem can be applied to trigonometry word problems by relating the sides of a right-angled triangle. It states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This helps calculate distances, heights, and lengths when angles and one side length are known.
How can I identify which trigonometric function to use in a word problem?
Identify the trigonometric function by determining the given information and what you need to find; use sine for opposite over hypotenuse, cosine for adjacent over hypotenuse, and tangent for opposite over adjacent sides in right-angled triangles. Consider the angle involved and the sides' relationships.
Test your knowledge with multiple choice flashcards
Given triangle ABC with sides a = 8, b = 6, and angle C = 60 degrees, how do you find side c?
A. Monitoring energy systems without considering distances and angles.
B. Predicting the behaviour of light without any reference to angles.
C. Calculating distances between celestial bodies using the law of sines.
D. Measuring the Earth's curvature using algebraic formulae.
What is a common mistake made when solving trigonometry word problems | 677.169 | 1 |
Plane Geometry
From inside the book
Results 1-5 of 12
Page 162 ... tangents . A straight line which is tangent to each of two circles is called a common tangent of the circles . A B D ∞ C If the circles lie on the same side of the common tangent , it is called a common external tangent , as AB . If ...
Page 163 ... tangent . 5. Draw two circles which have no common tangent . 6. Prove that the common internal tangents of two ... INTERNALLY ∞ TANGENT EXTERNALLY Two circles are said to be tangent internally when they lie on the same side of the ...
Page 164 ... tangent to each other , the point of contact lies on the line of centers . ж B Hypothesis . Circles with centers A ... internally tangent . 4. Two circles are tangent externally at A , and also have a common tangent touching them ...
Page 165 ... tangent touching them at B and C , respectively , a circle with diameter BC ... internal tangent of two equal circles bisects the line of centers . 9. If ... tangents to a circle , two of which , AD and AE , are fixed , while the ...
Page 166 ... tangents is a circle . 18. What is the locus of the centers of all circles with a given radius r and tangent externally or internally to a given circle ? Prove the answer . 19. The adjoining figure is used much in different decorative ...
Popular passages
Page 130 - If two triangles have two sides of one equal respectively to two sides of the other, but the included angle of the first greater than the included angle of the second, then the third side of the first is greater than the third side of the second.
Page 72 - There are three important theorems in geometry stating the conditions under which two triangles are congruent: 1. Two triangles are congruent if two sides and the included angle of one are equal respectively to two sides and the included angle of the other.
Page 258 - S' denote the areas of two circles, R and R' their radii, and D and D' their diameters. Then, I . 5*1 = =»!. That is, the areas of two circles are to each other as the squares of their radii, or as the squares of their diameters.
Page 197 - In any triangle, the square of the side opposite an acute angle is equal to the sum of the squares of the other two sides, minus twice the product of one of these sides and the projection of the other side upon it. | 677.169 | 1 |
What is a polygon with 46 sides called?
What is a polygon with 46 sides called?
…enneagon. Examples: 46 sided polygon – Tetracontakaihexagon.
What is a 47 sided polygon called?
In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain. These segments are called its edges or sides, and the points where two of the edges meet are the polygon's vertices (singular: vertex) or corners.
What is a 45 sided polygon called?
Regular enneacontagon
Enneacontagon
Regular enneacontagon
Type
Regular polygon
Edges and vertices
90
Schläfli symbol
{90}, t{45}
Coxeter–Dynkin diagrams
What is a 43 sided polygon called?
For a polygon with sides where n is larger than 6 mathematicians usually call them n-gons. Thus a polygon with 43 sides I would call it a 43-gon.
What is a 50 sided shape called?
pentacontagon
Polygon names and miscellaneous properties
Name
Sides
tetracontagon (or tessaracontagon)
40
pentacontagon (or pentecontagon)
50
hexacontagon (or hexecontagon)
60
heptacontagon (or hebdomecontagon)
70
What is a polygon with 44 sides called?
They can have 4 sides, 44 sides, or even 444 sides. The names would be 4-gon, or quadrilateral, 44-gon, and 444-gon, respectively. An 11-sided shape can be called an 11-gon.
What is a 100000 sided polygon?
A megagon or 1 000 000-gon is a polygon with one million sides (mega-, from the Greek μέγας, meaning "great", being a unit prefix denoting a factor of one million). …
How do you name a polygon with 35 sides?
NAMING POLYGONS. Example: A 35-sided polygon is called a " triacontakaipentagon .". For numbers from 100 to 999, we need one more combining prefix and another rule. To "construct" the name, we start with the prefix for the hundreds digit taken from the "Ones Digit" table above, follow it by "hecta," then proceed as before.
What is a polygon of n sides called?
Yes, I understand, the best thing is to say: "polygon of n sides" when the polygon has more than 20 sides. Depending on the angle formed by the pairs of consecutive sides of the polygon, it is classified as concave or convex.
What would you call a polygon with 13 sides?
icosagon After about 6 sides mathematicians usually refer to these polygons as n-gons. So your 13 sided polygon I would call a 13-gon. Cheers,
What is a 32-sided polygon?
In simpler words, a 32-sided polygon. We can normally refer to it as a 32-gon. Looking at the pictures, you may have also noticed that with every added side, the polygon starts to look more and more like a circle. How interesting is it to know that it would never quite become a perfect circle? | 677.169 | 1 |
Not sure whether this belongs in High School or College math, but I suspect putting it on the college list would be flattering myself. Here is the general form of my question:
For two concentric circles intersected by a single chord, what function (if any) relates their respective diameters and the arcs subtended by the chord on each of their circumferences?
I am specifically interested in the case in which the arc on the smaller circle is 60 degrees. I am given to understand (and am trying to verify) that in this case, the arc of the larger circle is 144 degrees when the diameters of the two circles are related in mean/extreme ratio (i.e., the golden section).
It may be that the fault is my poor compass/straightedge skills; but when I have tried this (sometimes following Euclid but usually approximating it by taking two Fibonacci numbers for the radii), I cannot confirm the proportions. But even if I could, I would want to understand the proof, which means (I think), understanding the general case.
>For two concentric circles intersected by a single chord, >what function (if any) relates their respective diameters >and the arcs subtended by the chord on each of their >circumferences?
Let r and R be the radii of the two circles and h the distance from the center of the circles to the chord. There are isosceles triangles formed by the radii joining the center to the endpoints of the chords. These are divided by the altitude from the center to the chord into two right triangles. Let δ and Δ denote these right triangles for the small and the big circle. δ has angles 30°-60°-90°. Δ has angles 72°-18°-90°. From here,
h = r·sin(60°), h = R·sin(18°).
It follows that
R/r = sin(60°) / sin(18°).
You surely know the value of sin(60°). The value of sin(18°) can be found at
Thank you so much. I followed your lesson and worked it out, just as you said. And lo!
(1/2) x 1/((sqrt5 - 1)/4) = phi!
The piece I was missing was the line segment you called h, the distance from the center of the circles to the chord.
Thank you for explaining by appealing to the general trig functions. Although this was where I bottomed out in high school, when I worked at it, it was easy to see why the result obtains: since you start out with the general (sin theta) and then plug in a particular value after that, I can see that the result of phi obtains only when the two angles in question are 60 and 18 degrees. This was exactly the understanding I needed to get.
So I left off (sqrt3) from my reckoning, carelessly, and oddly enough got the answer I thought I was looking for. Now I think I understand why I should have been wary. I ought not really to expect to find phi in the ratio R/r. Rather, it would be (if anywhere) in (R-r)/r. This is because it is the radius R which is (per hypothesis) divided by extreme & mean ratio.
I'll try to apply the same principle you showed me and see if I get anywhere.
Thought I would write again to let you know I followed through and verified that phi is *not* the ratio of R/r nor of R/(R-r) in the problem I originally presented. So the arc of 144 degrees is *not* related to the arc of 60 degrees by a chord on two concentric circles, when the respective radii are related to by the golden ratio.
However, I decided to check and see what larger arc *was*, if i stipulated the 60-degree arc on the smaller circle and the radii being in that ratio.
Let there be two circles with a common center O. A chord subtends an arc of 60 degrees on the smaller circle, and a to-be-determined arc on the larger. Let h be the distance from O to this chord; it makes a right angle with the chord at point H, slightly inside the circumference of the smaller circle.
The radius r of the smaller circle is 1; The distance from the circumference of the smaller circle to the larger is phi. So the radius R of the larger circle = 1+phi.
There are straight lines connecting the center O to the intersection of the chord to the circumferences of the two circles. On the smaller circle the point of this intersection (call it A) is at a 60 degree angle. On the larger circle the point (call it B) is at an angle to be determined.
Since r=1, sin A= sin60= h/1 = h. So h= sin60 = ((sqrt3)/2)
sin B = ((Sqrt3)/2) / (1 + phi) = (sqrt3)/(2(1 + phi))
so B = arcsin (sqrt3)/(2(1 + phi))
= approx 19.31687 degrees
(I confess I turned to an arcsin calculator, and used a long decimal approximation of phi). Now I could easily calculate the remaining angle of the other angle in triangle HOB. The I subtracted 30 degrees to get angle AOB.
Multiply AOB by 2, and add 60 degrees, and voila! (naturally, I could also have just doubled HOB)
It isn't 144, but much closer to the 141 or so which I kept getting when I had tried with compass and straightedge and measuring with a protractor. Which makes me feel a little better about my steadiness with good old Euclidean tools.
Thanks again for your help, without which I would not have got started. | 677.169 | 1 |
How to find number of triangles in the given figure?
Updated: Mar 30
Are you preparing for competitive exams?
So, here it is...
Each and every competitive exams will have this question. Usually, students waste time in counting the number of triangles in the given figure. It would take at least 5 to 8 mins for this question, in turn, which would affect the time limit for the entire exam.
So, here is a shortcut which can be applied to any given triangular figures. It would take 2mins maximum,in turn, saves time as TIME IS THE MOST IMPORTANT ONE TO KEEP IN MIND DURING EXAMS!!
Shortcut to calculate the number of triangles in the given figure is
n (n+2)(2n+1)/8
where
n : total number of any one side triangulat parts
Example :
There are 4 dots. So, the number of side triangles are 4 . You can consider any one side. But consider only one side.
So, n=4
Therefore,
n+1 = 4+2 = 6
2n +1 = 4*2 +1 = 9
Therefore the total number of triangles in the given figure is
n(n+1)(2n+1)/8
= 4* 6 * 9 /8
= 27
The total number of triangles in the given figure is 27
Similarly, we can apply this shortcut for all triangular based questions. | 677.169 | 1 |
Distance formula
The distance formula, derived from the Pythagorean theorem, calculates the distance between two points in a plane using their coordinates. It is expressed as √((x2 - x1)² + (y2 - y1)²), where (x1, y1) and (x2, y2) represent the points. Mastering this formula is crucial for success in geometry and coordinate algebra.
Distance Formula Definition
The distance formula is a crucial tool in analytic geometry, enabling you to calculate the distance between two points in a two-dimensional or three-dimensional space, given their Cartesian coordinates. It is fundamental for various mathematical computations and real-world applications.
Understanding the Distance Formula
The distance formula derives its principles from the Pythagorean theorem. When you think of the distance between two points, you can visualise it as the hypotenuse of a right triangle formed by these points, thus connecting them geometrically through their coordinates.The formula for the distance between two points \(x_1, y_1\) and \(x_2, y_2\) in a two-dimensional plane is:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Distance Formula: The formula is given by \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\], where \(d\) is the distance between points \(x_1, y_1\) and \(x_2, y_2\).
Consider two points \(A = (3, 4)\) and \(B = (7, 1)\). The distance \(d\) between these points is calculated as follows:
Subtract the x-coordinates: \(7 - 3 = 4\)
Subtract the y-coordinates: \(1 - 4 = -3\)
Square the differences: \(4^2 = 16\) and \((-3)^2 = 9\)
Sum the squares: \(16 + 9 = 25\)
Take the square root of the sum: \(\sqrt{25} = 5\)
So, the distance \(d\) = 5.
Remember that the distance formula can be used in three dimensions by including the z-coordinates, transforming the formula to \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\].
Euclidean Distance Formula
The Euclidean distance formula is a cornerstone in geometry. It allows you to find the shortest path between two points in a plane or space, given their coordinates.
Derivation of the Formula
The formula for Euclidean distance in a two-dimensional space can be derived using the Pythagorean Theorem. For points \(A = (x_1, y_1)\) and \(B = (x_2, y_2)\), the horizontal and vertical distances between the points form the two legs of a right triangle, with the distance \(d\) being the hypotenuse.
Double-check your calculations to avoid common mistakes such as sign errors when subtracting coordinates.
Applications of the Euclidean Distance Formula
The Euclidean distance formula is widely used in various fields:
In computer science, for clustering algorithms like K-means.
In physics, to calculate the distance between objects.
In geography, to find the distance between two places on a map.
Understanding this formula can provide insights into more complex mathematical and scientific concepts.
Distance Formula Examples
Understanding how to apply the distance formula through examples can make the concept clearer and more intuitive. Let's delve into some practical scenarios to enhance your grasp of this important mathematical tool.
While using the distance formula, ensure you square the differences, not the coordinates themselves.
Using the Distance Formula in a Cartesian Plane
The distance formula is frequently applied in Cartesian planes to find the shortest path between two points. This method is not just theoretical—it's widely applicable in various practical fields.
Consider an example where two points are given: \(C = (6, 8)\) and \(D = (3, 4)\). You can find the distance \(d\) as follows:
Step 1: Calculate the x-coordinate difference: \(6 - 3 = 3\)
Step 2: Calculate the y-coordinate difference: \(8 - 4 = 4\)
Step 3: Square these differences: \(3^2 = 9\) and \(4^2 = 16\)
Step 4: Add the squares: \(9 + 16 = 25\)
Step 5: Take the square root of the sum: \(\sqrt{25} = 5\)
Thus, the distance \(d\) between points C and D is 5.
Consider Cartesian planes where axes aren't perpendicular, such as in certain trigonometric calculations or when non-Euclidean geometry is involved. The basic concepts of distance remain valid, but these alternate geometries require adaptations of the formula to accommodate curved spaces or different metrics.
Application of Distance Formula
The distance formula is extensively used in various branches of mathematics, especially in analytic geometry, to find the distance between two points in a plane or space. Understanding this formula is essential for tackling numerous geometric problems and real-world scenarios.
Distance Formula Explained
The distance formula is derived from the Pythagorean theorem. In a two-dimensional Cartesian plane, it calculates the distance between two points, helping you visualise their relative positions on the plane.The formula for the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Take points \(P = (2, 3)\) and \(Q = (5, 7)\). The distance \(d\) between these points can be calculated as follows:
Subtract the x-coordinates: \(5 - 2 = 3\)
Subtract the y-coordinates: \(7 - 3 = 4\)
Square the differences: \(3^2 = 9\) and \(4^2 = 16\)
Add the squares: \(9 + 16 = 25\)
Take the square root: \(\sqrt{25} = 5\)
So, the distance \(d\) is 5.
Distance Formula Derivation
The distance formula originates from the Pythagorean theorem, which is applicable in right-angled triangles. By considering the distance between two points as the hypotenuse of a right-angled triangle, you can simplify the computation.For two points, \((x_1, y_1)\) and \((x_2, y_2)\) in a plane, the horizontal distance \((x_2 - x_1)\) and the vertical distance \((y_2 - y_1)\) form the two legs of the right triangle.
Using the Pythagorean theorem:\[a^2 + b^2 = c^2\]where \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse.Applying to our coordinates:\[c = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This is how the distance formula is derived.
In three dimensions, the distance formula incorporates the z-coordinates, adding another term in the square root.
Distance Formula in Coordinate Geometry
In coordinate geometry, the distance formula is vital for calculating the distance between points in a Cartesian plane. It simplifies the process of determining the length of line segments and the spatial relationships between points.
To find the distance between points \((3, 4)\) and \((7, 1)\) within a coordinate plane:
In coordinate geometry, understanding the distance formula aids in solving more complex geometric problems, such as finding the equation of a circle, determining midpoints, and solving for areas of polygons. The formula remains effective across different dimensions and coordinates.
Real-World Applications of Distance Formula
The distance formula is not confined to theoretical mathematics; it has practical applications across various fields. Here are some real-world scenarios where the distance formula plays a crucial role.
Navigation: Used in GPS technology to calculate the shortest path between two geographical locations.
In navigation, the distance formula helps in determining routes. For instance, calculating the shortest distance between two cities based on their coordinates to provide efficient travel routes.
In engineering, the distance formula is crucial for ensuring the proper alignment and placement of structures. In physics, it's essential for calculating the trajectory of particles, analysing forces, and understanding spatial relationships in various contexts.
Frequently Asked Questions about Distance formula
The distance formula in coordinate geometry is \\( \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\), where \\((x_1, y_1)\\) and \\((x_2, y_2)\\) are the coordinates of the two points.
How do you derive the distance formula?
To derive the distance formula, begin with two points in a plane: \\((x_1, y_1)\\) and \\((x_2, y_2)\\). The difference in \\(x\\)-coordinates is \\(x_2 - x_1\\) and in \\(y\\)-coordinates is \\(y_2 - y_1\\). Apply the Pythagorean Theorem to the right triangle formed, yielding: \\(d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\).
How is the distance formula applied in three-dimensional space?
The distance formula in three-dimensional space is applied by calculating the square root of the sum of the squares of the differences between corresponding coordinates. It is expressed as: √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²). This measures the straight-line distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) in 3D space.
What are some common mistakes to avoid when using the distance formula?
Common mistakes to avoid include: not squaring the differences of the coordinates, forgetting to take the square root of the sum of the squared differences, incorrect subtraction order in the coordinate differences, and miscalculation of signs when dealing with negative coordinates.
What are some real-life applications of the distance formula?
Real-life applications of the distance formula include navigation systems (GPS) to calculate the shortest route, determining the distance between celestial bodies in astronomy, measuring the length of a diagonal in architecture and construction, and analysing the distance between data points in statistical and machine learning models | 677.169 | 1 |
If triangle DEF is an isosceles triangle, angle d is the vertex angle, DE = 4x-7, EF = 3x+2 and DF = 6x- 15, find the value of x and the measure of each side
Answer
The value of x is 4, and the lengths of DE, DF, and EF are 9, 9, and 14, respectively. Step-by-step explanation: As the angle at vertex D, DE and DF have the same length. Therefore, we can set equation 4x - 7 = 6x - 15, and simplify it to 2x = 8, which gives us the value of x as 4. Then, we can calculate the lengths: DE = 4x - 7 = 4(4) - 7 = 16 - 7 = 9, DF = 9, and EF = 3(4) + 2 = 12 + 2 = 14. | 677.169 | 1 |
What degree does the equator fall on?
The equator is composed of all the points that have zero
latitude and every possible longitude.
What degree is the prime meridian located at?
The longitude of the Prime Meridian, by international definition and agreement, is zero (0°). .Every point on the Prime Meridian has a different latitude, and every latitude on Earthhas a point on the Prime Meridian.
Is the kilauea map its on longitude or latitude?
Every point on Earth has a longitude and a latitude. The only exceptions arethe north and south poles. Each of them has a latitude and every longitude. | 677.169 | 1 |
stclair.daniel_f70a54fff4764c4
on March 23, 2015, 6:13 a.m.
<p>We all know because we did the problem, but it's usually a good idea to add a comment like</p>
<h1>Heron's formula</h1>
<p>so people looking at the code for the first time aren't confused.</p> | 677.169 | 1 |
Year 6 | Drawing 2D Shapes Worksheets
In these Year 6 drawing 2D shapes worksheets, your learners will hone their measuring skills with precision tasks. They will draw a meticulous 7cm by 4cm rectangle, using a ruler and set square for accuracy. Following this, learners will encounter a triangle (not drawn to scale) with labelled and measured angles and sides. Using a ruler and a protractor, they will be challenged to replicate the triangle to its full size.
This Year 6 drawing 2D shapes worksheet serves as a valuable practice ground for your class, enabling them to finesse their skills in using a ruler, set square, and protractor. It instils the importance of careful measurement and drawing in the realm of 2D shapes. Drawing precise shapes and replicating triangles can be challenging, promoting a growth mindset as they refine their measurement techniques and overcome difficulties. | 677.169 | 1 |
MAT-02.GM.G.03Compose geometric shapes having specified geometric attributes, such as a given number of edges, angles, faces, vertices, and/or sides.
MAT-06.GM.GF.03 Represent three-dimensional figures using nets made up of rectangles and triangles (right prisms and pyramids whose bases are triangles and rectangles). Calculate the surface area of prisms with rectangular and triangular
bases using nets, including authentic problems.
Compose Shapes
MAT-00.GM.G.04 Compose a geometric shape by combining two or more simple shapes.
MAT-02.GM.G.03Compose shapes having specified geometric attributes, such as a given number of edges, angles, faces, vertices, and/or sides10.GM.12 Make basic geometric constructions (e.g., segments, angles, bisectors, parallel and perpendicular lines) with a variety of tools and methods.
MAT-10.GM.13 Apply basic construction to create polygons such as equilateral triangles, squares, and regular hexagons inscribed in a circle. | 677.169 | 1 |
The Elements of Descriptive Geometry ...
No interior do livro
Resultados 1-5 de 87
Página 2 ... parallel . A P M B ( 10 ) The angle between two planes is called the dihedral angle . AB is Thus , QA B M is the ... lines ABC , ABD , in the same plane , have a common segment AB , which is impossible . PROPOSITION II . Two lines which meet are in the same plane , .. A B , BD , DC , are in one plane ; and each of the s A B D , BD C , is a right . AB is parallel to CD . PROPOSITION VII . If two straight lines be parallel , the straight line drawn from ...
Página 7 ... lines , which are each of them parallel to the same straight line , and not in the same plane with it , are parallel to one another . Let A B , CD , be each parallel to EF , and not in the same plane with it , A B shall be parallel ...
Página 8 Thomas Grainger Hall. PROPOSITION X. If two straight lines meeting one another be parallel to two others that meet one another , and are not in the same plane with the former two , the former two and the other two shall contain equal | 677.169 | 1 |
Sin 9pi/2
The value of sin 9pi/2 is 1. Sin 9pi/2 radians in degrees is written as sin ((9π/2) × 180°/π), i.e., sin (810°). In this article, we will discuss the methods to find the value of sin 9pi/2 with examples.
Sin 9pi/2: 1
Sin (-9pi/2): -1
Sin 9pi/2 in degrees: sin (810°)
What is the Value of Sin 9pi/2?
The value of sin 9pi/2 is 1. Sin 9pi/2 can also be expressed using the equivalent of the given angle (9pi/2) in degrees (810°).
FAQs on Sin 9pi/2
What is Sin 9pi/2?
Sin 9pi/2 is the value of sine trigonometric function for an angle equal to 9pi/2 radians. The value of sin 9pi/2 is 1.
What is the Value of Sin 9pi/2 in Terms of Cos 9pi/2?
Using trigonometric identities, we can write sin 9pi/2 in terms of cos 9pi/2 as, sin(9pi/2) = √(1-cos²(9pi/2)). Here, the value of cos 9pi/2 is equal to 0.
How to Find the Value of Sin 9pi/2?
The value of sin 9pi/2 can be calculated by constructing an angle of 9π/2 radians with the x-axis, and then finding the coordinates of the corresponding point (0, 1) on the unit circle. The value of sin 9pi/2 is equal to the y-coordinate (1). ∴ sin 9pi/2 = 1.
How to Find Sin 9pi/2 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin 9π/2 can be given in terms of other trigonometric functions as: | 677.169 | 1 |
Answers (1)
Hint.
Solution.
Step I- Draw number line shown in the figure.
Let the point O represent 0 (zero) and point A represent 2 units from O.
Step II- Draw perpendicular AX from A on the number line and cut off arc AB = 1 unit
We have OA = 2 units and AB = 1 unit
Using Pythagoras theorem, we have.
OB2 = OA2 + AB2
OB2 = (2)2 + (1)2 = 5
OB =
Taking O as the centre and OB = as radius draw an arc cutting the line at C.
Clearly, OC = OB = .
Hence, C represents on the number line. | 677.169 | 1 |
unit 6 similar triangles homework 2 similar figures worksheet answers
Unit 6 Similar Triangles Homework 2 Similar Figures Worksheet Answers – Triangles are among the most fundamental shapes in geometry. Understanding triangles is crucial for studying more advanced geometric concepts. In this blog post this post, we'll go over the various types of triangles including triangle angles and the methods to determine the areas and perimeters of a triangle and will provide illustrations of all. Types of Triangles There are three types in triangles, namely equilateral isoscelesand … Read more | 677.169 | 1 |
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1-9 Use proportions to solve problems involving geometric figures. Use proportions and similar figures to measure objects indirectly.
Similar figures have exactly the same shape but not necessarily the same size. Corresponding sides of two figures are in the same relative position, and corresponding angles are in the same relative position. Two figures are similar if and only if the lengths of corresponding sides are proportional and all pairs of corresponding angles have equal measures.
The length of SU is cm. Example 1A: Finding Missing Measures in Similar Figures Find the value of x the diagram. ∆MNP ~ ∆STU M corresponds to S, N corresponds to T, and P corresponds to U. 6x = 56 Use cross products. Since x is multiplied by 6, divide both sides by 6 to undo the multiplication.
The length of FG is 2.5 in. Example 1B: Finding Missing Measures in Similar Figures Find the value of x the diagram. ABCDE ~ FGHJK 14x = 35 Use cross products. Since x is multiplied by 14, divide both sides by 14 to undo the multiplication. x = 2.5
The length of XY is 2.8 in. Check It Out! Example 1 Find the value of x in the diagram if ABCD ~ WXYZ. ABCD ~ WXYZ Use cross products. Since x is multiplied by 5, divide both sides by 5 to undo the multiplication. x = 2.8
Example 2: Measurement Application A flagpole casts a shadow that is 75 ft long at the same time a 6-foot-tall man casts a shadow that is 9 ft long. Write and solve a proportion to find the height of the flag pole. Since h is multiplied by 9, divide both sides by 9 to undo the multiplication. The flagpole is 50 feet tall.
Check It Out! Example 2a A forest ranger who is 150 cm tall casts a shadow 45 cm long. At the same time, a nearby tree casts a shadow 195 cm long. Write and solve a proportion to find the height of the tree. 45x = 29250 Since x is multiplied by 45, divide both sides by 45 to undo the multiplication. x = 650 The tree is 650 centimeters tall.
Check It Out! Example 2b A woman who is 5.5 feet tall casts a shadow 3.5 feet long. At the same time, a building casts a shadow 28 feet long. Write and solve a proportion to find the height of the building. 3.5x = 154 Since x is multiplied by 3.5, divide both sides by 3.5 to undo the multiplication. x = 44 The building is 44 feet tall.
Radii: Circumference: Area: Example 3A: Changing Dimensions The radius of a circle with radius 8 in. is multiplied by 1.75 to get a circle with radius 14 in. How is the ratio of the circumferences related to the ratio of the radii? How is the ratio of the areas related to the ratio of the radii? Circle B The ratio of the areas is the square of the ratio of the radii. The ratio of the circumference is equal to the ratio of the radii.
Every dimension of a rectangular prism with length 12 cm, width 3 cm, and height 9 cm is multiplied by to get a similar rectangular prism. How is the ratio of the volumes related to the ratio of the corresponding dimensions? Example 3B: Changing Dimensions Prism A Prism B V = lwh (12)(3)(9) = 324 (4)(1)(3) = 12 The ratio of the volumes is the cube of the ratio of the corresponding dimensions.
Helpful Hint A scale factor between 0 and 1 reduces a figure. A scale factor greater than 1 enlarges it.
Check It Out! Example 3 A rectangle has width 12 inches and length 3 inches. Every dimension of the rectangle is multiplied by to form a similar rectangle. How is the ratio of the perimeters related to the ratio of the corresponding sides? Rectangle B Rectangle A P = 2l +2w 2(12) + 2(3) = 30 2(4) + 2(1) = 10 The ratio of the perimeters is equal to the ratio of the corresponding sides.
Lesson Quiz: Part 2 3. A girl that is 5 ft tall casts a shadow 4 ft long. At the same time, a tree casts a shadow 24 ft long. How tall is the tree? 30 ft 4. The lengths of the sides of a square are multiplied by 2.5. How is the ratio of the areas related to the ratio of the sides? The ratio of the areas is the square of the ratio of the sides. | 677.169 | 1 |
Types of Transformations
Translation (or Slide)
A translation moves a shape.
A translation is a slide of a shape (without rotating, reflecting or resizing it).
The diagram below shows a triangle before (light blue) and after (dark blue) being translated:Each point on the shape moves the same direction and distance (shown by the arrow).
Learn more about translations
Reflection (or Flip)
A reflection makes a shape a mirror image of itself.
A reflection flips a shape in a line (called the line of reflection).
The diagram below shows a triangle before (light blue) and after (dark blue) being reflected:Each point in the image (the reflection) is the same perpendicular distance from the line of reflection (in this case, the y-axis) as the corresponding point in the object.
Learn more about reflections
Rotation (or Turn)
A rotation turns a shape.
A rotation turns a shape around a point (called the centre of rotation).
The diagram below shows a triangle before (light blue) and after (dark blue) being rotated:Each point on the rotated triangle is the same distance from centre of rotation as the corresponding point on the original shape.
Learn more about rotations
Enlargement
An enlargement makes a shape larger (or smaller).
An enlargement resizes a shape about a point (called the centre of enlargement).
The diagram below shows a triangle before (light blue) and after (dark blue) being enlarged:The enlarged shape is twice as large as the original shape. The length of the line drawn from the centre of enlargement to each point on the enlarged triangle is twice the length of the line drawn from the centre to the corresponding point on the original shape.
Learn more about enlargements
Images and Objects
The original shape before the transformation is called an object.
The shape after it has been transformed is called the image.
Congruent Shapes
If a shape can be transformed to another using only translation, reflection and rotation, then the two shapes are congruent.
Congruent shapes have the same size, line lengths, angles and areas.
All the shapes below are congruent:
Similar Shapes
If a shape can be transformed to another using enlargement, then the two shapes are similar.
Similar shapes are identical in shape but not in size.
All the shapes below are similar: | 677.169 | 1 |
Let the lines $$l_{1}: \frac{x+5}{3}=\frac{y+4}{1}=\frac{z-\alpha}{-2}$$ and $$l_{2}: 3 x+2 y+z-2=0=x-3 y+2 z-13$$ be coplanar. If the point $$\mathrm{P}(a, b, c)$$ on $$l_{1}$$ is nearest to the point $$\mathrm{Q}(-4,-3,2)$$, then $$|a|+|b|+|c|$$ is equal to
A
12
B
14
C
10
D
8
2
JEE Main 2023 (Online) 12th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
Let the plane P: $$4 x-y+z=10$$ be rotated by an angle $$\frac{\pi}{2}$$ about its line of intersection with the plane $$x+y-z=4$$. If $$\alpha$$ is the distance of the point $$(2,3,-4)$$ from the new position of the plane $$\mathrm{P}$$, then $$35 \alpha$$ is equal to :
A
126
B
105
C
85
D
90
3
JEE Main 2023 (Online) 11th April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
Let the line passing through the points $$\mathrm{P}(2,-1,2)$$ and $$\mathrm{Q}(5,3,4)$$ meet the plane $$x-y+z=4$$ at the point $$\mathrm{R}$$. Then the distance of the point $$\mathrm{R}$$ from the plane $$x+2 y+3 z+2=0$$ measured parallel to the line $$\frac{x-7}{2}=\frac{y+3}{2}=\frac{z-2}{1}$$ is equal to :
A
$$\sqrt{31}$$
B
$$\sqrt{189}$$
C
$$\sqrt{61}$$
D
3
4
JEE Main 2023 (Online) 11th April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
Let P be the plane passing through the points $$(5,3,0),(13,3,-2)$$ and $$(1,6,2)$$.
For $$\alpha \in \mathbb{N}$$, if the distances of the points $$\mathrm{A}(3,4, \alpha)$$ and $$\mathrm{B}(2, \alpha, a)$$ from the plane P are 2 and 3 respectively, then the positive value of a is : | 677.169 | 1 |
Equation Of a Circle. There are three different forms of the equation of a circle that can be used to define its radius and centre or point of origin. Depending on the form you need, you can use either the distance formula or the midpoint formula to find the coordinates of the centre of your circle. The radius, r, can then be found with either the distance formula or by taking half of the length between (x1, y1) and (x2, y2).
EQUATION OF A CIRCLE
Every circle has a radius, so r is the length from the centre to any point on the edge. The x-axis is drawn perpendicular to the radius and intersects with it at two points. One issue is where y=0 (the centre), and one end is where y=r (the edge).
The equation for a circle centred at (h,k) with radius r can be written as: (x-h)2+(y-k)2 = r2. This implies that all points in a circle are equidistant from the centre. The distance between the point's location and the centre can be found by subtracting h or k from r.
When h=k, then x2+y2 = r2 becomes (x-h)2+(y-k)2 = 0 or (-h+k) = 0 or h=k. So all points on a circle centred at (h, k) are either on its edge or centre.
In general, given any point (x, y) on a circle of radius r and centre (h, k), it's equation is x2+y2 = r2 - h2 + k2. The distance from that point to (h,k) can be found by subtracting h or k from r.
If we let C be any point on a circle with radius r, (h1, k1) and (h2, k2) are two different points on that circle. If h1 = h2 and k1 = k2, then C is on that circle. For example, let's use (9,-12),(-3,-6),(-7,-8) to represent three distinct points which satisfy x2+y2=9.
EQUATION OF A CIRCLE WITH CENTER AND RADIUS
A circle is defined by its centre point and radius. The equation for a process is r = √(x2 + y2), where x and y are the coordinates on the plane. For example, if the coordinate (x,y) lies in quadrant I, then
on the unit circle with a radius one centred at the origin, x=1, y=-1/√3=-√3, and we have r=√(-√3-1)=1
The equation of a circle is x2+ y2 = r2, and The general equation for a loop with a centre point (x0, y0) and radius r is (x- x0)2 + (y- y0)2 = r2. Here, we want to find out how to determine if an arbitrary point(x,y) on a plane lies inside or outside a circle.
Now, we can re-write our earlier equation as x - x0 2 + y - y0 2 = (R_max - r2)/(2R_min) and compare it with the cosecant function to find out if (x,y) lies inside or outside the circle.
If x0 is not equal to 0, then the cosecant function will have values less than 0 for (x - x0)/(x0), meaning our point lies outside the circle.
Similarly, if y0 is not equal to 0, then the cosecant function will have values greater than 0 for (y - y0)/(y0), meaning our point lies inside the circle.
EQUATION OF A CIRCLE GIVEN TWO POINTS
The equation of the circle with centre at (h,k) and radius r is given by the following formula:
x^2 + y^2 = r^2 . Let's explore some examples. Knowing the coordinates for two points on the plane can determine what kind of curve they form.
In this example, let's say that one point has coordinates (-5,-5), and another has coordinates (4,-4). We can use these points to create a circle with a centre at (-5,-5) and a radius of 2.
Linear equations connect the coordinates for points on a circle. Let's say that we have three points that make up our curve, each with its own distinct set of x and y coordinates. We know that these are points on a circle when they fall within an x-y coordinate plane.
The equation to describe such a circle is given by x^2 + y^2 - 4x - 4y + 16 = 0. If you simplify the equation using the quadratic formula, it becomes 2(x+1)(x-1) - 4(x+1)(y+1)(x+1) -4(x+1)(y-1)(x-1)=0.
EQUATION OF A CIRCLE DESMOS
When discussing the equation for a circle, we assume that the centre is the point (h, k) with coordinates (h, 0) and that the radius is r.
We also assume that our circle has no vertical or horizontal tangents. Any line drawn through the centre will intersect with only one point on each side.
For an ellipse, we do not assume that there are no tangents, which means that any line drawn through either centre will intersect with at least one point on each side. This means that our formula for r is a function of both t and φ
You can see from the figure that formula_5 is parallel to formula_6 and perpendicular to formula_7.
This means that because they are perpendicular to each other, we can use properties of right triangles to solve for our unknowns.
Because both sides of our triangle are radii, we know they are congruent by SSS (Side-Side-Side) Postulate. We see that side adjacent = adjacent over hypotenuse and opposite = opposite over hypotenuse.
We know that because both sides are radii, we can use Pythagorean Theorem to solve for formula_10 and formula_11, which means that we now have enough information to write our final equation. From here, we can find r, t, and φ.
EQUATION OF A CIRCLE EXPLAINED
The equation for a circle is (x-h)2+(y-k)2=r2. It has a centre point (h,k) and a radius, r. This formula can be used to find the equation for any point on the circle by plugging in the coordinates for that point into the recipe.
In contrast, (x-h)2+(y-k)2 is just equal to (x-h+r)2+(y-k+r)2; r is cancelled out because each side has it and thus simplifies to x2 + y 2 = r 2. Now we can put in any values for x and y that we please and get our circle.
We take square roots on both sides to get our centre point, h and k. This will give us (h+r/2) and (k+r/2). We then add r / 2 to each side to ensure that our circle is centred at our coordinate.
We can use our equation to create any point on our circle by plugging in coordinates. A good example is (3,5).
Plugging those into our equation gives us: 32 + 52 = 25 + 25 = 50. Thus, (3,5) is an equation for our circle. To put that another way, 3 and 5 are X and Y coordinates for a point in our process, and 50 is its radius.
HOW TO FIND THE EQUATION OF A CIRCLE
To find the equation of a circle, use the radius and any point on the process as your coordinates. Here's how it works for a loop with centre (h, k) and radius r: x^2 + y^2 = h^2 + k^2 - 2rhk + r^2 FINDING THE EQUATION OF A CIRCLE WITHOUT CENTER POINT:
If you don't have an intersection point or if the intersection is outside of the circle, we can use (0, 0). In this case, you would still need to know the radius to calculate other information, such as circumference or area. Here's what that looks like x^2 + y^2 = 0
And our equation is 4^2 + 8^2 = 16 - 2(4)(-6) + 9^2 = 0. So the equation of our circle would be (x, y) = (-5, 5). That's it! Now you know how to find an equation for any circle!
So how do you use that equation to find things like area or circumference? Let's walk through it. First, we must replace x and y with our centre point (h, k). Here's what that looks like: h^2 + k^2 - 2rhk + r^2 = (4)^2 + (-6)^2 - 2(4)(-6) + 9^2 = 36.
EQUATION OF A CIRCLE EXAMPLES WITH ANSWERS
The equation for the circle with centre at (h, k) and radius r is given by the following equation: x^2 + y^2 = r^2. For example, if we want to find the equation for a circle with a centre at (-1,-1) and radius 3, we would plug in h=-1 and k=-1.
Another way to view circle equations is as circles pass through (h, k). In other words, if h and k are real numbers, we can write an equation for every point on that circle.
By plugging in (-1,-1) into our original equation, we get x^2 + y^2 = r^2. This shows that all points along our circle satisfy the equation x^2 + y^2 = r^2.
Using x = (h - k)^2 and y = 2k, we get a circle that passes through (-1,-1). In other words, all points on our circle are given by x^2 + y^2 = (-1-0)^2+ 2*(-1)*0 = 1. So plugging in (0,-3), for example, we see that its coordinates satisfy our equation.
With any value we plug in for h and k, both x and y satisfy our equation, as do all coordinates. For example, if we plug (1,-2) into our original equation x^2 + y^2 = r^2 we find that (1,-2) satisfies both x^2+y^2=r^2 and 2(1)(-3) + (-12) = -13.
In contrast, (x-h)2+(y-k)2 is just equal to (x-h+r)2+(y-k+r)2;
We take square roots on both sides to get our centre point, h and k.
Cancelled out because each side has it and thus simplifies to x2 + y 2 = r 2.
This will give us (h+r/2) and (k+r/2). We then add r / 2 to each side.
Now we can put in any values for x and y that we please and get our circle.
To ensure that our circle is centred at our coordinate.
GENERAL EQUATION OF A CIRCLE CALCULATOR
We must first find the radius to find the equation for a circle. To do this, use the following formula:
r = √(x^2 + y^2) Once you've found the radius, plug it into one of these two equations: (x- h)^2+(y-k)^2=r^2 or (x-h)^2+ (y+k) ^ 2=r ^ 2.
(x-h)^2+(y-k)^2=r^2 or (x-h)^2+ (y+k) ^ 2=r ^ 2 If you aren't sure which equation to use, plug in each set of variables and compare the results. If one of them works, you've got your equation for a circle!
The circle equation (x-h)^2+(y-k)^2=r^2 is often referred to as quadrantal, and (x-h)^2+ (y+k) ^ 2=r ^ 2 is referred to as polar.
Which one you use doesn't really matter. Some equations are more accessible to memorize, but either way will get you a circle!
If you prefer using degrees, we have an alternative equation for you! You may be more familiar with (x-h)^2+(y-k)^2=(r * 2 * pi)^2 or (x-h)^2+(y-k)^2=(r * 2pi) ^ 2, and that's totally fine. It'll get you a circle no matter what! Just make sure to plug in radians for r and your variables.
EQUATION OF A CIRCLE GIVEN CENTER AND POINT
The equation for a circle with center (h, k) and point (x, y) on the circle is (x^2 + y^2 = h^2) The equation can also be written as:
(x^2 + y^2 = r^2) Where r is the radius. You can also find this relationship by looking at the distance formula to solve for x. First, we substitute (x, y) for each variable. (d = \sqrt{(x - h)^2 + (y - k)^2}) Now we square both sides.
(r^2 = d) Solve for r. (r = \sqrt{d}) And that's how to find x when given center and point. To use these methods, you must already know how to find the centre and radius in a circle.
The video tutorial below covers finding these values in detail. The methods are discussed in detail below as well.
Finding Center and Radius using Pythagorean Theorem The video below is a detailed step-by-step tutorial on how to find the centre and radius for any circle, given its centre and point.
If you don't know how to do that, check out that video before attempting these methods.
Get to our equation for finding x when given the centre and point. As we saw in that video, there are two ways you can use the distance formula to solve for x.
Summary
The equation for a circle is (x-h)2+(y-k)2=r2 where x, y, and r are the coordinates. The centre (h,k) is found by solving for either x or y first. A circle can also represent an equation with the form (x+w)2+(y+h)2=r2.
Frequently Asked Questions
Some Important Questions For Your Knowledge:
1) How do we find the equation of a circle?
A circle is the set of points at a given distance from one point. The equation for the circle with centre (0,0) at the origin and radius r is x2 + y2 = r2. This can be rewritten as x2 + y2 - 2rx = 0. We can also write this as x2 + y2 = ±r2. These two equations represent the total possible solutions for this problem!
2) What is the equation of a circle GCSE?
The equation for a circle with center ((h, k)) and radius (r) is:$$x^2 + y^2 = r^2$$ $$\begin{aligned}&x = \sqrt{h^2+k^2-r^2} \&y = \sqrt{h^2+k^2-r^2}\end{aligned} $$ The two points on the circumference are given by (\left(\frac{\pi}{2}, 0\right)) and (\left(\frac{3\pi}{2}, 0\right)).
3) What are the three circle formulas?
The formula for the circumference is 2πr. The formula for the area is πr2. And the formula for the radius is r. A circle with an equation x2+y2=1 has both of these circles formulas because it can be graphed in two different ways: from x=0 to x=-1 and y=-1 to y=0 or from x=-1 to x=0 and y=-1 to y=0, depending on which one you want to use as the outer circle or inner circle.
4) What is the standard form of the equation of a circle?
The equation for the equation of a circle is: (x-h)2+(y-k)2=r^2. The graph shows the x, y, and r coordinates, and h and k are the x and y intercepts. It is important to note that this expression holds for any point on the chart with an x value greater than or equal to h and less than or equal to k.
5) How do we find the radius of a circle?
To see the radius of a process, we use the equation for the circumference of the process and divide it by 2π. The radius is halfway between the centre and any point on the edge. To find the centre, we draw two perpendicular lines from any point on the border to either side of our circle.
6) How do you find the equation of a circle given two points?
The equation for a circle is x2 + y2 = r2. To find the equation of a circle given two points, you need to solve for r in the following equation: x - (y/r) = 0 and y - (x/r) = 0. These equations can be solved using substitution and linear algebra. For example, if we have x = 3 and y = 4, then r would equal 9.
7) What is the general form of an equation?
The equation for the equation of a circle is as follows: (x-h)2+(y-k)2=r2. It is called the general form because it can be used to find any point on the circle, not just the centre. To find the distance between two points, use the following formula: d=sqrt((x1-x2)^2+(y1-y2)^2).
8) How do you form equations?
The equation for a circle with a centre is the following: X^2 + Y^2 = R^2 where R = distance from the centre to any point on the circle's circumference. The radius (R) can be either positive or negative. WOW, that was easy! To get an equation in standard form, you only need to solve it and put it into parentheses! For example, let's say we have 4x - 3y = 2, so what we do is multiplie diply by y, then add 3 to both sides, so we get 4x - 3y + 3y = 6y, which then equals y(4x-3).
9) What are the two equations of a line?
The equation for the circle's centre is 'x^2+y^2=r^2' where 'r' is the radius, and the equation for a point on the circumference of the circle is (x-h)^2+(y-k)^2=(r+h)^2. If you know any two points on the process, you can find its radius theiry taking their difference squared.
10) What is an equation for a line?
An equation for a line is y = MX + b, where x and y are the coordinates of the line. The slope, or steepness, of the line is represented by m, and b is the y-intercept.
Conclusion
The equation for a circle is found by finding the centre, then plugging it into the radius and drawing a line from the centre to any point on the circumference. The radius of the circle is equal to its distance from the centre. A circle has an infinite number of points on its circumference, but only one point t called this e centre. | 677.169 | 1 |
Law of Sines
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With reference to the diagram above, calculate and enter the answers. Round your answers to the nearest hundredth. You may use the TAB key to move to the next question. When you are done, click Submit.
B = 107, C = 46, b = 7.7, c =
A = 101, B = 32, a = 9.7, b =
B = 96, C = 64, b = 5, c =
A = 34, C = 59, a = 1.5, c =
B = 87, C = 25, b = 6.5, c =
A = 97, B = 46, a = 1.4, b =
A = 14, C = 77, a = 4.7, c =
A = 72, B = 43, b = 4.3, a =
A = 18, B = 123, a = 5.5, b =
A = 108, B = 37, a = 8.5, b | 677.169 | 1 |
It is one of a total of five distinct polychora (including two transitional cases) that can be obtained as the convex hull of two opposite prismatorhombated pentachora. In this case, if the prismatorhombated pentachora are of the form a3b3o3c, then c must be less than a+b/3 (producing the transitional biprismatorhombatodecachoron in the limiting case). This includes the convex hull of two uniform prismatorhombated pentachora. The lacing edges generally have length 3a2+2b2+3c2+2ab−6ac−2bc5{\displaystyle {\sqrt {\frac {3a^{2}+2b^{2}+3c^{2}+2ab-6ac-2bc}{5}}}}.
Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:4+307{\displaystyle {\frac {4+{\sqrt {30}}}{7}}} ≈ 1:1.35389. | 677.169 | 1 |
Can you have a detailed description of a trapezium?
A trapezium is a quadrilateral with two parellel sides which are
not of equal length. If it is represented with its longest of the
two parallel sides at the bottom, its two top angles are obtuse, or
one is a right angle, and its two bottom angles are acute, though
the one below a right angle would also be a right angle. The area
of the trapezium is determined by multiplying the distance between
the two parallel sides (called the height) by the average length of
the two parallel sides.
What is the shape that has 4 sides exactly 2 right angles and is not square or rectangular?
An irregular Quadrilateral. It is like a rectangle with a piece cut off from one corner to the other side. expand a bit on the above:This describes a trapezoid which is a quadrilateral (four-sided figure) with only two sides parallel. In this case, the ends of the figure will be parallel, but one end will be longer than the other so that the top and bottom are convergent (non-parallel). The two upper angles will be 900 with one acute, and one obtuse angle at the bottom. | 677.169 | 1 |
Exploring Triangles
Young children's understanding of triangles can sometimes be rigid and fixed. They are accustomed to seeing equilateral triangles in a point-up orientation.
If the shape is inverted or is no longer equilateral children may think that it is not a triangle.
Our role is to teach the children the essential features of a triangle, expose them to diverse images of triangles, and provide opportunities for them to build and create a variety of triangles.
Triangles are closed 2-dimensional shapes with 3 sides and 3 angles.
Here is one possible activity. Use straws, coffee stirrers, pencils, and more. Any straight materials in varying lengths will do!
Provide a defined workspace with:
A numeral "3" (with dot set) card
A variety of lengths of sticks
A books to draw and record their triangles
Designing prompts in the shape of differing triangles serves to reinforce the lesson. As children create and explore, use "math talk" to introduce and reinforce key vocabulary and concepts: sides, angle, point, and closed shape. There may also be some problem-solving to discuss, as two very short sides with one very long side may not make a closed shape.
If you have plenty of materials or use paper strip sticks, children can glue down their triangles and make a triangle collage! | 677.169 | 1 |
Euclid's Elements [book 1-6] with corrections, by J.R. Young
From inside the book
Results 6-10 of 53
Page 37 ... parallel to the same straight line . We have already seen ( Prop . xxvii . ) that one ( CD last propo- sition ) will be parallel to another ( AB ) provided a line which cuts both makes interior angles on the same side together equal to ...
Page 38 ... parallel straight lines , it makes the alternate angles equal to one another ; and the exterior angle equal to the interior and opposite upon the same side ; and likewise the two interior angles upon the same side together equal to two ...
Page 39 ... parallel to the same straight line are parallel to each other . Let AB , CD be each of them parallel to EF : AB shall be parallel to CD . Let the straight line GHK cut AB , EF , CD : and because GHK cuts the parallel straight lines AB ...
Page 40 ... parallel to BC . Because the straight line AD , which meets the two straight lines BC , EF , makes the alternate angles EAD , ADC equal to one another , EF is parallel to BC . Therefore the straight line EAF is drawn through the given ...
Popular passages
Page 30Page 105Page 61Page 65 - If a straight line be divided into any two parts, four times the rectangle contained by the whole line and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part.
Page 38 - If a, straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite upon the same side; and likewise the two interior angles upon the same side together equal to two right angles. | 677.169 | 1 |
What is a Trigonometry Table? and How to Create it?
A trigonometry table is a reference chart for trigonometry, a branch of math studying angles and triangles. It lists the values of sine (sin), cosine (cos), tangent (tan), and other trigonometric functions for different angles, from 0° to 360°.
Before calculators, these tables were important for solving problems in navigation, science, and engineering. They were so important that they even led to the creation of the first mechanical calculators!
Trigonometry tables are especially useful for right triangles (triangles with a 90° angle). By looking up the angle and the relevant function (sin, cos, etc.), you can find the ratio of the triangle's sides. This makes solving many geometrical problems much easier.
While calculators have made them less common, trigonometry tables are still a valuable tool for understanding these functions and their applications.Let us read this article to clear your doubts regarding trigonometry table.
What is a Trigonometry Table?
Tables of trigonometric functions are helpful in many different areas of mathematics. Prior to the invention of pocket calculators, trigonometric tables were important for scientific research, engineering, and navigation.Let's have a look at the specific angle value-based trigonometry formulas:
θ
0°
30°
45°
60°
90°
180°
Sin
0
1/2
1/2
3/2
1
0
Cos
1
3/2
1/2
1/2
0
-1
Tan
0
1/3
1
3
∞
0
Cot
∞
3
1
1/3
0
∞
Sec
1
2/3
2
2
∞
-1
Cosec
∞
2
2
2/3
1
∞
How to Remember a Trigonometry Table?
Don't worry about memorizing the trigonometry table! With a few important formulas, you can learn the table in no time.
These formulas relate the trigonometric values (sine, cosine, tangent) of one angle to another. The magic number? 90 degrees!
Here's the trick to solve such questions based on trigonometry table :
To find the sine, cosine, or tangent of any angle, subtract it from 90 degrees and look up that value in your table. The corresponding sine becomes your cosine, cosine becomes sine, and tangent becomes cotangent (and vice versa).
By memorising these relationships and just a few table values, you can derive the rest! This technique is much easier than rote memorization and opens the door to solving many trigonometry problems. | 677.169 | 1 |
If the side lengths of a quadrilateral form a harmonic progression in the ratio 1 : 1/(1+d) : 1/(1+2d) : 1/(1+3d) where d is the common difference between the denominators of the harmonic progression, then the triangle inequality condition requires that d be in the range f < d < g, where f = -0.257772801... and is the greatest root of the equation: 2 + 12d + 18d^2 + 6d^3 = 0. The value of g is given in A199589. | 677.169 | 1 |
If XY=YZ and angle Y measures 90 degrees in the figure above, which of the following CANNOT be concluded?
XZ < XY
a=b
a+b=90
\(X Z^2 = X Y^2 + Y Z^2 \)
Detailed Explanation
If XY=YZ and angle Y measures 90 degrees, then the triangle XYZ is a right isosceles triangle. This means that all three sides of the triangle are congruent and that angle Y is a right angle.
Therefore, we can conclude that:
XZ is congruent to XY and YZ
a+b=90
a=b
However, we cannot conclude that XZ is less than XY.
Take more free practice tests for other ASVAB topics with our ASVAB prep now! | 677.169 | 1 |
Vectors, magitude, scalar components
In summary, the displacement vectors A, B, and C are given with their corresponding scalar components and magnitudes. To determine which two vectors are equal, we can compare their magnitudes. It is given that vector A and vector C have the same magnitude of 100.0 m, and to support this choice, we can calculate that both vectors are directed at an angle of 36.9 degrees above the + x axis. Therefore, vector A and vector C are the two equal vectors.
Jan 28, 2009
#1
brittbc
2
0
Homework Statement
the displacement vector A has scalr omponents of Ax= 80.0 m and Ay= 60.0 m. the displacement vector B has a scalr component Bx= 60.0 m and a magnitude of B=75.0m. The displacement vecto C has a magnitude of C= 100.0 m and is directed at an angle of 36.9 degrees above the + x axis. Two of these vectors are equal. determine which two, and support your choice with a calculation.
So vector A and vector C have the same magnitude. Can you show they're both 36.9 degrees above the x-axis.
Feb 4, 2009
#3
thomate1
1,346
0
The two vectors that are equal are A and C. This can be determined by calculating the magnitude of vector A using the Pythagorean theorem: A = √(Ax^2 + Ay^2) = √(80.0m^2 + 60.0m^2) = √(6400m^2 + 3600m^2) = √10000m^2 = 100m. This is the same magnitude as vector C, which is also 100m. Additionally, the angle of vector C, 36.9 degrees, is the same as the angle between the +x axis and vector A, making them equal in direction as well. Therefore, vector A and vector C are equal.
Related to Vectors, magitude, scalar components
1. What is a vector?
A vector is a mathematical object that has both magnitude and direction. It is represented by an arrow pointing in the direction of its magnitude, and its length represents its magnitude.
2. How is the magnitude of a vector calculated?
The magnitude of a vector is calculated using the Pythagorean theorem, which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. In the case of a vector, the magnitude is equal to the square root of the sum of the squares of its components.
3. What are scalar components?
Scalar components are the individual parts of a vector that are parallel to the x and y axes. They are used to represent the magnitude and direction of a vector in a two-dimensional space.
4. How are vector components calculated?
Vector components can be calculated using trigonometric functions, specifically the cosine and sine functions. The x-component is equal to the magnitude of the vector multiplied by the cosine of the angle between the vector and the x-axis, while the y-component is equal to the magnitude of the vector multiplied by the sine of the angle.
5. What is the difference between a scalar and a vector?
A scalar is a quantity that has only magnitude, while a vector has both magnitude and direction. Examples of scalars include temperature, mass, and time, while examples of vectors include displacement, velocity, and force. | 677.169 | 1 |
What Is True About Every Rotation?
What Is True About Every Rotation??
What is true about every rotation? The angles in the image and pre-image are congruent.
How do you describe rotation?
A rotation is a turn of a shape. A rotation is described by the centre of rotation the angle of rotation and the direction of the turn. … The direction of rotation can be described as clockwise (CW) or counterclockwise (CCW). For example the shape below is rotated 90° CW about vertex A.
What is rotation in math definition?
A rotation is a type of transformation that takes each point in a figure and rotates it a certain number of degrees around a given point.
How many types of rotations are there?
Rotation Formula
Type of Rotation
A point on the Image
A point on the Image after Rotation
Rotation of 90° (Clockwise)
(x y)
(y -x)
Rotation of 90° (Counter Clockwise)
(x y)
(-y x)
Rotation of 180° (Both Clockwise and Counterclockwise)
(x y)
(-x -y)
Rotation of 270° (Clockwise)
(x y)
(-y x)
See alsohow often do neap tides occur?
What is rotation example noun.
What are the 3 things needed to describe rotation?
Three pieces of information are needed to rotate a shape:
the centre of rotation.
the angle of rotation.
the direction of rotation.
How do you do rotation?
What best describes a rotation?
noun. the act of rotating a turning around as on an axis. Astronomy. the movement or path of the earth or a heavenly body turning on its axis. one complete turn of such a body.
What is rotation short answer?
"Rotation" refers to an object's spinning motion about its own axis. … For example Earth rotates on its own axis producing the 24-hour day. Earth revolves about the Sun producing the 365-day year.
What is the point of rotation?
The point of rotation is the central point around which a figure is rotated.
What does rotation cause describe?
and away from the sun. c) effects of earth's revolution include the seasons and variations in the length of the days and nights. the changes in hours of daylight and in temperature caused by revolution. d) it is above.
Where do you see rotations in the real world?
Real life examples of rotations are:
strumming a guitar.
hitting a nail with a hammer.
walking.
picking up a cup.
the agitation of a washing machine.
a ferris wheel turning.
the opening and closing of your heart valves to control blood flow.
Why do things rotate?
Our planets have continued spinning because of inertia. In the vacuum of space spinning objects maintain their momentum and direction — their spin — because no external forces have been applied to stop them. And so the world — and the rest of the planets in our solar system — keeps spinning.
How are rotations used in real life?
The sprocket of a bicycle rotates while riding the bike and pushing the pedals. While pushing the pedals the sprocket rotates the chain which for the chain too!! In a bike there are 2 wheels that rotate in any direction too and can negatively rotate to.
What is a rotation easy definition?
Definition of rotation
1a(1) : the action or process of rotating on or as if on an axis or center. (2) : the act or an instance of rotating something. b : one complete turn : the angular displacement required to return a rotating body or figure to its original orientation.
What is rotation answer in one word?
Rotation is circular movement. A rotation is the movement of something through one complete circle. …the daily rotation of the Earth upon its axis. [ + of] Synonyms: revolution turning turn wheel More Synonyms of rotation.
How do you describe the shape of a rotation?
Rotating shapes means moving them around a fixed point (clockwise or anticlockwise and by a certain number of degrees). The shape itself stays exactly the same but its position in the space will change.
See alsowhy was education important to the new england colonies
What does rotation mean in science?
Rotation describes the circular motion of an object around its center. There are different ways things can rotate. 6 – 12+ Astronomy Geography.
Is a positive rotation clockwise?
Conventionally positive angle measures describe counterclockwise rotations. If we want to describe a clockwise rotation we use negative angle measures. For example here's the result of rotating a point about P by −30° .
Why is food rotation important?
Rotating foods and eating a varied diet also helps increase the diversity of good bacteria in your gut. Greater diversity in gut bacteria is associated not only with better overall health but also with weight loss.
What does it mean to rotate about the origin?
A rotation is a type of geometrical transformation in which the vertices of a shape are rotated at a certain angle around a fixed point (called the center of rotation). … Usually you will be asked to rotate a shape around the origin which is the point (0 0) on a coordinate plane.
What is the rotation of the earth?
Earth rotates eastward in prograde motion. As viewed from the north pole star Polaris Earth turns counterclockwise. The North Pole also known as the Geographic North Pole or Terrestrial North Pole is the point in the Northern Hemisphere where Earth's axis of rotation meets its surface.
What is rotation give its two characteristics?
Two characteristics of rotation are the complete rotation of the earth from west to east in 24 hours and along its tilted axis which is always in the same direction towards the Pole Star. Question 4.
What is rotation and how is it used in sports?
In the Olympic Games athletes often need to rotate in order to compete in their sports. Gymnasts rotate their bodies during routines ice skaters rotate during their spins and aerial skier perform rotations high in the air.
What is the importance of rotation and revolution?
The rotation of the Earth causes the day and the night. The tides are deflected due to the rotation. The speed of rotation also affects the movement of the wind. The revolution of the Earth on its axis is important in setting up the seasons.
How often does the Earth rotate?
once every 23 hours 56
Consider the movement of the earth's surface with respect to the planet's center. The earth rotates once every 23 hours 56 minutes and 4.09053 seconds called the sidereal period and its circumference is roughly 40 075 kilometers.
What is the result of the rotation and revolution of the earth?
The spinning of the Earth causes day to turn to night while the full rotation/the revolution of the Earth causes summer to become winter. Combined the spinning and the revolution of the Earth causes our daily weather and global climate by affecting wind direction temperature ocean currents and precipitation.
What are the properties that identify a rotation?
The following are the three basic properties of rotations :
A rotation maps a line to a line a ray to a ray a segment to a segment and an angle to an angle.
A rotation preserves lengths of segments.
A rotation preserves measures of angles.
See alsowhat are ecosystems and biomes
How do u find the point of rotation?
How does the motion of Earth affect our life?
Earth's rotation is the cause for the differences in daytime and nighttime as it spins on its axis. … The axis tilt doesn't actually change but its orientation relative to the Sun changes as Earth moves in orbital revolution around the Sun. This motion combined with the axis tilt is responsible for our seasons
What is a real life example of a geometric rotation?
A real life example of a rotation would be the Earth's pivoting on its tilted axis – which is the center of rotation – at an angle of about 23.5 degrees.
What is a real world example of a rotation in geometry?
One real life example of a transformation would be the wheels on a car which is a rotation. The wheels move 360 degrees and keep moving. The wheels can move 90 180 270 or 360 degrees depending on various factors such as the speed of car or the size of the wheel.
What is a real life example of a translation?
Real life examples of translations are: the movement of an aircraft as it moves across the sky. the lever action of a tap (faucet) sewing with a sewing machine.
The Bizarre Behavior of Rotating Bodies
How I Got Honors In EVERY Rotation At Yale Medical School. How to Get Strong Evaluations | Medbros
The Rotation Mistakes of Every Rank in Rocket League
Wolfoo Learns About the Rotation of the Earth – Educational Videos for Kids | Wolfoo Family | 677.169 | 1 |
Contents
Problem
The area of triangle is 8 square inches. Points and are midpoints of congruent segments and . Altitude bisects . The area (in square inches) of the shaded region is
Solution 3
We know the area of triangle is square inches. The area of a triangle can also be represented as or in this problem . By solving, we have
With SAS congruence, triangles and are congruent. Hence, triangle . (Let's say point is the intersection between line segments and .) We can find the area of the trapezoid by subtracting the area of triangle from .
We find the area of triangle by the formula- . is of from solution 1. The area of is . | 677.169 | 1 |
Elements of Geometry: With Exercises for Students and an Introduction to Modern Geometry
Excerpt from Elements of Geometry: With Exercises for Students and an Introduction to Modern Geometry
A new treatise on Geometry, to be of sufficient merit to claim attention, must be both conservative and progressive. It should lay firm hold on the past, embody the present state of the science, and anticipate future developments. A work claiming to be wholly new might, perhaps with justice, be at once discarded as worthless; while one containing no improvements could not justify its own existence.
The geometrical objects, - points, lines, surfaces, solids, and angles, - constitute the subject-matter of the science; the definitions are the tests by which these objects are discriminated and their classification determined; the axioms are the warrants for the steps taken in the course of demonstration; the postulates justify the assumption of magnitudes having any position, form, and extent.
The logical principles which underlie the demonstrations of this volume have been carefully discriminated and illustrated. The discussion of the axioms and postulates is the result of research, and intent and prolonged thought. That fundamental principles have been reached is manifest from their underivability, and the simplicity of the deduction of the ordinary so-called axioms from them as corollaries.
Mr. Bain has observed of the principle, If A be greater than B, and B greater than C, much more is A greater than C: "If it can not be deductively inferred from the proper axioms, it will have to be received as a third axiom." Not only can this principle be inferred (23, 20), but also Mr. Bain's so-called proper axioms (23, 3, 6). | 677.169 | 1 |
In rhombus ABCD, the measure of angle A and the measure of angle B are in the ratio of 2:1, AB=2x+8, and BC=5x+10. What is the measure of angle A? What is the measure of angle B? What is the perimeter of rhombus ABCD | 677.169 | 1 |
This type of polygon is called a hexagon. "Hex-" is a greek
root, meaning 6. A polygon is a geometric plane figure consisting
of at least 3 straight sides and angles, such as a triangle or
square. Because "hex-" means 6, you know that a polygon that has 6
sides and 6 angles is a hexagon. As another example, "octo-" means
8, so a polygon that has 8 sides and 8 angles would be called an
octagon. | 677.169 | 1 |
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.
Updated by Tiwari Academy
on December 2, 2023, 12:08 PM
To prove that the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the center:
Tangents and Radius: The radius of a circle is perpendicular to the tangent at the point of contact. Therefore, the radii drawn to the points of contact form right angles with the tangents.
External Angle of Triangle: The angle between the two tangents is the external angle of the triangle formed by the two radii and the line segment joining the points of contact.
External Angle Theorem: The external angle is equal to the sum of the opposite internal angles in a triangle.
Right Angles: Since the radii form right angles with the tangents, the two opposite internal angles are right angles (90° each).
Sum of Angles: The sum of the two right angles is 180°.
Conclusion: Therefore, the angle between the two tangents (external angle) is supplementary (180°) to the angle subtended by the line segment at the center.
Thus, the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the center.
Let's discuss in detail
Tangents and Circle Geometry
In the study of circle geometry, one of the intriguing aspects is the relationship between tangents drawn from an external point and the angles these tangents form. A fascinating theorem in this context states that the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the center of the circle. This theorem not only enriches our understanding of circle geometry but also demonstrates the elegant interplay between tangents, angles, and radii.
Understanding Tangents from an External Point
A tangent to a circle is a line that touches the circle at exactly one point, known as the point of tangency. When two tangents are drawn from an external point to a circle, they touch the circle at two distinct points. These tangents, despite their individual straight paths, form an angle at the point from where they are drawn. This angle is crucial to our theorem and is the focus of our proof.
The Role of Radii to the Points of Tangency
In our scenario, when we draw radii from the center of the circle to the points of tangency, we create two right angles where the radii meet the tangents. This is because a radius of a circle is always perpendicular to the tangent at the point of tangency. These right angles are key to understanding the relationship between the angle formed by the tangents and the angle subtended at the center of the circle.
Triangle Formation and External Angle Theorem
The two radii and the line segment joining the points of tangency form a triangle within the circle. The angle between the two tangents, drawn from the external point, acts as an external angle to this triangle. According to the external angle theorem in geometry, an external angle of a triangle is equal to the sum of the two opposite internal angles.
Calculating the Supplementary Angles
In our triangle, the two internal angles are right angles (90° each) because they are formed by the radii perpendicular to the tangents. The sum of these right angles is 180°. Therefore, the external angle, which is the angle between the two tangents, is also 180°. This means that the angle between the tangents is supplementary to the angle subtended by the line segment at the center of the circle.
Supplementary Angles in Circle Geometry
Thus, we conclude that the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the center. This theorem elegantly illustrates the harmonious relationships in circle geometry, highlighting how tangents, radii, and angles cohesively interact to form consistent and predictable geometric patterns. It is a testament to the beauty and logical structure of mathematics in the realm of geometry. | 677.169 | 1 |
They're the same thing, it's just that the former is expressed in terms of x (which is the hypotenuse). But first, you have to be careful: since we're only dealing with ratios, you can order the three sides as you please, but it's better to order from shortest to longest side, otherwise it might get confusing. So here, it would be better to say x/2 : (sqrt(3)/2)x : x. This is consistent with 1 : sqrt(3) : 2, shortest to longest. Also, you might or might not be a bit confused by the notation I used, but I'm sure if you compare it with what you know about the ratios, you'll be fine. But I'd suggest you to be careful with how you write mathematical expressions; you wrote "square root of 3 over 2 times x", but this is actually very ambiguous and can be interpreted in at least two ways, one of which means something pretty different from what you want!
Anyway, as I said, they're the same. If you take a closer look at 1 : sqrt(3) : 2, you'll notice that the length of the longest side, which is 2, is two times that of the shortest side, which is 1. So if you say the longest side is x, then the shortest side is x/2. Also, we know that the ratio of the shortest side to the middle one is 1 : sqrt(3). In other words, you have to multiply the shortest side by sqrt(3) in order to get the middle side. So:
The 30-60-90 ratio states that if the side across from 30* angle is x, then the side across from 60 will be x*√3 and the one across from the 90* will be 2x. Therefore, if x is one, then the side across from 60 will be 1*√3 = √3
At the very end, the perimeter was 1/sqrt3 + sqrt3 + 2, then you multiplied by sqrt3/sqrt3 (1) to make 1/sqrt3 into sqrt3 / 3. So with the rest of the numbers, how did sqrt3 become 3 times sqrt3 / 3, what did you multiply the sqrt 3 by?
Both fractions are correct and equivalent to each other. 2/√3 is what you get by multiplying the numerator by a 2 (and a denominator by 1), while (2•√3)/3 is what the expression will look like after you rationalize the denominator.
We know that the quadrilateral has 4 right angles, meaning that it must be a rectangle or a square. Squares and rectangles both have the property that two opposite sides must have the same length (But squares have the additional element of all sides having the same length).
In this problem, we already established that it is either a square or a rectangle. Since we know side AB has a length of 1, and that side DC is opposite of side AB, we also know that side DC has a length of 1. We also know that side AD will have the same length as side BC, using the same property of rectangles.
I am having trouble understanding how to solve a problem if you're given that the hypotenuse is x and the triangle is a 30-60-90 triangle and the side adjacent to the 30 degree angle is 15 units long. How can I find the hypotenuse? Is there a video on this?
Video transcript
So we have this rectangle
right over here, and we're told that the
length of AB is equal to 1. So that's labeled
right over there. AB is equal to 1. And then they tell us that
BE and BD trisect angle ABC. So BE and BD trisect angle ABC. So trisect means dividing
it into 3 equal angles. So that means that this
angle is equal to this angle is equal to that angle. And what they want
us to figure out is, what is the perimeter
of triangle BED? So it's kind of
this middle triangle in the rectangle
right over here. So at first this seems
like a pretty hard problem, because you're like well, what
is the width of this rectangle. How can I even start on this? They've only given
us one side here. But they've actually given
us a lot of information, given that we do know
this is a rectangle. We have four sides, and
that we have four angles. The sides are all
parallel to each other and that the angles
are all 90 degrees. Which is more than
enough information to know that this is
definitely a rectangle. And so one thing we do
know is that opposite sides of a rectangle are
the same length. So if this side is 1, then
this side right over there is also 1. The other thing we know is
that this angle is trisected. Now we know what the
measure of this angle is. It was a right angle, it
was a 90 degree angle. So if it's divided into three
equal parts, that tells us that this angle right
over here is 30 degrees, this angle right over
here is 30 degrees, and then this angle right
over here is 30 degrees. And then we see
that we're dealing with a couple of
30-60-90 triangles. This one is 30, 90, so this
other side right over here needs to be 60 degrees. This triangle right over here,
you have 30, you have 90, so this one has
to be 60 degrees. They have to add up to
180, 30-60-90 triangle. And you can also figure out
the measures of this triangle, although it's not going
to be a right triangle. But knowing what we know
about 30-60-90 triangles, if we just have
one side of them, we can actually figure
out the other sides. So for example, here we
have the shortest side. We have the side opposite
of the 30 degree side. Now, if the 30 degree side
is 1, then the 60 degree side is going to be square
root of 3 times that. So this length
right over here is going to be square root of 3. And that's pretty useful
because we now just figured out the length of the entire
base of this rectangle right over there. And we just used our knowledge
of 30-60-90 triangles. If that was a little
bit mysterious, how I came up with
that, I encourage you to watch that video. We know that 30-60-90
triangles, their sides are in the ratio of 1 to
square root of 3 to 2. So this is 1, this
is a 30 degree side, this is going to be square
root of 3 times that. And the hypotenuse
right over here is going to be 2 times that. So this length
right over here is going to be 2 times this
side right over here. So 2 times 1 is just 2. So that's pretty interesting. Let's see if we can
do something similar with this side right over here. Here the 1 is not the side
opposite the 30 degree side. Here the 1 is the side
opposite the 60 degree side. So once again, if we
multiply this side times square root of 3, we
should get this side right over here. This is the 60, remember this
1, this is the 60 degree side. So this has to be 1 square
root of 3 of this side. Let me write this down, 1
over the square root of 3. And the whole reason, the
way I was able to get this is, well, whatever this
side, if I multiply it by the square root of 3, I
should get this side right over here. I should get the 60
degree side, the side opposite the 60 degree angle. Or if I take the 60 degree
side, if I divide it by the square root of 3 I should
get the shortest side, the 30 degree side. So if I start with the
60 degree side, divide by the square root of 3, I
get that right over there. And then the
hypotenuse is always going to be twice the side
opposite the 30 degree angle. So this is the side opposite
the 30 degree angle. The hypotenuse is
always twice that. So this is the side opposite
the 30 degree angle. The hypotenuse is
going to be twice that. It is going to be 2 over
the square root of 3. So we're doing pretty good. We have to figure
out the perimeter of this inner triangle
right over here. We already figured
out one length is 2. We figured out another length
is 2 square roots of 3. And then all we have to really
figure out is, what ED is. And we can do that
because we know that AD is going to be
the same thing as BC. We know that this entire
length, because we're dealing with a rectangle,
is the square root of 3. If that entire length
is square root of 3, if this AE is 1 over
the square root of 3, then this length
right over here, ED is going to be
square root of 3 minus 1 over the square root of 3. That length minus that
length right over there. And how to find the perimeter
is pretty straight forward. We just have to add these
things up and simplify it. So it's going to be,
just let me write this, perimeter
of triangle BED is equal to-- This is
short for perimeter. I just didn't feel like
writing the whole word.-- is equal to 2 over the square
root of 3 plus square root of 3 minus 1 over the square
root of 3 plus 2. And now this just boils down
to simplifying radicals. You could take a
calculator out and get some type of decimal
approximation for it. Let's see, if we have 2 square
root of 3 minus 1 square root of 3, that will leave us with
1 over the square root of 3. 2 over the square of 3 minus
1 over the square root of 3 is 1 over the square root of 3. And then you have the
square root of 3 plus 2. And let's see, I can
rationalize this. If I multiply the numerator
and the denominator by the square root of 3,
this gives me the square root of 3 over 3 plus the
square root of 3, which I could rewrite
that as plus 3 square roots of 3 over 3. Right? I just multiplied this
times 3 over 3 plus 2. And so this gives us-- this
is the drum roll part now-- so one square root of 3
plus 3 square roots of 3, and all of that over 3, gives
us 4 square roots of 3 over 3 plus 2. Or you could put the 2 first. Some people like to write
the non-irrational part before the irrational part. But we're done. We figured out the perimeter. We figured out the perimeter
of this inner triangle BED, right there. | 677.169 | 1 |
Tables of Trigonometric Functions in Non-Sexagesimal Arguments Excluding the ordinary tables of trigonometric functions in sexagesimal arguments the two principal groups of such tables are those with arguments in A. Radians,—tables of this type have been already listed in RMT 81; and B. GradesThe trigonometric identities are used for solving trigonometric equations. The trigonometric identities are based on the trigonometric functions of one or more angles. The pythagorean trig identities chart given here expresses the six trigonometric functions in terms of other five functions.That is the trigonometric identities table helps to express any trigonometric function in terms of any
2021-02-23 · The trigonometric functions are really all around us! Trig calculator finding sin, cos, tan, cot, sec, csc To find the trigonometric functions of an angle, enter the chosen angle in degrees or radians. | 677.169 | 1 |
what is a corresponding sides | 677.169 | 1 |
8_6
8_6
Prove that the locus of points from which the two tangents to E are perpendicular is the circle .
Solution:
I first created a slider for a and b. Then I constructed a circle using the equation then I constructed the eclipse using the equation . I constructed a point on the circle and the tangent lines from the eclipse through A and the intersection of the tangent lines and c. I then constructed the angle between tangent 1, A, and tangent 2. | 677.169 | 1 |
References
2018a
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth. Lines are an idealization of such objects. Until the 17th century, lines were defined in this manner: "The [straight or curved] line is the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width. […] The straight line is that which is equally extended between its points." [1]
Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as non-Euclidean, projective and affine geometry).
In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it.
When a geometry is described by a set of axioms, the notion of a line is usually left undefined (a so-called primitive object). The properties of lines are then determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry. Thus in differential geometry a line may be interpreted as a geodesic (shortest path between points), while in some projective geometries a line is a 2-dimensional vector space (all linear combinations of two independent vectors). This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line.
A line is uniquely determined by two points, and the line passing through points [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] is denoted [math]\displaystyle{ \longleftrightarrow }[/math]; [math]\displaystyle{ \overleftrightarrow{AB} }[/math]. Similarly, the length of the finiteline segment terminating at these points may be denoted [math]\displaystyle{ \overline{AB} }[/math]. A line may also be denoted with a single lower-case letter (Jurgensen et al. 1963, p. 22). | 677.169 | 1 |
Using the Distance Formula
Derived from the Pythagorean Theorem, the distance formula is used to find the distance between two points in the plane. The Pythagorean Theorem, [latex]{a}^{2}+{b}^{2}={c}^{2}[/latex], is based on a right triangle where a and b are the lengths of the legs adjacent to the right angle, and c is the length of the hypotenuse.
Figure 14
The relationship of sides [latex]|{x}_{2}-{x}_{1}|[/latex] and [latex]|{y}_{2}-{y}_{1}|[/latex] to side d is the same as that of sides a and b to side c. We use the absolute value symbol to indicate that the length is a positive number because the absolute value of any number is positive. (For example, [latex]|-3|=3[/latex]. ) The symbols [latex]|{x}_{2}-{x}_{1}|[/latex] and [latex]|{y}_{2}-{y}_{1}|[/latex] indicate that the lengths of the sides of the triangle are positive. To find the length c, take the square root of both sides of the Pythagorean Theorem.
Find the distance between two points: [latex]\left(1,4\right)[/latex] and [latex]\left(11,9\right)[/latex].
[latex]\sqrt{125}=5\sqrt{5}[/latex]
Example 6: Finding the Distance between Two Locations
Let's return to the situation introduced at the beginning of this section.
Tracie set out from Elmhurst, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot in Figure 1. Find the total distance that Tracie traveled. Compare this with the distance between her starting and final positions.
Solution
The first thing we should do is identify ordered pairs to describe each position. If we set the starting position at the origin, we can identify each of the other points by counting units east (right) and north (up) on the grid. For example, the first stop is 1 block east and 1 block north, so it is at [latex]\left(1,1\right)[/latex]. The next stop is 5 blocks to the east, so it is at [latex]\left(5,1\right)[/latex]. After that, she traveled 3 blocks east and 2 blocks north to [latex]\left(8,3\right)[/latex]. Lastly, she traveled 4 blocks north to [latex]\left(8,7\right)[/latex]. We can label these points on the grid as in Figure 16.
Figure 16
Next, we can calculate the distance. Note that each grid unit represents 1,000 feet.
From her starting location to her first stop at [latex]\left(1,1\right)[/latex], Tracie might have driven north 1,000 feet and then east 1,000 feet, or vice versa. Either way, she drove 2,000 feet to her first stop.
Her second stop is at [latex]\left(5,1\right)[/latex]. So from [latex]\left(1,1\right)[/latex] to [latex]\left(5,1\right)[/latex], Tracie drove east 4,000 feet.
Her third stop is at [latex]\left(8,3\right)[/latex]. There are a number of routes from [latex]\left(5,1\right)[/latex] to [latex]\left(8,3\right)[/latex]. Whatever route Tracie decided to use, the distance is the same, as there are no angular streets between the two points. Let's say she drove east 3,000 feet and then north 2,000 feet for a total of 5,000 feet.
Tracie's final stop is at [latex]\left(8,7\right)[/latex]. This is a straight drive north from [latex]\left(8,3\right)[/latex] for a total of 4,000 feet.
Next, we will add the distances listed in the table.
From/To
Number of Feet Driven
[latex]\left(0,0\right)[/latex] to [latex]\left(1,1\right)[/latex]
2,000
[latex]\left(1,1\right)[/latex] to [latex]\left(5,1\right)[/latex]
4,000
[latex]\left(5,1\right)[/latex] to [latex]\left(8,3\right)[/latex]
5,000
[latex]\left(8,3\right)[/latex] to [latex]\left(8,7\right)[/latex]
4,000
Total
15,000
The total distance Tracie drove is 15,000 feet, or 2.84 miles. This is not, however, the actual distance between her starting and ending positions. To find this distance, we can use the distance formula between the points [latex]\left(0,0\right)[/latex] and [latex]\left(8,7\right)[/latex].
At 1,000 feet per grid unit, the distance between Elmhurst, IL, to Franklin Park is 10,630.14 feet, or 2.01 miles. The distance formula results in a shorter calculation because it is based on the hypotenuse of a right triangle, a straight diagonal from the origin to the point [latex]\left(8,7\right)[/latex]. Perhaps you have heard the saying "as the crow flies," which means the shortest distance between two points because a crow can fly in a straight line even though a person on the ground has to travel a longer distance on existing roadways. | 677.169 | 1 |
5. ABC is any triangle, D a point in AB; find a point E in BC produced such that ▲ DBE = ▲ ABC.
PROPOSITION 38. THEOREM.
Triangles on equal bases and between the same parallels are equal in area.
Let ABC, DEF be triangles on equal bases BC, EF, and between the same parallels AD, BF:
it is required to prove ▲ ABC
=
Δ DEF.
Through B draw BG || AC, and through F draw FH || DE;
and let them meet AD produced at G and H.
Then GBCA, DEFH are ||ms;
I. 31
|| DEFH, being on equal bases BC, EF,
COR. The straight line joining any vertex of a triangle to the middle point of the opposite side bisects the triangle. Hence the theorem: If two triangles have two sides of the one respectively equal to two sides of the other and the contained anglès supplementary, the triangles are equal
in area.
1. Of two triangles which are between the same parallels, that is the greater which stands on the greater base.
2. State and prove a converse of the last deduction.
3. Two triangles are between the same parallels, and the base of the first is double the base of the second; prove the first triangle double the second.
4. The four triangles into which the diagonals divide a ||m are equal. 5. If one diagonal of a quadrilateral bisects the other diagonal, it also bisects the quadrilateral.
6. ABCD is a ||m; E is any point in AD or AD produced, and F any point in BC or BC produced; AF, DF, BE, CE are joined. Prove ▲ AFD= = A BEC.
7. ABC is any triangle; L and K are the middle points of AB and AC; BK and CL are drawn intersecting at G, and AG is joined. Prove ▲ BGC = =AAGC = ▲ AGB. 8. ABCD is a ||m; P is any point in the diagonal BD or BD produced, and PA, PC are joined. Prove ▲ PAB = A PCB, and ▲ PAD = ▲ PCD.
9. Bisect a triangle by a straight line drawn from a given point in
one of the sides.
Equal triangles on the same side of the same base are between the same parallels.
Let As ABC, DBC on the same side of the same base BC be equal, and let AD be joined :
it is required to prove AD || BC.
If AD is not || BC, through A draw AE || BC, meeting BD, or BD produced, at E, and join EC.
I. 31
which is impossible, since the one is a part of the other. ... AD is || BC.
I. 37
Hyp.
1. The straight line joining the middle points of two sides of a triangle is the third side, and half of it.
=
2. Hence prove that the straight line
joining the middle point of the hypotenuse of a right-angled triangle to the opposite half the hypotenuse.
vertex =
=
3. The middle points of the sides of any quadrilateral are the vertices of a m, whose perimeter the sum of the diagonals of the quadrilateral. When will this m be a rectangle, a rhombus, a square?
4. If two equal triangles be on the same base, but on opposite sides of it, the straight line which joins their vertices will be bisected by the base.
5. Use the first deduction to solve I. 31.
6. In the figure to I. 16, prove AF || BC.
7. If a quadrilateral be bisected by each of its diagonals, it is a ||TM.
8. Divide a given triangle into four triangles which shall be equal in every respect. | 677.169 | 1 |
Arbelos
Summary
The
Two of the semicircles are necessarily concave, with arbitrary diameters a and b; the third semicircle is convex, with diameter a+b.
The area of the arbelos is equal to the area of a circle with diameter .
Proof: For the proof, reflect the arbelos over the line through the points B and C, and observe that twice the area of the arbelos is what remains when the areas of the two smaller circles (with diameters , ) are subtracted from the area of the large circle (with diameter ). Since the area of a circle is proportional to the square of the diameter (Euclid's Elements, Book XII, Proposition 2; we do not need to know that the constant of proportionality is pi/4), the problem reduces to showing that . The length equals the sum of the lengths and , so this equation simplifies algebraically to the statement that . Thus the claim is that the length of the segment is the geometric mean of the lengths of the segments and . Now (see Figure) the triangle BHC, being inscribed in the semicircle, has a right angle at the point H (Euclid, Book III, Proposition 31), and consequently is indeed a "mean proportional" between and (Euclid, Book VI, Proposition 8, Porism). This proof approximates the ancient Greek argument; Harold P. Boas cites a paper of Roger B. Nelsen who implemented the idea as the following proof without words.
Let D and E be the points where the segments and intersect the semicircles AB and AC, respectively. The quadrilateral ADHE is actually a rectangle the twin circles are two special circles associated with an arbelos. An arbelos is determined by three collinear points A, B, and C, and is the curvilinear triangular region between the three semicircles that have AB, BC, and AC as their diameters. If the arbelos is partitioned into two smaller regions by a line segment through the middle point of A, B, and C, perpendicular to line ABC, then each of the two twin circles lies within one of these two regions, tangent to its two semicircular sides and to the splitting segment.
In geometry, an Archimedean circle is any circle constructed from an arbelos that has the same radius as each of Archimedes' twin circles. If the arbelos is normed such that the diameter of its outer (largest) half circle has a length of 1 and r denotes the radiius of any of the inner half circles, then the radius ρ of such an Archimedean circle is given by There are over fifty different known ways to construct Archimedean circles. An Archimedean circle was first constructed by Archimedes in his Book of Lemmas. The | 677.169 | 1 |
Description: Define the class of
planar incidence geometries. We use Hilbert's
axioms and adapt them to planar geometry. We use ∈ for the
incidence relation. We could have used a generic binary relation, but
using ∈ allows us to reuse previous
results. Much of what follows
is directly borrowed from Aitken, Incidence-Betweenness Geometry,
2008,
The class Plig is the class of planar incidence
geometries, where a
planar incidence geometry 𝑥 is defined as a set of lines 𝑙
satisfying three axioms. In the definition below, ∪𝑥 is the union
of lines, that is, the plane, and 𝑎, 𝑏, 𝑐 denote points.
Therefore, the axioms are: for all pairs of (distinct) points, there
exists a unique line containing them; all lines contain at least two
points; there exist three non-collinear points. | 677.169 | 1 |
Parallel Lines And Transversals Worksheet Answer Key With Work
Parallel lines and transversals unit vocabulary assignment and puzzles this is a introductory vocabulary assignment for a unit on parallel lines and transversals. In the diagram on the next page line t is a transversal of lines q and r.
Pin On Parallel Lines
These angles and specific pairings of these angles are given special names.
Parallel lines and transversals worksheet answer key with work. Then make a conjecture about their angle measures. Answer sheet for parallel cut by a transversal displaying top 8 worksheets found for this concept. Word problems on sets and venn diagrams.
Identify the pairs of angles in the diagram. Some of the worksheets for this concept are chapter 3 parallel lines and transversals answer pdf 3 parallel lines and transversals geometry answer key parallel lines and transversals pdf work section 3 1 parallel lines and transversals parallel lines and transversals answer key lesson. Your students will love working with these task cards.
Non parallel lines and transversals worksheet 1 in the diagram below t is a transversal that intersects ℓ1 and ℓ2. Name another pair of lines and their transversal. 5 3 non parallel lines and transversals.
It asks the students to provide the definition and any prior knowledge that they have for each term. Parallel lines unit 3. Parallel lines and transversals worksheet answers with work a line that intersects two or more coplanar lines at two different points is called a transversal.
Use these 40 task cards with your students to help them practice solving word problems that include parallel lines cut by a transversal. 2 in the diagram below identify the relationship between each pair of angles. Time and work word problems.
Identify the pairs of angles in the diagram. Notice that line t forms a total of eight angles with lines q and r. | 677.169 | 1 |
Difference Between Heptagon and Hectagon
Table of Contents
Key Differences
A heptagon is a polygon with seven sides and seven angles. Each internal angle in a regular heptagon - one where all sides and angles are equal - measures approximately 128.57 degrees. In contrast, the term "hectagon" is not widely recognized in geometry and seems to be a misinterpretation or a misspelling of "hectogon," which is occasionally used to refer to a 100-sided polygon, more commonly known as a hectogon.
Heptagons can be regular or irregular. A regular heptagon has sides of equal length and equal angles between each side, whereas an irregular heptagon does not have uniform sides or angles. A hectogon (if one interprets "hectagon" as such) would have 100 sides and, if regular, each internal angle would measure 176.4 degrees, indicative of its much larger number of sides compared to the heptagon.
In terms of symmetry, a regular heptagon has seven lines of symmetry, one for each side. The hectogon, being a polygon with 100 sides, would have 100 lines of symmetry if it is regular. The complexity of constructing a heptagon with straightedge and compass is not nearly as great as that of a hectogon, which is significantly more complex and not commonly constructed or utilized in practical applications.
The heptagon is often seen in everyday objects such as tables or decorations. There is rarely any mention of a "hectagon" in daily usage; this could lead to confusion with a hectogon, which due to its large number of sides is more a mathematical curiosity than a shape used in design or architecture.
Finally, when considering the etymology, "hepta" is a Greek root meaning 'seven,' aligning with the heptagon's seven sides. On the other hand, "hecta" or "hecto" signifies 'hundred,' which would imply that a "hectagon" intends to reference a shape with a hundred sides.
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Comparison Chart
Number of Sides
Seven
One hundred (in the case of a hectogon)
Angle Measures
~128.57 degrees (if regular)
176.4 degrees (if regular hectogon)
Lines of Symmetry
Seven (if regular)
One hundred (if regular hectogon)
Common Usage
More common in everyday life
Rarely used; mostly in mathematical context
Etymology
Greek 'hepta' for seven
Greek 'hecto' for hundred
Compare with Definitions
Heptagon
Heptagon shapes have seven angles.
Each angle of a regular heptagon measures approximately 128.57 degrees.
Hectagon
Hectogon shapes are mainly theoretical and not commonly seen in practical use.
The architect's design incorporated a hectogon, which was a topic of interest for mathematicians.
Heptagon
A regular heptagon features seven lines of symmetry.
The children's puzzle was shaped like a perfect heptagon, with all sides equal.
Hectagon
Hectogon, often miswritten as 'hectagon', is a complex polygon for construction.
The complexity of constructing a hectogon intrigued the math students.
Heptagon
Heptagons can be regular or irregular in geometry.
She drew an irregular heptagon, with none of its sides matching in length.
Hectagon
A hectogon is a polygon with one hundred sides.
In geometry class, we learned that a regular hectogon is practically impossible to draw by hand.
Heptagon
A heptagon is a polygon with seven sides.
The stop sign was mistakenly designed as a heptagon instead of an octagon.
Hectagon
A regular hectogon has internal angles of 176.4 degrees each.
The artist's abstract painting featured a pattern that resembled a hectogon.
Heptagon
In geometry, a heptagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of septua-, a Latin-derived numerical prefix, rather than hepta-, a Greek-derived numerical prefix; both are cognate) together with the Greek suffix "-agon" meaning angle.
Hectagon
The term hectogon comes from the Greek word for 'hundred'.
A hectogon is named after the Greek root 'hecto' indicating its hundred sides.
Heptagon
A plane figure with seven straight sides and angles.
Hectagon
A polygon with 100 edges, vertices, and angles.
Heptagon
A polygon with seven sides and seven angles.
Hectagon
Misspelling of hexagon
Heptagon
(geometry) A polygon with seven sides and seven angles.
Hectagon
Misspelling of heptagon
Heptagon
A plane figure consisting of seven sides and having seven angles.
Heptagon
A seven-sided polygon
Heptagon
Heptagon is derived from the Greek word for 'seven'.
The heptagon was often used in ancient cultures, symbolizing perfection and integration.
Common Curiosities
Can a heptagon be irregular?
Yes, a heptagon can be irregular with unequal sides and angles.
How many sides does a heptagon have?
A heptagon has seven sides.
What is a hectagon?
"Hectagon" is likely a misspelling of hectogon, which has one hundred sides.
What are the lines of symmetry in a regular heptagon?
A regular heptagon has seven lines of symmetry.
What is the internal angle of a regular heptagon?
The internal angle of a regular heptagon is approximately 128.57 degrees.
How is a heptagon different from a hexagon?
A heptagon has seven sides, while a hexagon has six.
Where does the name heptagon originate?
The name heptagon comes from the Greek word 'hepta' meaning seven.
Are hectogons commonly used in design?
No, hectogons are not common in design due to their complexity.
Is a heptagon always symmetrical?
A regular heptagon is symmetrical, with seven lines of symmetry.
Do hectogons have practical applications?
Hectogons are mostly theoretical and have little practical application.
Is it possible to construct a perfect heptagon with a compass and straightedge?
It is not possible to construct a perfect regular heptagon with just a compass and straightedge.
Are all heptagons convex?
Heptagons can be either convex or concave, but regular heptagons are always convex.
Where does the name hectogon come from?
The name hectogon comes from the Greek word 'hecto' meaning hundred.
What are the lines of symmetry in a regular hectogon?
A regular hectogon has one hundred lines of symmetry.
Is "hectagon" a commonly accepted geometric term?
No, "hectagon" is not standard; the term should be hectogon for a hundred-sided polygon | 677.169 | 1 |
Constructing the inverse of a point with respect to a circle
Definition. Consider a circle with centre \(O\) and radius \(r\). Given any point \(P\) in the plane, the inverse of \(P\) with respect to the circle is the unique point \(P'\) on the ray \(OP\) such that \(|OP| |OP'| = r^2\).
Constructing the inverse using tangent lines
The first picture below illustrates how to construct the inverse of a point \(P\) using tangent lines. Draw a line through \(P\) which is orthogonal to the ray \(OP\). Let \(X\) and \(Y\) be the two points where this line intersects the circle. Now draw tangents to the circle at \(X\) and \(Y\) and let \(P'\) be the point where these tangents intersect. Then \(P'\) is the inverse of \(P\).
Angle bisector construction of the inverse
The next picture explains a different construction of the inverse. The circle centred at \(O\) and the point \(P\) are given, as well as a point \(X\) on the circle. Let \(Z\) be the intersection of the circle and the ray \(OP\). Choose a point \(P'\) on the ray \(OP\) such that \(XZ\) is the angle bisector of \(\angle PXP'\). Then \(P'\) is the inverse of \(P\) with respect to the circle.
Isosceles triangle construction of the inverse
A third construction of the inverse uses isosceles triangles. In the diagram below, the circle is centred at \(O\), the point \(P\) is given and we want to construct the inverse \(P'\) of \(P\). Let \(X\) be the intersection of the circle with the perpendicular bisector of \(OP\). Now let \(P'\) be the intersection of the ray \(OP\) with the perpendicular bisector of \(OX\). Then \(P'\) is the inverse of \(P\).
All three constructions
In the final diagram, we see all three constructions at once. The lines in the tangent construction are coloured orange, the angle bisector construction is coloured red and the isosceles triangle construction is coloured green. You can move the points \(O\), \(P\) and \(R\) around to convince yourself that all three constructions do indeed give the same point \(P'\). | 677.169 | 1 |
Are all isosceles trapezoids similar?
Are all isosceles trapezoids similar?
Alternatively, it can be defined as a trapezoid in which both legs and both base angles are of equal measure. Note that a non-rectangular parallelogram is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry….
Isosceles trapezoid
Properties
convex, cyclic
Are 2 isosceles trapezoids similar?
For two triangles to be similar the angles in one triangle must have the same values as the angles in the other triangle. The sides must be proportionate. Hence it is not always true that isosceles triangles are similar.
Is all isosceles triangles are similar?
All isosceles triangles are not similar for a couple of reasons. The length of the two equal sides can stay the same but the measure of the angle between the two equal side will change, as will the base and the base angles.
Are two trapezoids always similar?
All regular trapezoids are not similar. Perhaps the simplest case to see this is the special case of rectangles. All rectangles share four congruent right angles but they are not all similar.
Are two Rhombi always similar?
In a Rhombus, the opposite sides are parallel, and hence the opposite angles are equal. But the value of those angles can be anything. So, it can very much happen that two rhombuses have different angles. Hence, all rhombuses are not similar.
Is an isosceles trapezoid congruent?
An isosceles trapezoid has two congruent legs and one pair of parallel sides. The base angles are congruent to one another, and by same side interior angles, the upper angles are supplementary to the respective base angles, meaning that they are both 180° – (the measure of the base angle).
Which figures are always similar?
Answer: The two geometrical figures which are always similar are circles, squares or line segment.
Are all Rhombi similar?
Which of the following statement is true a all isosceles triangles are similar?
(a) All isosceles triangles are similar – untrue Statement A is False. All isosceles triangles are not similar. An isosceles triangle is one in which 2 out of the 3 sides are equal in length and that their 2 base angles are equal.
How do you know if trapezoids are similar?
In order for two trapezoids to be similar their corresponding sides must have the same ratio. Since the largest base length in the image is and the corresponding side is , the other base must also be times greater than the corresponding side shown in the image.
Which of the following statement is correct all isosceles triangles are similar?
Statement A is False. All isosceles triangles are not similar. An isosceles triangle is one in which 2 out of the 3 sides are equal in length and that their 2 base angles are equal.
What shapes are all similar?
Specific types of triangles, quadrilaterals, and polygons will always be similar. For example, all equilateral triangles are similar and all squares are similar. If two polygons are similar, we know the lengths of corresponding sides are proportional. | 677.169 | 1 |
8th Grade Practice Msa that you have studied in algebra so far in the 8th grade is about to be put to test in the end of year exam. How ready do you feel you are to tackle the questions? Take up the practice test below on every topic and find out. Use of calculators is allowed.
Questions and Answers
1.
1. What is the ninth term in this sequence?5, 14, 23, 32, .....
A.
A. 41
B.
B. 68
C.
C. 77
D.
D. 86
Correct Answer C. C. 77
Explanation The pattern is add nine. Continue to add nine to each term until you have nine numbers.
5, 14, 23, 32, 41, 50, 59, 68, 77
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2.
2. In this diagram line a is parallel to line b, and line t is a transversal. Which pair of angles are corresponding angles?
A.
A. angle 1 and angle 8
B.
B. angle 4 and angle 6
C.
C. angle 2 and angle 8
D.
D. angle 2 and angle 6
Correct Answer D. D. angle 2 and angle 6
Explanation Corresponding angles are angles that are in the same position on each of the parallel lines. Angle 2 and angle 6 are both above and right of the parallel lines and transversal. Other corresponding angle pairs are: 1 and 5, 3 and 7, and 4 and 8.
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3.
3. This drawing shows how to ....
A.
A. construct a perpendicular to a line segment through a point on the segment.
B.
B. construct a bisector of a segment.
C.
C. construct a perpendicular bisector of a segment.
D.
D. construct a segment congruent to a given segment.
Correct Answer A. A. construct a perpendicular to a line segment through a point on the segment.
Explanation Looking at the picture, the lines are perpendicular (at right angles). So answers b and d are wrong. The lines are not bisecting eachother because R is not the midpoint of the horizontal segment. So answer a is the correct answer.
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4.
4. A circle has a diameter of 12 inches. Which is the best estimate of the area of the circle?
A.
A. 19 square inches
B.
B. 38 square inches
C.
C. 113 square inches
D.
D. 452 square inches
Correct Answer C. C. 113 square inches
Explanation The area of a circle is pi times radius squared. Since the diameter is 12, the radius is 6. The radius squared is 36. Pi (3.14) times 36 is closest to answer C.
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5.
5. The table below shows the distance in kilometers between some towns. How much farther is it from Edgertown to Parker than from Aurora to Broomfield?
A.
A. 2.4 kilometers
B.
B. 3.7 kilometers
C.
C. 4.7 kilometers
D.
D. 13.1 kilometers
Correct Answer B. B. 3.7 kilometers
Explanation Looking on the table, Edgertown to Parker is 8.4 km, and Aurora to Broomfield is 4.7 km. "How much farther..?" means subtract.
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6.
6. Identify the hypotenuse of this right triangle.
A.
A. side PR
B.
B. side RQ
C.
C. sides PQ
D.
D. sides PQ and RQ
Correct Answer A. A. side PR
Explanation The hypotenuse of a right triangle is the side that doesn't touch the right angle (PR). Sides PQ and RQ meet to form the right angle, so they are the legs of the triangle.
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7.
7. What is the area of this figure?
A.
A. 36 square units
B.
B. 45 square units
C.
C. 54 square units
D.
D. 66 square units
Correct Answer B. B. 45 square units
Explanation The area of a triangle is one-half times base times height. The top of the figure is a 3 x 6 triangle, so the area is 1/2(3 x 6) = 9. The left and right sides of the figure are 3 x 4 triangles. Each has an area of 1/2(3 x 4) or 6. The center of the figure is a 6 x 4 rectangle, which has an area of 24. Add all the parts: 9 + 6 + 6 + 24 = 45.
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8.
8. This scatterplot compares the height and weight of 9 elementary school students. What conclusion can you draw from this scatter plot?
A.
A. As height increases, so does weight.
B.
A. As height increases, weight decreases.
C.
A. As height increases, weight remains about the same.
D.
A. There is no relationship between the two sets of data.
Correct Answer A. A. As height increases, so does weight.
Explanation The scatterplot shows a positive correlation between height and weight. As the height of the students increases, their weight also tends to increase.
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9.
9. What is the total number of outcomes of drawing a card from a deck of 10 different cards, flipping a coin, and rolling a number cube?
A.
A. 18
B.
B. 60
C.
C. 120
D.
D. 240
Correct Answer C. C. 120
Explanation The fundamental counting principle says to multiply the number of outcomes for each event. So 10 x 2 x 6 = 120.
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10.
10. Which of the following best describes this graph?
A.
A. linear
B.
B. nonlinear
C.
C. both linear and nonlinear
D.
D. neither linear nor nonlinear
Correct Answer B. B. nonlinear
Explanation Linear means a straight line, and nonlinear means everything but a straight line. Answers c and d don't make sense. Since the parabola is curved, it is nonlinear.
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11.
11. In this diagram, line a is parallel to line b, and line t is a transversal. If the measure of angle 8 = 41 degrees, what is the measure of angle 1?
A.
A. 41 degrees
B.
B. 49 degrees
C.
C. 82 degrees
D.
D. 139 degrees
Correct Answer A. A. 41 degrees
Explanation Angles 1 and 8 are alternate exterior angles, so they are congruent (equal).
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12.
12. Which of the following shows 7.4 x 10-4 in standard notation?
A.
A. 0.000074
B.
B. 0.0074
C.
C. 0.00074
D.
D. 74,000
Correct Answer C. C. 0.00074
Explanation The number 7.4 x 10-4 in standard notation is equivalent to 0.00074. In scientific notation, the number is written as 7.4 multiplied by 10 raised to the power of -4. To convert it to standard notation, we move the decimal point four places to the left, resulting in 0.00074.
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13.
13. This year Lucy joined a mail order book club. She paid a basic membership fee of $7.50. Each book she bought cost $4.95. If n stands for the number of books she has bought through the club, which expression shows the total amount she has spent in the book club?
A.
A. $7.50 x $4.95n
B.
B. ($7.50 + $4.95) x n
C.
C. $7.50 + $4.95n
D.
D. $7.50n + $4.95
Correct Answer C. C. $7.50 + $4.95n
Explanation The expression $7.50 + $4.95n represents the total amount Lucy has spent in the book club. The $7.50 represents the basic membership fee that she paid, which is a fixed cost. The $4.95n represents the cost of each book she bought, multiplied by the number of books she has bought (n). So, adding the fixed cost and the variable cost gives the total amount spent in the book club.
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14.
14. Which linear equation corresponds to this graph?
A.
A. y = x + 3
B.
B. y = x - 3
C.
C. y = 3x
D.
D. y = 3 - x
Correct Answer B. B. y = x - 3
Explanation The y-intercept is - 3. The line is going up one, over one, so the rise over run is 1/1 or 1. You don't need to write the 1 in front of the x.
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15.
15. Harvey bought a stereo that cost $325. The sales tax in his state is 6%. How much tax did Harvey have to pay on the stereo?
A.
A. $1.95
B.
B. $19.50
C.
C. $195
D.
D. $305.50
Correct Answer B. B. $19.50
Explanation First you need to change the 6% to a decimal 0.06. Then multiply 0.06 by the price.
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16.
16. Ada and Beth met at a corner cafe for lunch. From there, Ada took a bus 5 km north to her house, and Beth took a bus 12 km east to her house. How far is Ada's house from Beth's house?
A.
A. 7 km
B.
B. 13 km
C.
C. 17 km
D.
D. 26 km
Correct Answer B. B. 13 km
Explanation Since Ada is traveling north and Beth is traveling east, their paths are perpendicular. The distance between the houses would be the hypotenuse of the right triangle. Use the pythagorean thereom to find the distance. Five-squared + 12-squared = hypotenuse squared = 25 + 144 = 169. Then take the square root of 169 = 13.
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17.
17. What is the slope of this line?
A.
A. -3/2
B.
B. -2/3
C.
C. 2/3
D.
D. 3/2
Correct Answer A. A. -3/2
Explanation Slope is the rise over the run. Since the line is pointing down as you go from left to right, the slope is negative. The rise is -3 and the run is 2. So the slope is -3/2.
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18.
18. Which expression is the same as (-7 x 8) + (-7 x 2)?
A.
A. -7 x (8 + 2)
B.
B. -7 x (8 x 2)
C.
C. (-7 + 8) x 2
D.
D. (-7 x 8) + 2
Correct Answer A. A. -7 x (8 + 2)
Explanation This problem shows the distributive property in reverse.
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19.
19. Which is the best estimate of the volume of a cylinder whose radius is 6 cm and whose height is 8 cm?
A.
A. 151 cubic centimeters
B.
B. 301 cubic centimeters
C.
C. 904 cubic centimeters
D.
D. 3,617 cubic centimeters
Correct Answer C. C. 904 cubic centimeters
Explanation The volume of a cylinder is the area of the circular base times the height. The area of the circular base is pi x r-squared = pi x 36 = 113. Then multiply 113 x 8 = 904.
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20.
20. Shawna paid $7.40 for 4 gallons of gasoline. At that rate, how much would she pay for 6 gallons of gasoline?
A.
A. $1.85
B.
B. $22.20
C.
C. $14.80
D.
D. $11.10
Correct Answer D. D. $11.10
Explanation Set up a proportion.
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21.
ECR21. Kyle researched the annual budget of his town and discovered that it was 28,600,000 dollars. Part A Write this number in scientific notation.Part BUse what you know about writing numbers in scientific notation to explain why your answer is correct.Suppose Kyle made a mistake and the budget was 10 times as great as the number he recorded. How would you have to change your answer in Part B to show the new number in scientific notation? Explain your steps. | 677.169 | 1 |
Euclid's Elements [book 1-6] with corrections, by J.R. Young
To describe an equilateral triangle upon a given finite straight line.
Let AB be the given straight line; it is required to describe an equilateral triangle upon AB.
From the centre A, at the distance AB,
describe the circle BCD, and * 3 Pos- from the centre B, at the dis
tulate.
tance BA, describe the circle D A BE ACE; and from the point C, in which the circles cut one another, draw the straight lines CA, CB, to the
#1 Post.
finition.
points A, B; ABC shall be an equilateral triangle. Because the point A is the centre of the circle BCD, AC is equal to AB; and because the point B is the cen12 De tre of the circle ACE, BC is equal to BA: But it has been proved that CA is equal to AB; therefore CA, CB, are each of them equal to AB; but things which are equal to the same thing are equal to one another; therefore CA is equal to CB: wherefore CA, AB, BC are equal to one another: and the triangle ABC is therefore equilateral, and it is described upon the given straight line AB. Which was required to be done.
*1st Axiom.
PROP. II. PROB.
From a given point to draw a straight line equal to a given straight line.
Let A be the given point, and BC the given straight line; it is required to draw from the point A a straight line equal to BC.
#1 Post. From the point A to B draw* the straight line *Prop. 1. AB; and upon it describe the equilateral triangle DAB. From the centre B, at the distance BC, #3 Post. describe the circle CGH, and produce DB to meet *2 Post. it in G. From the centre D, at the distance DG, describe the circle GKL, and produce DA to meet it in L; AL shall be equal to BC.
*12 Def.
Because the point B is the centre of the circle CGH, BC is equal to BG; and because D is the centre of the *By con- circle GKL, DL is equal to DG; struction and DA, DB, parts of them, are *3 Ax. equal; therefore the remainder* AL
is equal to the remainder BG: but
H
+ 1 Ax.
it has been shewn, that BC is equal to BG; wherefore AL and BC are each of them equal to BG: and things which are equal to the same thing are equalf to one another; therefore the straight line AL is equal to BC. Wherefore, from the given point A, a straight line AL has been drawn equal to the given straight line BC. Which was to be done.
The triangle DBA, employed in this problem, might evidently have been isosceles only, having merely DB equal to DA; for no consequence is deduced from the equality of BA with each of these. Euclid has not, however, shewn how to construct a triangle on BA except under the restrictions that all three of its sides are equal; and as his plan is never to assume any construction as practicable, till he has, by aid of the postulates, actually shewn how to effect it, we at once see the reason, and the propriety, of his directing an equilateral triangle to be described upon AB; and we subjoin these remarks solely for the purpose of pointing out to the student the restrictions which are non-essential to the argument.
PROP. III. PROB.
From* 2. 1.
From the point A draw the straight line AD equal to C; and
D
A
VE B
F
*3 Post. from the centre A, and at the distance AD, describe* the circle DEF: AE shall be equal to C.
Because A is the centre of the circle DEF, AE is +12 Def, equal to AD; but the straight line C is likewise + Const. equal to AD; whence AE and C are each of them #1 Ax. equal to AD:wherefore the straight lineAE is equal to* C; and from AB the greater of two straight lines, a part AE has been cut off equal to C the less. Which was to be done.
C
PROP. IV. THEOREM.
If euch, viz. those to which the equal sides are opposite.
A
Let ABC, DEF be two triangles, which have the two sides AB, AC, equal to the two sides DE, DF, each to each, viz. AB to DE, and AC to 'DF; and the angle BAC equal to the angle EDF, the base BC shall be equal to the base EF; and the triangle ABC to the triangle DEF; and the other angles to which the equal sides are opposite, shall be equal, each to each, viz. the angle ABC to the angle DEF, and the angle ACB to DFE.
B
CE
For, if the triangle ABC be applied to DEF, so that the point A may be on D, and the straight line AB upon DE; the point B shall coincide with the point E, because AB is
',
+Hyp.
equal to DE: and AB coinciding with DE, AC Hyp shall fall upon DF, because the angle BAC is equal to the angle EDF; wherefore also the point C shall coincide with the point F, because the straight line AC+ is equal to DF: but the point B +Hyp. was proved to coincide with the point E; wherefore the base BC shall coincide with the base EF: because, the point B coinciding with E, and C with F, if the base BC did not coincide with the base EF, two straight lines would enclose a space, which is impossible. Therefore the base BC coincides with the base EF, and therefore is equalt to it. Wherefore the whole triangle ABC coincides with the whole triangle DEF, and is equal to it; and the other angles of the one coincide with the remaining angles of the other, and are equal to them, viz. the angle ABC to the angle DEF, and the angle ACB to
#10 Ax.
18 Ax.
DFE. Therefore, if two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles contained by those sides equal to one another, their bases shall likewise be equal, and the triangles shall be equal, and their other angles to which the equal sides are opposite shall be equal, each to each. Which was to be demonstrated.
PROP. V. THEOR.
The angles at the base of an isosceles triangle are equal to one another; and if the equal sides be produced, the angles upon the other side of the base shall be equal.
Let ABC be an isosceles triangle, of which the side AB is equal to AC, and let the straight lines AB, AC be produced to D and E: the angle ABC shall be equal to the angle ACB; and the angle CBD to the angle BCE.
+Const.
A
In BD take any point F, and from AE the greater, #3. 1. cut off AG equal to AF the less, and join FC, GB. Because AF is equal to† AG, and AB to AC, ¡Hyp. the two sides FA, AC are equal to the two GA, AB, each to each; and they contain the angle at A common to the two triangles AFC, AGB; therefore the base FC is equal #4. 1. to the base GB, and the triangle AFC, to the triangle AGB; and the remaining angles of the one are equal* to the remaining angles of the other, each to each, to which the equal sides are opposite; viz. the angle ACF to the angle D
#4. 1.
B
F
ABG, and the angle AFC to the angle AGB: and because the whole AF is equal to the whole AG, of which the parts AB, AC, are equal; the remainder BF is equal* to the remainder CG; and FC was proved to be equal to GB; therefore the two sides BF, FC are equal to the two CG, GB, each to each; and the angle BFC was proved to be equal to the angle CGB, wherefore the two triangles BFC, CGB; are equal, and their remaining angles each to each, to which the equal sides are opposite: therefore the angle FBC is equal to the angle GCB, and the angle BCF
to the angle CBG. And, since it has been demonstrated, that the whole angle ABG is equal to the whole ACF, the parts of which, the angles CBG, BCF are also equal; therefore
the remaining angle ABC is equal† to the remaining +3 Ax. angle ACB, which are the angles at the base of the triangle ABC: and it has also been proved that the angle FBC is equal to the angle GCB, which are the angles upon the other side of the base. Therefore the angles at the base, &c. Q. E. D.
lar.
COR.-Hence every equilateral triangle is also equiangu
PROP. VI. THEOR.
If two angles of a triangle be equal to one another, the sides also which subtend or are opposite to the equal angles, shall be equal to one another.
#3. 1.
Let ABC be a triangle having the angle ABC equal to the angle ACB: the side AB shall be equal to the side AC. For, if AB be not equal to AC, one of them is greater than the other: let AB be the greater; and from it cut off BD equal to AC, the less, and join DC: therefore, because in the triangles DBC, ACB, DB is equal to AC, and BC common to both, the two sides, DB, BC are equal to the two AC, CB, each to each; and the angle DBC is equal to the anglet ACB; +Hyp. therefore the base DC is equal to the base AB, and the triangle DBC is equal to the triangle ACB, the less to the greater, which is absurd. Therefore AB is not unequal to AC, that is, it is equal. to it. Wherefore, if two angles, &c. Q.E.D. | 677.169 | 1 |
To find one interior angle of a regular pentagon, you can divide the sum of the interior angles by the number of sides, in this case, 5.
See also: Interior angles of a polygon
Exterior angles of a pentagon are the angles between the pentagon and the extended line from the next side.
The sum of the exterior angles of any polygon is always 360^{\circ}.
To find each of the exterior angles of a regular pentagon, you can use the formula:
\text { Exterior angle of a regular polygon }=\cfrac{360}{n}, where n is the number of sides.
Alternatively, if you know the size of the exterior angle of a polygon, you can use a rearrangement of the above formula to identify the number of sides the polygon has.
\text { Number of sides }=\cfrac{360}{\text { exterior angle }}
See also: Sum of exterior angles of a polygon
What are pentagon angles?
Common Core State Standards
How does this relate to 7 th grade maths solve problems involving pentagon angles
In order to solve problems involving pentagon angles:
Identify if the pentagon is regular or irregular.
Identify what the question is asking for.
Solve the problem using the information you have gathered.
Pentagon angles examples
Example 1: finding the exterior angle of a regular pentagon
Below is a regular pentagon. Find the angle marked x.
Identify if the pentagon is regular or irregular.
The question tells us that this is a regular pentagon. This means the sides of the pentagon are equal and the interior angles are equal.
2Identify what the question is asking for.
The question wants us to find the angle x, which is one of the five exterior angles of this pentagon.
3Solve the problem using the information you have gathered.
To find the size of the exterior angles of a regular pentagon, you can use the formula: \text { Exterior angle of a regular polygon }=\cfrac{360}{n}, where n is the number of sides.
So, the missing angle can be calculated as,
x=\cfrac{360}{5}=72^{\circ}.
So, the exterior angle of a pentagon equals 72^{\circ}.
Example 2: finding the sum of the interior angles of a pentagon
What is the sum of the interior angles of a pentagon?
Identify if the pentagon is regular or irregular.
The question does not clarify whether the pentagon is regular or irregular, however, this doesn't matter in the context of the question. The sum of the interior angles of a pentagon will be the same whether it is regular or irregular.
Identify what the question is asking for.
The question asks us to calculate the sum of the internal angles of a pentagon.
Solve the problem using the information you have gathered.
To calculate the sum of the interior angles of a pentagon, octagon or94+125+118)=111111=69^{\circ}.
Example 6: solving angle problems involving pentagons
The diagram shows a shape formed by two regular polygons.
Find the size of the angle labeled x.
Identify if the pentagon is regular or irregular.
The question states that both polygons in the diagram are regular polygons.
Identify what the question is asking for.
The question asks us to find the size of the angle x, which is formed on the exterior of the two regular pentagons.
Solve the problem using the information you have gathered.
To find the size of the angle, x, you need to find the missing interior angles that meet at the point.
To find the size of an interior angle of any pentagon, you need to first calculate the sum of the interior angles.
To calculate the sum of the interior angles of two interior angles are 108^{\circ} each, you can use the angle fact 'angles around a point add up to 360^{\circ} to find the angle x.
So,
x=360-(108+108)=144^{\circ}.
Teaching tips for pentagon angles
Provide worksheets with a variety of problems involving different types of pentagons, including finding missing angles in irregular pentagons and solving word problems.
Discuss how changing the side lengths in irregular pentagons affects the angles.
Suggested study materials include a handout showing the basic properties of polygons and the specific characteristics of pentagons as well as examples of pentagons in real life with corresponding angles and side lengths labeled.
Easy mistakes to make
Not correctly identifying regular and irregular polygons It is important to correctly identify whether a polygon is regular or irregular, using the information given in the question or on the diagram. Do not assume one or the other.
Using the formulae incorrectly Whether a polygon is regular or irregular will determine how the formulae for working with interior and exterior angles can be used. For example, the formula \text { Exterior angle of a regular polygon }=\cfrac{360}{n}, where n is the number of sides, can only be used to find the size of exterior angles of regular polygons, where all exterior angles are equal.
Now that you know the angles inside the pentagon add up to 540^{\circ}, you can subtract the interior angles you know, to find the missing value.
x=540-(54+92+135+136)=123^{\circ}.
4. The diagram shows a pentagon. Calculate the missing angle x.
540^{\circ}
108^{\circ}
72^{\circ}
116^{\circ}
The interior and exterior angles of any polygon add up to 180^{\circ}, so, x=180-64=116^{\circ}.
5. The diagram shows a pentagon. Calculate the missing angle x.
108^{\circ}
72^{\circ}
57^{\circ}
540^{\circ}
To find the size of the exterior angle, x, you need to find the missing interior angle that corresponds to it. To find the size of an interior angle of any pentagon you need to first calculate the sum of the interior angles.54+135+136)=123123=57^{\circ}.
6. The diagram below is formed by three regular polygons. Find the size of the missing value x.
36^{\circ}
108^{\circ}
72^{\circ}
144^{\circ}
To find the size of the angle, x, you need to find the missing interior angles that meet it at the point.
To find the size of an interior angle of any pentagon you need to first calculate the sum of the interior angles three interior angles are 108^{\circ} each, you can use the angle fact 'angles around a point add up to 360^{\circ} to find the angle x.
So,
x=360-(108+108+108)=36^{\circ}.
Pentagon angles FAQs
What are pentagon angles?
A pentagon is a type of polygon with 5 sides. In a regular pentagon, all interior angles are equal and measure 108 degrees. In an irregular pentagon, the angles can vary. The sum of the interior angles of any pentagon always adds up to 540 degrees.
What is the difference between a convex pentagon and a concave pentagon?
A convex pentagon has all interior angles less than 180^{\circ} and no vertices pointing inward, while a concave pentagon has at least one interior angle greater than 180^{\circ} and at least one vertex pointing inward.
What is the polygon interior angle sum theorem?
The polygon interior angle sum theorem states that the sum of the interior angles of a polygon with n sides is (n-2) \times 180^{\circ}.
What is the polygon exterior angle sum theorem?
The polygon exterior angle sum theorem states that the sum of the measures of the exterior angles of a polygon, one at each vertex, is always 360^{\circ}.
Can a pentagon have any right angles?
A pentagon has five sides. In a regular pentagon, all interior angles are 108 degrees. An irregular pentagon can have up to three right angles, as a fourth would leave no degree measure for the final angle, resulting in a straight | 677.169 | 1 |
How many jobs use the Pythagorean Theorem?
There are 59 jobs that use Pythagorean Theorem.
How do farmers use the Pythagorean Theorem?
Agriculturists, which are farmers, gardeners and environmentalists all use the Pythagorean Theorem. These Agriculturists use this formula to measure where precise lines need to be drawn to determine growing spaces. Geologist use the Pythagorean Theorem when they track earthquake activity.
How do cartographers use the Pythagorean Theorem?
The Pythagorean Theorem is used to calculate the steepness of slopes of hills or mountains. A surveyor looks through a telescope toward a measuring stick a fixed distance away, so that the telescope's line of sight and the measuring stick form a right angle.
How is Pythagoras theorem used in architecture and construction?
Architects use the Pythagorean Theorem to calculate the heights of buildings and the lengths of walls. Athletes even use the Pythagorean Theorem when they are calculating distances, which are important in determining how fast they can run or where a ball needs to be thrown.
Is Pythagorean Theorem used for construction?
The Pythagorean theorem can be used to build staircases, roofs, and can even be used to calculate the angle for safely placing a ladder when you need to work in high areas. It's one of the most popular mathematical rules out there because it comes in handy any time you need to create a 90 degree angle.
How the Pythagorean Theorem is used for surveying the land?
The Pythagorean Theorem is used to calculate the steepness of slopes of hills and mountains. A surveyor looks through a telescope toward a measuring stick a fixed distance away, so the telescope's line of sight and the measuring distance form a right triangle.
What jobs would use the Pythagorean Theorem?
There are many relevant applications that require the use of the Pythagorean Theorem. Engineers and astronomers use the Pythagorean Theorem to calculate the paths of spacecraft, including rockets and satellites. Architects use the Pythagorean Theorem to calculate the heights of buildings and the lengths of walls.
Do engineers use Pythagorean Theorem?
There are many relevant applications that require the use of the Pythagorean Theorem. Pythagoras' theorem is used by engineers in various calculations such as for building construction work, on tunnelworks, bridge work, designing maps etc.
Why do we use Pythagoras in construction and building projects?
What does Pythagorean Theorem help us in the field of math and construction?
Pythagoras' theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root operation. The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer.
What types of jobs use the Pythagorean theorem?
Many positions that fall under the umbrella term of management use the Pythagorean Theorem regularly. Computer and information systems managers, construction managers, and engineering and natural sciences managers all need this age-old formula in the day-to-day business of their respective fields.
Can the Pythagorean theorem be used on other types of triangles?
While its impossible to use the Pythagorean theorem on anything but a right triangle, it is possible to use other theorems and corollaries available to better understand different types of triangles.
Why is the Pythagorean theorem useful for two dimensional navigation?
The Pythagorean Theorem is useful for two-dimensional navigation. You can use it and two lengths to find the shortest distance.
Can the Pythagorean theorem be used to find the distance between two objects?
While its impossible to use the Pythagorean theorem on anything but a right triangle, it is possible to use other theorems and corollaries available to better understand different types of triangles. These theories and corollaries combined with the Pythagorean Theorem will allow you to find the distance between any two | 677.169 | 1 |
If the distance between the centers of two circles is equal to the sum of their radii, the circles are ____.
Touching internally
Touching externally
Intersecting
Non-touching
If the distance between the centers of two circles is equal to the sum of their radii, the circles are touching externally. This means they touch at exactly one point.
3 / 66
The common tangent to two circles that does not intersect the line segment joining their centers is called ____.
Internal tangent
External tangent
Direct tangent
Indirect tangent
The common tangent to two circles that does not intersect the line segment joining their centers is called an external tangent. This type of tangent lies outside both circles and touches them at exactly one point each.
4 / 66
A circle has a radius of 7 units. Calculate the length of a tangent from a point 25 units away from the center.
A tangent is a line that touches a circle at exactly one point. This point is called the point of tangency. Unlike a secant which intersects the circle at two points, a tangent only touches the circle at one point. This unique property makes it a crucial concept in circle geometry. The tangent is perpendicular to the radius at the point of contact.
6 / 66
In a cyclic quadrilateral, the sum of opposite angles is ____.
90°
180°
270°
360°
In a cyclic quadrilateral, the sum of the measures of opposite angles is always 180°. This is known as the supplementary property of cyclic quadrilaterals and is used in various geometric proofs and problems.
7 / 66
If the distance between the centers of two circles is equal to the difference of their radii, the circles are ____.
Touching internally
Touching externally
Intersecting
Non-touching
If the distance between the centers of two circles is equal to the difference of their radii, the circles are touching internally. This means they touch at exactly one point inside the smaller circle.
8 / 66
A line touches a circle at exactly one point. This line is called a ____.
Chord
Secant
Tangent
Diameter
A line that touches a circle at exactly one point is called a tangent. This point is called the point of tangency. The tangent is perpendicular to the radius at the point of contact.
9 / 66
If the radius of a circle is 5 units, what is the length of a tangent from a point 13 units away from the center?
The angle subtended by a chord at the center of a circle is called ____.
Central angle
Inscribed angle
Tangential angle
Exterior angle
The angle subtended by a chord at the center of a circle is called the central angle. This angle is significant in circle geometry because it is directly related to the arc it intercepts and the properties of the chord.
11 / 66
If a wheel of a bicycle touches the ground at a point and the distance from the center of the wheel to the point of contact is 70 cm, what is this distance called?
Diameter
Radius
Secant
Tangent
The distance from the center of the wheel to the point of contact with the ground is called the radius. The radius of a circle is a line segment from the center to any point on the circumference, including the point where it touches the ground.
12 / 66
In a right triangle, the circle that is tangent to all three sides is called ____.
Circumcircle
Incircle
Excircle
None of the above
The incircle of a triangle is a circle that is tangent to all three sides of the triangle. In a right triangle, the incircle touches each side at exactly one point. The radius of the incircle is related to the area and the semiperimeter of the triangle.
13 / 66
A flagpole is 10 units tall. If a person standing 24 units away from the base of the flagpole observes the top of the pole, what is the length of the hypotenuse formed?
The tangent at any point on a circle is perpendicular to the radius at that point. This perpendicularity is a defining characteristic of tangents and is used to solve many geometric problems. It is also a key concept in understanding the properties of circles and their tangents.
15 / 66
The length of the tangent from a point 10 units away from the center of a circle is 6 units. What is the radius of the circle?
If two tangents are drawn to a circle from an external point, then ____.
They are parallel
They intersect at the circle
They are perpendicular to each other
They are equal in length
When two tangents are drawn from an external point to a circle, they are equal in length. This property is a consequence of the fact that the tangents form two congruent right triangles with the radii to the points of tangency. The equality of these tangents is used in various geometric problems and proofs.
18 / 66
A tree 9 units tall casts a shadow of 12 units. What is the length of the hypotenuse formed?
15 units
16 units
17 units
18 units
Using the Pythagorean theorem, we get (distance from top to tip of shadow)^2 = (height)^2 + (shadow length)^2. So, hypotenuse = √((9)^2 + (12)^2) = √(81 + 144) = √225 = 15 units. This hypotenuse is the direct distance from the tree top to the shadow tip.
19 / 66
If two tangents are drawn to a circle from an external point P, and the points of tangency are A and B, then the angle APB is ____.
45°
60°
90°
180°
When two tangents are drawn from an external point to a circle, the angle between the tangents at the external point (angle APB) is 90 degrees. This is because the tangents are equal in length and form two congruent right triangles with the radii to the points of tangency.
20 / 66
A circle has a radius of 9 units. Calculate the length of a tangent from a point 15 units away from the center.
A statue 10 units tall casts a shadow of 7 units. What is the distance from the top of the statue to the tip of the shadow?
11 units
12 units
13 units
147)^2) = √(100 + 49) = √149 = 12 units. This hypotenuse represents the direct distance from the statue top to the shadow tip.
22 / 66
If the lengths of two tangents drawn from an external point to a circle are 9 units and 12 units, what can be concluded about the point?
It lies inside the circle
It lies on the circle
It lies outside the circle
It is the center of the circle
If the lengths of two tangents drawn from an external point to a circle are given, the point lies outside the circle. This is because tangents can only be drawn from points outside the circle, and they are equal in length. However, there seems to be a misunderstanding in the question since both tangents should have equal lengths. The correct length should be equal.
23 / 66
The angle between a tangent and a radius at the point of tangency is always ____.
0°
45°
90°
180°
The angle between a tangent and a radius at the point of tangency is always 90°. This perpendicularity is a fundamental property of tangents and circles.
24 / 66
In a cyclic quadrilateral, the sum of the measures of the opposite angles is ____.
90°
180°
270°
360°
In a cyclic quadrilateral, the sum of the measures of the opposite angles is always 180°. This property is used in various geometric proofs and problems involving cyclic quadrilaterals.
25 / 66
The point where a tangent touches a circle is called the ____.
Vertex
Focus
Point of tangency
Center
The point where a tangent touches a circle is called the point of tangency. This point is significant in geometry because it is the only point where the tangent and the circle intersect, and it is where the radius is perpendicular to the tangent.
26 / 66
The perpendicular drawn from the center of a circle to a chord ____.
Bisects the chord
Is tangent to the chord
Is equal to the radius
Intersects the chord at an acute angle
The perpendicular drawn from the center of a circle to a chord bisects the chord. This property is useful in solving many geometric problems involving circles and chords. It helps in finding the length of the chord and other related measurements.
27 / 66
In a right triangle inscribed in a circle with one side as the diameter, what is the measure of the angle opposite to the diameter?
30°
45°
60°
90°
In a right triangle inscribed in a circle with one side as the diameter, the angle opposite to the diameter is 90°. This is a consequence of the Thales' theorem.
28 / 66
The distance from a point outside the circle to the point of tangency can be found using the ____.
Pythagorean Theorem
Tangent-Secant Theorem
Circle Theorem
Area Theorem
The Tangent-Secant Theorem states that the square of the length of the tangent segment is equal to the product of the lengths of the entire secant segment and its external segment. This relationship helps in calculating the distance from a point outside the circle to the point of tangency.
29 / 66
If two tangents are drawn from an external point to a circle, then the angle between the two tangents is equal to ____.
The angle between the radii to the points of tangency
The angle at the center subtended by the chord joining the points of tangency
The angle subtended by the chord at the center
None of the above
The angle between two tangents drawn from an external point to a circle is equal to the angle at the center subtended by the chord joining the points of tangency. This property is useful in various geometric problems and constructions involving tangents and circles.
30 / 66
The length of a tangent from a point 13 units away from the center of a circle is 12 units. What is the radius of the circle?
The line segment that touches a circle at one endpoint and extends outside the circle is called ____.
Radius
Chord
Secant
Tangent
A tangent is a line segment that touches a circle at one endpoint and extends outside the circle. It is distinct from a chord, which lies entirely inside the circle, and a secant, which intersects the circle at two points. The unique property of a tangent is that it touches the circle at exactly one point.
35 / 66
A ladder 16 units long is placed against a wall such that its base is 12 units away from the wall. How high does the ladder reach on the wall?
A circle is described with center O and radius r. If P is an external point, and PA and PB are tangents from P to the circle, then the triangle PAB is ____.
Isosceles
Equilateral
Right-angled
Scalene
In the triangle PAB, where PA and PB are tangents from P to the circle, the triangle is isosceles because PA = PB. This is due to the equality of tangent segments from an external point to the points of tangency.
37 / 66
If two circles touch each other externally, the distance between their centers is equal to ____.
The difference of their radii
The sum of their radii
The product of their radii
The quotient of their radii
If two circles touch each other externally, the distance between their centers is equal to the sum of their radii. This is because the point of tangency lies on the line segment joining the centers of the two circles, making the distance equal to the sum of the radii.
38 / 66
In a cyclic quadrilateral, the product of the lengths of its diagonals is equal to ____.
The product of the lengths of its opposite sides
The product of the lengths of its adjacent sides
The sum of the lengths of its opposite sides
The sum of the lengths of its adjacent sides
In a cyclic quadrilateral, the product of the lengths of its diagonals is equal to the product of the lengths of its opposite sides. This property is known as Ptolemy's theorem, which is a fundamental result in circle geometry.
39 / 66
A tangent to a circle forms a 30° angle with a chord that it intersects. The angle subtended by the chord at the center of the circle is ____.
30°
60°
90°
120°
When a tangent to a circle forms a 30° angle with a chord, the angle subtended by the chord at the center of the circle is twice the angle between the chord and the tangent. Therefore, it is 2 × 60° = 120°.
40 / 66
A pole 12 units tall casts a shadow of 5 units. What is the distance from the top of the pole to the tip of the shadow?
13 units
14 units
15 units
165)^2) = √(144 + 25) = √169 = 13 units. This represents the hypotenuse in the right triangle formed.
41 / 66
If the radius of a circle is doubled, the length of the tangent from an external point to the circle ____.
Doubles
Remains the same
Is halved
Increases by 4 times
If the radius of a circle is doubled, the length of the tangent from an external point to the circle remains the same. The length of the tangent depends on the distance from the external point to the center and the original radius, but not directly on the radius alone.
42 / 66
The circle that passes through all the vertices of a polygon is called ____.
Incircle
Circumcircle
Excircle
None of the above
The circle that passes through all the vertices of a polygon is called the circumcircle. The center of this circle is known as the circumcenter, and it is equidistant from all the vertices of the polygon.
43 / 66
A circle is inscribed in a triangle if ____.
Each side of the triangle is a tangent to the circle
The circle passes through each vertex of the triangle
The circle is tangent to each side of the triangle
The circle intersects each side of the triangle
A circle is inscribed in a triangle if it is tangent to each side of the triangle. This means that each side of the triangle is tangent to the circle at a single point. The points of tangency divide the sides of the triangle into segments that are used in various geometric proofs and constructions.
44 / 66
The distance between the centers of two circles is 17 units. If the radii of the circles are 10 units and 7 units respectively, find the length of the common external tangent.
15 units
10 units
12 units
1717)^2 – (3)^2) = √(289 – 9) = √280 = 15 units.
45 / 66
A vertical wall is 8 units high and casts a shadow of 6 units on the ground. What is the distance from the top of the wall to the tip of the shadow?
The common tangent to two circles that does not intersect the line segment joining their centers is called ____.
Direct common tangent
Indirect common tangent
Internal common tangent
External common tangent
The common tangent to two circles that does not intersect the line segment joining their centers is called an external common tangent. This type of tangent is outside both circles and touches them at exactly one point each.
47 / 66
If two tangents are drawn to a circle from an external point, the tangents are ____.
Equal in length
Unequal in length
Parallel
Perpendicular
When two tangents are drawn from an external point to a circle, they are equal in length. This property is used in many geometric proofs and problems.
48 / 66
A person stands 12 units away from the base of a lighthouse. If the lighthouse is 20 units tall, what is the length of the shadow cast by the lighthouse on the ground?
A tree casts a shadow of 15 units on the ground. If the tree is 9 units tall and the angle of elevation of the sun is 45 degrees, what is the distance from the top of the tree to the tip of the shadow?
The point where a tangent touches a circle is called the point of tangency. This is the only point where the tangent and the circle intersect.
51 / 66
A water tank casts a shadow of 9 units when the height of the tank is 12 units. What is the distance from the top of the tank to the tip of the shadow?
12 units
15 units
18 units
219)^2) = √(144 + 81) = √225 = 15 units. This distance represents the hypotenuse of the right triangle formed.
52 / 66
A building 18 units high casts a shadow of 24 units. What is the distance from the top of the building to the tip of the shadow?
30 units
32 units
34 units
36 units
Using the Pythagorean theorem, we get (distance from top to tip of shadow)^2 = (height)^2 + (shadow length)^2. So, distance = √((18)^2 + (24)^2) = √(324 + 576) = √900 = 30 units. The hypotenuse in this right triangle is the distance from the building top to the shadow tip.
53 / 66
A circle is circumscribed about a quadrilateral if ____.
The quadrilateral is a rectangle
The quadrilateral has one pair of parallel sides
The sum of opposite angles is 180°
The quadrilateral is a parallelogram
A circle can be circumscribed about a quadrilateral if and only if the sum of opposite angles of the quadrilateral is 180 degrees. This property is known as the supplementary angles property of cyclic quadrilaterals.
54 / 66
A flagpole 15 units high casts a shadow of 20 units. What is the distance from the top of the flagpole to the tip of the shadow?
20 units
25 units
30 units
35 units
Using the Pythagorean theorem, we get (distance from top to tip of shadow)^2 = (height)^2 + (shadow length)^2. So, distance = √((15)^2 + (20)^2) = √(225 + 400) = √625 = 25 units. The hypotenuse formed in this scenario represents the direct distance from the top of the flagpole to the shadow's tip.
55 / 66
A car tire touches the ground at a single point. This point is known as the ____.
Center
Circumference
Tangent
Radius
The point where a car tire touches the ground is known as the tangent point. This is because the tire, which can be modeled as a circle, touches the ground at exactly one point. The ground acts as a tangent to the tire at this point.
56 / 66
The angle between a tangent and a radius at the point of tangency is ____.
45°
60°
90°
180°
The angle between a tangent and the radius drawn to the point of tangency is always 90 degrees. This is because the tangent is perpendicular to the radius at the point of contact. This property is fundamental in many geometric proofs and constructions involving circles and tangents.
57 / 66
A building casts a shadow of 30 units on the ground. If the height of the building is 18 units, what is the distance from the top of the building to the tip of the shadow?
The length of the tangent from a point outside the circle to the point of tangency is called the ____.
Radius
Diameter
Secant
Tangent segment
The length of the tangent from a point outside the circle to the point of tangency is called the tangent segment. This segment is an important element in problems involving tangents and circles, as it is often equal to other tangent segments drawn from the same external point.
59 / 66
A triangle inscribed in a circle such that one of its sides is the diameter of the circle is called ____.
Scalene triangle
Equilateral triangle
Isosceles triangle
Right triangle
A triangle inscribed in a circle such that one of its sides is the diameter of the circle is called a right triangle. This is a consequence of the fact that the angle subtended by the diameter at the circumference is a right angle (90°).
60 / 66
In a circle, the radius to the point of tangency is always ____.
Equal to the diameter
Perpendicular to the tangent
Parallel to the chord
Tangent to the circle
In a circle, the radius to the point of tangency is always perpendicular to the tangent. This is a fundamental property of tangents and circles. The tangent is perpendicular to the radius at the point of contact, which is a critical concept in circle geometry.
61 / 66
The distance between the centers of two circles is 20 units. If the radii of the circles are 12 units and 8 units respectively, find the length of the common external tangent.
16 units
12 units
18 units
1020)^2 – (4)^2) = √(400 – 16) = √384 = 16 units.
62 / 66
A ladder 20 units long is placed against a wall such that its base is 16 units away from the wall. How high does the ladder reach on the wall?
A lamppost 14 units tall casts a shadow of 21 units. What is the distance from the top of the lamppost to the tip of the shadow?
21 units
24 units
25 units
28 units
Using the Pythagorean theorem, we get (distance from top to tip of shadow)^2 = (height)^2 + (shadow length)^2. So, distance = √((14)^2 + (21)^2) = √(196 + 441) = √637 = 25.24 units (approximated to 28 units). The hypotenuse is the distance from the lamppost top to the shadow tip.
64 / 66
A tree of height 10 units casts a shadow of 24 units. Find the distance from the top of the tree to the tip of the shadow.
26 units
27 units
28 units
2924)^2) = √(100 + 576) = √676 = 26 units. This distance forms the hypotenuse of the right triangle formed by the tree height and shadow length.
65 / 66
A statue is 7 units tall and casts a shadow of 24 units on the ground. What is the distance from the top of the statue to the tip of the shadow? | 677.169 | 1 |
Rand Q are points on the x-axis. What is the area
[#permalink]
30 Nov 2018, 10:48
5
Kudos
Answer is A alone is sufficient Drop a median ( PS) from P to X axis . since it is an equilateral triangle the median will be at 90 degrees on X axis and will be an angle bisector at P . Now this triangle PSR or PSQ are 30-60-90 triangles since we know PS - it is 2root3 we can find out QS and hence the base of the triangle . so we can find the area out
Coming to option B . well the equilaterla triange can have any base , it can be 8 , or 6 or 4 , or 7 and we have no clue about the height so it is not sufficient | 677.169 | 1 |
Relationships one to one function refer to relationships between any two items in which one can only belong to one other item. In a mathematical sense, these relationships are known as one-to-one functions, in which there are equal numbers of items or in which one item is paired with only one other item. The name … Read more In the number 0.1234, there … Read more
The axis of symmetry is a straight line that divides an object into two identical parts, thus creating one part as a mirror image of another. Folded along the axis of symmetry, the two parts become superimposed. In terms of symmetry, the mirror line is the straight line. The horizontal, vertical, or slanting line can … Read more
Parabolas are graphs of quadratic functions in mathematics. A parabola is a projective curve of a circle, according to Pascal. Galileo described a path called a parabolic path that is followed by projectiles falling under uniform gravity. In most cases, physical motions of bodies follow a curvilinear path in the shape of a parabola. Parabolas | 677.169 | 1 |
Trisecting Sentence Examples
The curve also permits the solution of the problems of duplicating a cube and trisecting an angle.
10
5
It became known as the "Delian problem" or the "problem of the duplication of the cube," and ranks in historical importance with the problems of "trisecting an angle" and "squaring the circle."
5
4
Although Hippocrates could not determine the proportionals, his statement of the problem in this form was a great advance, for it was perceived that the problem of trisecting an angle was reducible to a similar form which, in the language of algebraic geometry, is to solve geometrically a cubic equation.
3
4
He also considered the equation associated with the problem of trisecting an angle, namely a cubic equation.
0
1
His investigation of the properties of amicable numbers and of the problem of trisecting an angle, are of importance. | 677.169 | 1 |
Tag: midpoint
As the name implies midpoint values refers to the middle or center point somewhere, however, in geometry the midpoint specifically refers to the midpoint of a line segment. The midpoint of a line segment is a point that is present Read more… | 677.169 | 1 |
Centroid Calculator
Centroid Calculator
Welcome to our Centroid Calculator page, developed by Newtum. This tool simplifies the mathematical concept of centroid, making it accessible and easy to understand. We invite you to explore and learn more about this fascinating topic.
Unveiling the Concept Behind the Tool
The Centroid Calculator is a user-friendly tool that simplifies the process of calculating the centroid. It's an invaluable resource for anyone dealing with centroid calculations, making the process faster and more efficient.
Decoding the Formula of Centroid Calculator
The formula of the Centroid Calculator is essential for accurate calculations. Understanding this formula can significantly enhance your comprehension of the concept of centroid, thereby making your mathematical journey easier and more enjoyable.
X and Y represent the coordinates of the centroid, xi and yi are the coordinates of each point, and di is the weight or area of each point.
Step by Step Guide: How to Use Our Centroid Calculator
Our Centroid Calculator is designed to be user-friendly and easy to navigate. Simply follow the step-by-step instructions provided below to make the most of this efficient tool.
Enter the coordinates of each point in the respective fields.
Enter the weight or area of each point.
Click on 'Calculate' to get the centroid coordinates.
Why Our Centroid Centroid Calculator
Used in geometry to calculate the centroid of a triangle or polygon.
Helpful in engineering fields for determining the center of gravity.
Useful in computer graphics for object rotation and scaling.
Applying the Centroid Calculator Formula: Practical Examples
Example 1: Consider a triangle with vertices at points (2,3), (4,5), and (6,8). The centroid can be calculated using our tool quickly and accurately.
Example 2: Suppose you have a polygon with vertices at points (1,2), (3,4), (5,6), and (7,8). Our Centroid Calculator can efficiently calculate the centroid of this polygon.
Ensuring Your Data Security with Centroid Calculator
As we conclude, we'd like to emphasize the security and efficiency of our Centroid Calculator. Given that all calculations are processed on your device, without the need for server interaction, your data remains secure and confidential. This tool is a testament to our commitment to providing quality educational resources while prioritizing user security. We hope that you find our Centroid Calculator beneficial and that it enhances your understanding of the fascinating concept of centroid.
Frequently Asked Questions About Centroid Calculator
What is a Centroid Calculator?
A Centroid Calculator is a tool that helps calculate the centroid, or geometric center, of a plane figure or shape. | 677.169 | 1 |
A Comprehensive Guide to Polar Graphs: Types and Uses
Polar graphs, also known as polar coordinate graphs, are a type of graphing system used to represent mathematical functions in polar coordinates. They are created by plotting points based on the radius (distance from the origin) and the angle (measured in degrees or radians). Here are some common types of polar graphs:
1. Circle
A circle in polar coordinates is represented by a constant radius. The equation for a circle is r = a, where 'a' is the radius. When graphed in polar coordinates, this equation results in a circle with a fixed radius centered at the origin.
2. Cardioid
A cardioid is a heart-shaped curve in polar coordinates. It is represented by the equation r = a(1 + cos(θ)), where 'a' determines the size of the cardioid. As the angle θ increases, the distance from the origin, r, varies based on the cosine function, creating the characteristic shape of a cardioid.
3. Rose Curve
A rose curve, also known as a rhodonea curve, is a type of polar graph that resembles a flower with symmetric petals. It is represented by the equation r = a cos(nθ), where 'a' determines the size of the curve, and 'n' determines the number of petals. Different values of 'n' result in different numbers of petals, such as 2 for a simple rose with two petals or 5 for a five-petal rose.
4. Spiral
A spiral is a polar graph that continuously moves away from or towards the origin as the angle θ increases. It can be represented by equations such as r = aθ, r = ae^(bθ), or r = a/(θ – c), where 'a', 'b', and 'c' are constants that determine the size, shape, and direction of the spiral.
5. Lemniscate
A lemniscate is a figure-eight-shaped curve in polar coordinates. The most well-known lemniscate is the Lemniscate of Bernoulli, represented by the equation r^2 = a^2 cos(2θ), where 'a' determines the size of the curve. This curve has two loops and is symmetric about the origin.
These are just a few examples of the different types of polar graphs that can be represented in polar coordinates. Polar graphs offer a different perspective from the Cartesian coordinate system and can help visualize and understand various mathematical functions and curves. The beauty of polar graphs lies in their ability to represent complex patterns and symmetries in a unique and visually appealing way. | 677.169 | 1 |
A Complete Guide Covering The Different Types Of Angles
Angles are a crucial part of geometry and important to be studied to assist the students' understanding of geometry. An angle is a geometrical figure that is formed when two rays intersect at a point.
The two rays in the figure are known as sides of the angle, and the common endpoint is known as the vertex of the angle. The intersection of plane rays as well as curved rays is known as an angle. This blog post discusses different types of angles.
Different types of Angles
It can be said that angles are the first thing to be learned about geometry to understand the entire geometry. A student must realise angles well to make their way easily through geometry and mathematics.
Angles might seem to be very basic, but they prepare the foundation of the subject, which makes them vital. Learning angles at a deeper level helps students build a strong foundation. There are two components constituted by an angle, as follows:
Vertex
Side
The two sides can intersect in any possible way to form the vertex of the angle. Based on the intersection, angles are classified into four major types: acute angle, right angle, obtuse angle, and reflex angle. We'll now discuss these different types of angles in detail and their properties to know them better.
Acute angle and its properties
An acute angle is an angle that always measures less than 90°. An acute angle can also be formed when a right angle is divided into two parts; by this way, we will have two acute angles. For example, two rays A and B meet at a point O; if the angle formed at O is less than 90°, then it will be termed as an acute angle.
Properties of an acute angle:
The measurement lies between 0° to 90°.
Acute angles make an acute triangle in which all angles are acute. However, their sum will always be 180°.
As mentioned above, acute angles always measure less than 90°; hence, 30°, 45°, 60° are some of the degrees of acute angles. Pair of scissors we use, open at an acute angle.
Right angle and its properties
When the two rays of the angle meet at a measure of exactly 90°, the angle is classified as a right angle. No angle measuring less than or greater than 90° can be considered to be a right angle.
For instance, if there are two rays or sides of the angle named as side A and side B, then the intersection point of A and B that can be named O must form a 90° so that ∟AOB can be called a right angle.
Properties of a right angle:
The measurement of the point of intersection is always 90°.
A right-angle makes a right-angle triangle with the other two angles as acute angles.
A standing wall makes a right angle with the floor.
Obtuse angle and its properties
When the sides of an angle intersect to make an angle that measures more than 90°, it is known as an obtuse angle. The measurement of an obtuse angle has to be more than 90° and less than and 180°.
Obtuse angles are also said to be opposite angles of acute angles. If an angle has two sides named L and M, then the intersection angle of L and M at the vertex O must be <90° and >180° to make an obtuse angle.
Properties of an obtuse angle:
The measurements of the intersection point of two rays lie between 90° and 180°.
It is greater than the right angle and smaller than the straight angle.
An obtuse angle makes an obtuse angle triangle which has other two angles as acute angles.
There are several degrees that lie in the range of an obtuse angle, such as 95°, 110°, 120°, 165°, etc.
The angle between the minute hand and hour hand at 12:20 PM is obtuse.
Straight angle and its properties
When two rays of an angle lie in opposite directions and make an angle measuring 180°, it is a straight angle. The measurement of a straight angle can't be less than or more than 180°.
A straight angle makes a straight line through the vertex of the straight angle; that's why we call it a straight angle.
Properties of a Straight angle
The measurement lies exactly at 180°.
A straight angle consists of two right angles.
It's not possible to make a triangle using a straight angle.
You can see a straight angle in the clock when the time will be 6 o'clock, the hour and minute hand will make a straight angle.
Reflex Angle and its properties
A reflex angle is an angle that measures more than 180° but less than 360°. A reflex angle can lie anywhere between the range of 180° and 360°. It is more than a half rotation but not a complete rotation. A reflex angle always lies on the opposite side of an acute angle and an obtuse angle.
If you add any of the acute angles, right angle, and obtuse angle to a straight line or 180°, it will make a reflex angle.
Properties of a reflex angle
A Reflex angle is always greater than 180° and smaller than 360°.
A reflex angle always lies opposite to the acute angle and right angle.
180° right angle= reflex angle
180° acute angle= reflex angle
180° obtuse angle= reflex angle
A hinge joint can never make a reflex angle without being broken.
Conclusion
Geometry is just like a tree with so many branches; lines and angles are one of them. Above, we read about various angles, their way of formation, their measurement, properties, and where we can see them in real life.
However, there is a further classification of angles supplementary angles, complementary angles, and other angles formed by them.
Angles are part and parcel of our day-to-day lives — mechanical engineers use them to construct buildings, athletes use angles to throw Javelin, and more. We can't ignore the importance of angles in mathematics as well as life creating theoretical Exp | 677.169 | 1 |
Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with ...
Def. Ix. It is of the highest importance to attain a clear conception of an angle, and of the sum and difference of two angles. The literal meaning of the term angulus suggests the Geometrical conception of an angle, which may be regarded as formed by the divergence of two straight lines from a point. In the definition of an angle, the magnitude of the angle is independent of the lengths of the two lines by which it is included; their mutual divergence from the point at which they meet, is the criterion of the magnitude of an angle, as it is pointed out in the succeeding definitions. The point at which the two lines meet is called the angular point or the vertex of the angle, and must not be confounded with the magnitude of the angle itself. The right angle is fixed in magnitude, and, on this account, it is made the standard with which all other angles are compared.
Two straight lines which actually intersect one another, or which when produced would intersect, are said to be inclined to one another, and the inclination of the two lines is determined by the angle which they make with one another.
Def. x. It may be here observed that in the Elements, Euclid always assumes that when one line is perpendicular to another line, the latter is also perpendicular to the former; and always calls a right angle, ópðr γωνία ; but a straight line, εὐθεῖα γραμμή.
Def. XIX. This has been restored from Proclus, as it seems to have a meaning in the construction of Prop. 14, Book 11; the first case of Prop. 33, Book III, and Prop. 13, Book vi. The definition of the segment of a circle is not once alluded to in Book 1, and is not required before the discussion of the properties of the circle in Book III. Proclus remarks on this definition: "Hence you may collect that the center has three places: for it is either within the figure, as in the circle; or in its perimeter, as in the semicircle; or without the figure, as in certain conic lines."
Def. XXIV-XXIX. Triangles are divided into three classes, by reference to the relations of their sides; and into three other classes, by reference to their angles. A further classification may be made by considering both the relation of the sides and angles in each triangle.
In Simson's definition of the isosceles triangle, the word only must be omitted, as in the Cor. Prop. 5, Book 1, an isosceles triangle may be equilateral, and an equilateral triangle is considered isosceles in Prop. 15, Book IV. Objection has been made to the definition of an acute-angled triangle. It is said that it cannot be admitted as a definition, that all the three angles of a triangle are acute, which is supposed in Def. 29. It may be replied, that the definitions of the three kinds of angles point out and seem to supply a foundation for a similar distinction of triangles.
Def. xxx-XXXIV. The definitions of quadrilateral figures are liable to objection. All of them, except the trapezium, fall under the general idea of a parallelogram; but as Euclid defined parallel straight lines after he had defined four-sided figures, no other arrangement could be adopted than the one he has followed; and for which there appeared to him, without doubt, some probable reasons. Sir Henry Savile, in his Seventh Lecture, remarks on some of the definitions of Euclid, "Nec dissimulandum aliquot harum in manibus exiguum esse usum in Geometriâ." A few verbal emendations have been made in some of them.
A square is a four-sided plane figure having all its sides equal, and one angle a right angle: because it is proved in Prop. 46, Book 1, that if a parallelogram have one angle a right angle, all its angles are right angles.
An oblong, in the same manner, may be defined as a plane figure of four sides, having only its opposite sides equal, and one of its angles a right angle.
A rhomboid is a four-sided plane figure having only its opposite sides equal to one another and its angles not right angles.
Sometimes an irregular four-sided figure which has two sides parallel, is called a trapezoid.
Def. xxxv. It is possible for two right lines never to meet when produced, and not be parallel.
Def. A. The term parallelogram literally implies a figure formed by parallel straight lines, and may consist of four, six, eight, or any even number of sides, where every two of the opposite sides are parallel to one another. In the Elements, however, the term is restricted to four-sided figures, and includes the four species of figures named in the Definitions
XXX-XXXIII.
The synthetic method is followed by Euclid not only in the demonstrations of the propositions, but also in laying down the definitions. He commences with the simplest abstractions, defining a point, a line, an angle, a superficies, and their different varieties. This mode of proceeding involves the difficulty, almost insurmountable, of defining satisfactorily the elementary abstractions of Geometry. It has been observed, that it is necessary to consider a solid, that is, a magnitude which has length, breadth, and thickness, in order to understand aright the definitions of a point, a line, and a superficies. A solid or volume considered apart from its physical properties, suggests the idea of the surfaces by which it is bounded: a surface, the idea of the line or lines which form its boundaries: and a finite line, the points which form its extremities. A solid is therefore bounded by surfaces; a surface is bounded by lines; and a line is terminated by two points. A point marks position only: a fine has one dimension, length only, and defines distance: a superficies has two dimensions, length and breadth, and defines extension: and a solid has three dimensions, length, breadth, and thickness, and defines some portion of space.
It may also be remarked that two points are sufficient to determine the position of a straight line, and three points not in the same straight line, are necessary to fix the position of a plane.
ON THE POSTULATES.
THE definitions assume the possible existence of straight lines and circles, and the postulates predicate the possibility of drawing and of producing straight lines, and of describing circles. The postulates form the principles of construction assumed in the Elements; and are, in fact, problems, the possibility of which is admitted to be self-evident, and to require no proof.
It must, however, be carefully remarked, that the third postulate only admits that when any line is given in position and magnitude, a circle may be described from either extremity of the line as a center, and with a radius equal to the length of the line, as in Euc. 1, 1. It does not admit the description of a circle with any other point as a center than one of the extremities of the given line.
Euc. 1. 2, shews how, from any given point, to draw a straight line equal to another straight line which is given in magnitude and position.
ON THE AXIOMS.
AXIOMS are usually defined to be self-evident truths, which cannot be rendered more evident by demonstration; in other words, the axioms of Geometry are theorems, the truth of which is admitted without proof. It is by experience we first become acquainted with the different forms of geometrical magnitudes, and the axioms, or the fundamental ideas of their equality or inequality appear to rest on the same basis. The conception of the truth of the axioms does not appear to be more removed from experience than the conception of the definitions.
These axioms, or first principles of demonstration, are such theorems as cannot be resolved into simpler theorems, and no theorem ought to be admitted as a first principle of reasoning which is capable of being demonstrated. An axiom, and (when it is convertible) its converse, should both be of such a nature as that neither of them should require a formal demonstration.
The first and most simple idea, derived from experience is, that every magnitude fills a certain space, and that several magnitudes may successively fill the same space.
All the knowledge we have of magnitude is purely relative, and the most simple relations are those of equality and inequality. In the comparison of magnitudes, some are considered as given or known, and the unknown are compared with the known, and conclusions are synthetically deduced with respect to the equality or inequality of the magnitudes under consideration. In this manner we form our idea of equality, which is thus formally stated in the eighth axiom: " 'Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another."
Every specific definition is referred to this universal principle. With regard to a few more general definitions which do not furnish an equality, it will be found that some hypothesis is always made reducing them to that principle, before any theory is built upon them. As for example, the definition of a straight line is to be referred to the tenth axiom; the definition of a right angle to the eleventh axiom; and the definition of parallel straight lines to the twelfth axiom.
The eighth axiom is called the principle of superposition, or, the mental process by which one Geometrical magnitude may be conceived to be placed on another, so as exactly to coincide with it, in the parts which are made the subject of comparison. Thus, if one straight line be conceived to be placed upon another, so that their extremities are coincident, the two straight lines are equal. If the directions of two lines which include one angle, coincide with the directions of the two lines which contain another angle, where the points, from which the angles diverge, coincide, then the two angles are equal: the lengths of the lines not affecting in any way the magnitudes of the angles. When one plane figure is conceived to be placed upon another, so that the boundaries of one exactly coincide with the boundaries of the other, then the two plane figures are equal. It may also be remarked, that the converse of this proposition is not universally true, namely, that when two magnitudes are equal, they coincide with one another: since two magnitudes may be equal in area, as two parallelograms or two triangles, Euc. 1. 35, 37; but their boundaries may not be equal: and, consequently, by superposition, the figures could not exactly coincide: all such figures, however, having equal areas, by a different arrangement of their parts, may be made to coincide exactly.
This axiom is the criterion of Geometrical equality, and is essentially different from the criterion of Arithmetical equality. Two geometrical magnitudes are equal, when they coincide or may be made to coincide: two abstract numbers are equal, when they contain the same aggregate of units; and two concrete numbers are equal, when they contain the same number of units of the same kind of magnitude. It is at once obvious, that Arithmetical representations of Geometrical magnitudes are not admissible in Euclid's criterion of Geometrical Equality, as he has not fixed the unit of magnitude of either the straight line, the angle, or the superficies. Perhaps Euclid intended that the first seven axioms should be applicable to numbers as well as to Geometrical magnitudes, and this is in accordance with the words of Proclus, who calls the axioms, common notions, not peculiar to the subject of Geometry.
Several of the axioms may be generally exemplified thus: Axiom 1. If the straight line AB be equal A
to the straight line CD; and if the straight line EF be also equal to the straight line CD; E then the straight line AB is equal to the straight line EF.
Axiom II. If the line AB be equal to the line A CD; and if the line EF be also equal to the line GH: then the sum of the lines AB and EF is equal to the sum of the lines CD and GH.
Axiom III. If the line AB be equal to the A line CD; and if the line EF be also equal to the line GH; then the difference of AB and EF, is equal to the difference of CD and GH.
E
E
B
D
B C
D
F
G
H
B с
D
F G
H
Axiom Iv. admits of being exemplified under the two following forms:
2. If the line AB be equal to the line CD; A and if the line EF be less than the line GH; then the sum of the lines AB and EF is less than the sum of the lines CD and GH.
Axiom v. also admits of two forms of exemplification. 1. If the line AB be equal to the line CD; and if the line EF be greater than the line GH; then the difference of the lines AB and EF is greater than the difference of CD and GH.
A
B C
D
E
2. If the line AB be equal to the line CD; and if the line EF be less than the line GH; then the difference of the lines AB and EF is less than the difference of the lines CD and GH. The axiom, "If unequals be taken from equals, the remainders are
unequal," may be exemplified in the same manner. Axiom VI. If the line AB be double of the A line CD; and if the line EF be also double of the line CD;
then the line AB is equal to the line EF. Axiom VII. If the line AB be the half of the line CD; and if the line EF be also the half of the line CD;
then the line AB is equal to the line EF.
It may be observed that when equal magnitudes are taken from unequal magnitudes, the greater remainder exceeds the less remainder by as much as the greater of the unequal magnitudes exceeds the less.
If unequals be taken from unequals, the remainders are not always unequal; they may be equal: also if unequals be added to unequals the wholes are not always unequal, they may also be equal.
Axiom IX. The whole is greater than its part, and conversely, the part is less than the whole. This axiom appears to assert the contrary of the eighth axiom, namely, that two magnitudes, of which one is greater than the other, cannot be made to coincide with one another. Axiom x. The property of straight lines expressed by the tenth axiom, namely, "that two straight lines cannot enclose a space," is obviously implied in the definition of straight lines; for if they enclosed a space, they could not coincide between their extreme points, when the two lines are equal.
Axiom XI. This axiom has been asserted to be a demonstrable theorem. As an angle is a species of magnitude, this axiom is only a partieular application of the eighth axiom to right angles.
Axiom XII. See the notes on Prop. xxix. Book I.
ON THE PROPOSITIONS.
WHENEVER a judgment is formally expressed, there must be something respecting which the judgment is expressed, and something else which constitutes the judgment. The former is called the subject of the proposition, and the latter, the predicate, which may be anything which can be affirmed or denied respecting the subject.
The propositions in Euclid's Elements of Geometry may be divided into two classes, problems and theorems. A proposition, as the term imports, is something proposed; it is a problem, when some Geometrical construction is required to be effected: and it is a theorem when some Geometrical property is to be demonstrated. Every proposition is naturally divided into two parts; a problem consists of the data, or things given; and the quæsita, or things required: a theorem, consists of the subject or hypothesis, and the conclusion, or predicate. Hence the distinction between a problem and a theorem is this, that a problem consists of the data and the quæsita, and requires solution: and a theorem consists of the hypothesis and the predicate, and requires demonstration.
All propositions are affirmative or negative; that is, they either assert some property, as Euc. 1. 4, or deny the existence of some property, as Euc. I. 7; and every proposition which is affirmatively stated has a contradictory corresponding proposition. If the affirmative be proved to be true, the contradictory is false.
All propositions may be viewed as (1) universally affirmative, or universally negative; (2) aš particularly affirmative, or particularly negative.
The connected course of reasoning by which any Geometrical truth is established is called a demonstration. It is called a direct demonstration when the predicate of the proposition is inferred directly from the premisses, as the conclusion of a series of successive deductions. The demonstration is called indirect, when the conclusion shows that the introduction of any other supposition contrary to the hypothesis stated in the proposition, necessarily leads to an absurdity.
It has been remarked by Pascal, that " Geometry is almost the only subject as to which we find truths wherein all men agree; and one cause of this is, that Geometers alone regard the true laws of demonstration." | 677.169 | 1 |
Visualising Solid Shapes worksheet for class 7 Important Topics
Some important Facts about Visualising Solid Shapes worksheet for class 7
The circle, the square, the rectangle, the quadrilateral and the triangle are examples of plane figures; the cube, the cuboid, the sphere, the cylinder, the cone and the pyramid are examples of solid shapes.
Plane figures are of two-dimensions (2-D) and the solid shapes are of three-dimensions (3-D).
The corners of a solid shape are called its vertices; the line segments of its skeleton are its edges; and its flat surfaces are its faces.
A net is a skeleton-outline of a solid that can be folded to make it. The same solid can have several types of nets.
Solid shapes can be drawn on a flat surface (like paper) realistically. We call this 2-D representation of a 3-D solid.
Two types of sketches of a solid are possible:
An oblique sketchdoes not have proportional lengths. Still it conveys all important aspects of the appearance of the solid.
An isometric sketch is drawn on an isometric dot paper, a sample of which is given at the end of this book. In an isometric sketch of the solid the measurements kept proportional.
DRAWING SOLIDS ON A FLAT SURFACE
Your drawing surface is paper, which is flat. When you draw a solid shape, the images are somewhat distorted to make them appear three-dimensional. It is a visual illusion. You will find here two techniques to help you. | 677.169 | 1 |
Introduction to The Pythagorean Theorem
We've all heard of it in one context or another. It's the method we use to find the third leg of a right triangle, known as the hypotenuse.
If you are given any two legs of a right triangle (a triangle with one
right angle) you are able to find the third. a and b represent the legs of the
triangle, and c represents the hypotenuse. The hypotenuse is the leg of a right triangle that sits across from the right angle, and is also the longest side in a right triangle.
Let's say you're given triangle ABC. Angle B is a right angle, meaning side AC is the hypotenuse since it sits across from the right triangle. If you are given side AB = 3, side BC = 4, and side AC is unknown, you can find the unknown side using the Pythagorean Theorem. Knowing that side AC is the hypotenuse, we can make the equation:
32+ 42 = x²
So now, let's solve for the unknown:
9 + 16 = x²
25 = x²
5 = x
Using the Pythagorean Theorem, you can actually solve for any of the three legs — not just the hypotenuse — as long as you know the values of the other two legs! Simply plug in the two values that you do know, and solve algebraically to find the unknown leg.
Fun Facts About the Creator of the Pythagorean Theorem!
- The Pythagorean Theorem was named after Pythagoras, an ancient Greek philosopher and mathematician.
- It is believed that Pythagoras had synesthesia, a condition where one associates multiple senses with each other. For example, individuals with synesthesia might match a smell to a color, or a sound to a color.
- Pythagoras didn't allow his followers to eat beans.
- Pythagoras was most likely a vegetarian.
- Along with the Pythagorean Theorem, Pythagoras also discovered Pythagorean tuning, the five regular solids, the Theory of Proportions, and the sphericity of the Earth. | 677.169 | 1 |
But if you instead want to fit the original points into a rotated rectangle then you would have to rotate the rectangle by the negative of that angle.
The image below shows the face from your question together with a plane rotated by −50.005 degrees instead.
If you're also interested in the optimal size of the rectangle I guess it's still best to calculate the rotated points and then determine the smallest and largest coordinates like it's been done in the example script. | 677.169 | 1 |
How Many Squares
I do not have correct answer, I guess 65. what about yours?
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted \square ABCD.
The square is the n=2 case of the families of n-hyper cubes and n-orthoplexes | 677.169 | 1 |
Central Angles
A central angle in a circle is an angle whose vertex is at the centre of the circle and whose sides (radii) extend to the circumference. It is always measured in degrees and is closely linked to the arc it intercepts. Remember, the measure of a central angle is equal to the measure of its intercepted arc.
Central Angles: Definition and Basics
The study of central angles is a key component when learning about circles. Understanding central angles helps you grasp more complex geometry concepts. Let's dive into the basics to get a strong foundation.
Definition of Central Angle
A central angle is an angle whose vertex is at the centre of the circle and whose sides (or arms) extend to the circumference. Essentially, the central angle forms a pie slice of the circle.
You can think of a central angle as being like the pin in the centre of a pizza, forming slices.
Central Angle of a Circle: Key Concepts
The sum of all central angles in a circle is 360 degrees or 2π radians.
Consider the following properties and formulas:
The formula to find the measure of a central angle is given by the arc length divided by the radius: \left(\measuredangle = \frac{L}{R}\right).
For example, if a sector of a circle has an arc length of 5 cm and the radius is 2 cm, the central angle can be calculated as:
\[\measuredangle = \frac{5}{2} = 2.5\text{ radians}\]
In addition to the basic properties, central angles have unique characteristics in special circles such as unit circles. In a unit circle (which is a circle with a radius of 1), the measure of the arc length is equivalent to the measure of the central angle in radians. This relationship is incredibly useful in trigonometry.
Central Angles in a Circle: Examples
Examples help solidify your understanding by seeing the concepts in action. Let's look at a few examples to illustrate how central angles work:
Example 1: Find the central angle of a circle where the arc length is 8 cm, and the radius is 4 cm.
We use the formula:
\[\measuredangle = \frac{L}{R} = \frac{8}{4} = 2\, \text{radians}\]
When dealing with circles, it's also crucial to understand the relationship between central angles and inscribed angles. An inscribed angle is an angle formed by two chords in a circle that share a common endpoint. The inscribed angle is half the measure of the central angle that intercepts the same arc. This comes in handy in various geometry problems.
How to Find Central Angles
Understanding how to find central angles is essential in geometry. This section covers the formula, step-by-step guide, and practical uses.
Central Angle Formula
The central angle formula is crucial for finding the measure of an angle based on the arc length and radius of the circle. The formula is
\[\measuredangle = \frac{L}{R}\]
Example: To find the central angle for an arc length of 6 cm and a radius of 3 cm:
\[\measuredangle = \frac{6}{3} = 2\,\text{radians}\]
Remember that the central angle can be measured in degrees or radians. To convert radians to degrees, use the formula \(Degrees = Radians \times \frac{180}{\pi}\).
Step-by-Step Guide on How to Find Central Angle
Follow these steps to find a central angle:
Step 1: Identify the arc length (L).
Step 2: Measure the radius (R).
Step 3: Apply the formula \(\measuredangle = \frac{L}{R}\).
Using these steps simplifies the process.
A circle is a set of points in a plane that are equidistant from a given point called the centre.
Understanding central angles deeply involves exploring their relationship with other angles in the circle, such as inscribed angles. The inscribed angle theorem states that an inscribed angle is always half of the measure of the central angle that subtends the same arc. This theorem is vital in solving geometric problems and proofs.
Practical Uses of Central Angle Formula
Central angles have several practical applications:
Engineering: Designing gears and rotation parts.
Architecture: Creating rounded structures and domes.
Art: Constructing circular patterns and designs.
Let's consider an example in design:
Example in Architecture: To design a circular arch with an arc length of 10 m and a radius of 5 m, you find the central angle as:
\[\measuredangle = \frac{10}{5} = 2\,\text{radians}\]
This result helps in determining the curvature of the arch.
Central Angle Exercises
Practising with central angle problems will help you gain confidence and master this important geometry topic. Below are a series of exercises to test your understanding of central angles.
Practice Problems for Central Angles
Try to solve these problems to improve your skills:
Problem 1:
Find the central angle if the radius is 7 cm and the arc length is 14 cm.
Problem 2:
Calculate the central angle for a sector with an arc length of 10 cm and a radius of 4 cm.
Problem 3:
A circle has a central angle of 3 radians and a radius of 6 cm. Find the arc length.
As you practice, remember that central angles are essential in understanding circular motion in physics and engineering. They also help in solving real-life problems involving curves and rotations.
Understanding how to apply these solutions in different contexts will greatly enhance your problem-solving skills. Studying various scenarios can help you see how central angles relate to sectors and arcs in multiple disciplines.
Tips for Solving Central Angle Problems
Here are some tips to help you solve central angle problems more effectively:
Always check if your central angle is in degrees or radians to ensure correct calculations.
If stuck, try drawing the circle and marking the radius and arc length to visualise the problem better.
When dealing with more complex problems, remember that central angles are not only related to geometry but also integral in trigonometry and calculus. This interconnection highlights their importance in advanced mathematics and science.
Applications of Central Angles in Real Life
Central angles are not just theoretical concepts confined to mathematics textbooks. They have numerous practical applications in various fields. From designing buildings to understanding celestial bodies, central angles play crucial roles in our daily lives. Let's explore some of these applications.
Example: Consider a circular archway with a radius of 4 metres and an arc length of 6 metres. To find the central angle, we can use the formula:
\[\measuredangle = \frac{L}{R} = \frac{6}{4} = 1.5\, \text{radians}\]
Remember, in architecture, the accurate calculation of angles ensures that the final structure is both functional and harmonious in design.
Architects also consider the visibility and accessibility dictated by central angles when designing spaces like theatres or stadiums. The seating arrangement often revolves around the central angle to ensure a clear view of the stage or field from any point, maximising the audience experience.
Central Angles in Astronomy
Astronomers utilise central angles to measure the positioning and movement of celestial bodies. By calculating central angles, astronomers can determine the distance between planets, stars, and galaxies. This understanding is pivotal in charting astronomical maps and studying cosmic phenomena.
An astronomical unit (AU) is the average distance between the Earth and the Sun, approximately 149.6 million kilometers. Central angles help in expressing distances in space using AU.
Example: If two planets are observed with an angular separation of 0.01 radians from Earth, and we know the distance to the nearest planet is 1 AU, we can approximate the distance to the other planet using central angles.
In astrophysics, the concept of parallax relies on central angles. Parallax involves measuring the apparent shift of a star against the backdrop of more distant stars as observed from Earth at different times of the year. The angle of this shift, known as the parallax angle, helps determine the star's distance from Earth.
Real-Life Examples of Central Angles
Central angles are everywhere around you. Whether it's the slice of a pizza or the wheels of a car, understanding central angles helps you see the world through a geometric lens. Here are a few relatable examples:
Example: A pizza is cut into 8 equal slices. The central angle for each slice can be calculated as:
\[\measuredangle = \frac{360^{\text{o}}}{8} = 45^{\text{o}}\]
Each slice forms a 45-degree angle at the centre of the pizza.
When cutting pizzas or cakes into equal slices, you are essentially dividing the circle into equal central angles.
Even in sports, central angles play a role. For instance, in golf, the strategy involves understanding the angle at which to hit the ball to ensure it lands in the targeted area. By calculating the central angle, players can better predict the ball's trajectory, enhancing their chances of a successful shot.
Central Angles - Key takeaways
Definition of Central Angle: An angle whose vertex is at the centre of the circle and whose sides extend to the circumference.
Properties of Central Angles: Measured in degrees or radians; subtends an arc on the circle; sum of all central angles in a circle is 360 degrees or 2π radians.
Central Angle Formula: The measure of a central angle is given by the formula θ = L / R, where L is the arc length and R is the radius.
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Frequently Asked Questions about Central Angles
A central angle in a circle is an angle whose vertex is at the centre of the circle, and its sides (rays) extend to intersect the circumference of the circle.
How do you calculate the measure of a central angle?
To calculate the measure of a central angle, use the formula: Central Angle = (Arc Length / Radius) in radians, or Central Angle = (Arc Length * 360) / (2π * Radius) in degrees. Ensure the arc length and radius are in the same units.
How is a central angle different from an inscribed angle?
A central angle has its vertex at the centre of the circle and its sides are radii, while an inscribed angle has its vertex on the circle itself and its sides are chords. The measure of a central angle is equal to the arc it intercepts, whereas an inscribed angle is half the measure of the intercepted arc.
Can a central angle be greater than 180 degrees?
Yes, a central angle can be greater than 180 degrees. Such an angle is called a reflex angle.
How does the length of the arc relate to the central angle?
The length of the arc is directly proportional to the central angle. When the central angle is measured in radians, the length of the arc is the product of the radius and the central angle. Thus, doubling the central angle doubles the arc length.
Test your knowledge with multiple choice flashcards
How do central angles help in architecture?
A. They allow for precise calculations of arches, domes, and circular floors.
B. They assist in electrical wiring layout design.
C. They are only used for aesthetic purposes.
D. They help architects design rectangular frames more accurately.
What is the definition of a central angle?
A. An angle whose vertex is at the centre of the circle and whose sides extend to the circumference.
B. An angle that always measures 90 degrees.
C. An angle formed by two chords that meet inside the circle but not at the centre.
D. An angle located anywhere within a circle, not necessarily at the centre.
How do central angles assist astronomers?
A. They are involved in measuring the gravitational pull between galaxies.
B. They assist in determining the chemical composition of planets.
C. They help measure the positioning and movement of celestial bodies.
D. They are used to calculate the temperature of stars.
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3. How many sides does a regular polygon have if the measure of an exterior angle is 24°? Solution:
Each exterior angle = sum of exterior angles/Number of angles
24°= 360/ Number of sides
⇒ Number of sides = 360/24 = 15
Thus, the regular polygon has 15 sides.
4. How many sides does a regular polygon have if each of its interior angles is 165°?
Solution:
Interior angle = 165°
Exterior angle = 180° – 165° = 15°
Number of sides = sum of exterior angles/ exterior angles
⇒ Number of sides = 360/15 = 24
Thus, the regular polygon has 24 sides.
5. a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?
b) Can it be an interior angle of a regular polygon? Why?
Solution:
a) Exterior angle = 22°
Number of sides = sum of exterior angles/ exterior angle
⇒ Number of sides = 360/22 = 16.36
No, we can't have a regular polygon with each exterior angle as 22° as it is not divisor of 360.
b) Interior angle = 22°
Exterior angle = 180° – 22°= 158°
No, we can't have a regular polygon with each exterior angle as 158° as it is not divisor of 360.
6. a) What is the minimum interior angle possible for a regular polygon? Why?
b) What is the maximum exterior angle possible for a regular polygon?
Solution:
a) Equilateral triangle is regular polygon with 3 sides has the least possible minimum interior angle because the regular with minimum sides can be constructed with 3 sides at least.. Since, sum of interior angles of a triangle = 180°
10. Explain how this figure is a trapezium. Which of its two sides are parallel? (Fig 3.32)
Solution:
When a transversal line intersects two lines in such a way that the sum of the adjacent angles on the same side of transversal is 180° then the lines are parallel to each other. Here, ∠M + ∠L = 100° + 80° = 180°
Thus, MN || LK
As the quadrilateral KLMN has one pair of parallel line therefore it is a trapezium. MN and LK are parallel lines.
11. Find m∠C in Fig 3.33 if AB || DC ?
Solution:
m∠C + m∠B = 180° (angles on the same side of transversal)
⇒ m∠C + 120° = 180°
⇒ m∠C = 180°- 120° = 60°
12. Find the measure of ∠P and ∠S if SP || RQ ? in Fig 3.34. (If you find m∠R, is there more than one
method to find m∠P?)
Solution:
∠P + ∠Q = 180° (angles on the same side of transversal)
⇒ ∠P + 130° = 180°
⇒ ∠P = 180° – 130° = 50°
also, ∠R + ∠S = 180° (angles on the same side of transversal)
⇒ 90° + ∠S = 180°
⇒ ∠S = 180° – 90° = 90°
Thus, ∠P = 50° and ∠S = 90°
Yes, there are more than one method to find m∠P.
PQRS is a quadrilateral. Sum of measures of all angles is 360°.
Since, we know the measurement of ∠Q, ∠R and ∠S.
∠Q = 130°, ∠R = 90° and ∠S = 90°
∠P + 130° + 90° + 90° = 360°
⇒ ∠P + 310° = 360°
⇒ ∠P = 360° – 310° = 50°Solution:
(a) False.
Because, all square are rectangles but all rectangles are not square.
(b) True
(c) True
(d) False.
Because, all squares are parallelograms as opposite sides are parallel and opposite angles are equal.
(e) False.
Because, for example, a length of the sides of a kite are not of same length.
(f) True
(g) True
(h) True
2. Identify all the quadrilaterals that have.
(a) four sides of equal length (b) four right angles
Solution:
(a) Rhombus and square have all four sides of equal length.
(b) Square and rectangle have four right angles.
3. Explain how a square is.
(i) a quadrilateral (ii) a parallelogram (iii) a rhombus (iv) a rectangle Solution
(i) Square is a quadrilateral because it has four sides.
(ii) Square is a parallelogram because it's opposite sides are parallel and opposite angles are equal.
(iii) Square is a rhombus because all the (ii) are perpendicular bisectors of each other (iii) are equal Solution Solution
RectangleSolution
AD and DC are drawn so that AD || BC and AB || DC
AD = BC and AB = DC
ABCD is a rectangle as opposite sides are equal and parallel to each other and all the
interior angles are of 90°.
In a rectangle, diagonals are of equal length and also bisects each other.
Hence, AO = OC = BO = OD
Thus, O is equidistant from A, B and C.
Edu- Spot is an online educational portal which provides you smarter education in a simpler way. | 677.169 | 1 |
So I'm wondering if there is a more efficient way to do the above? I know I don't need the hypotenuse and crap (will remove those extra things), but I'm wondering about different techniques all together. | 677.169 | 1 |
This is for a real life situation not a theoretical one. I'm trying to check if a point exist in a Segment of a 2d circle. In other words, I need to know if a given point P(x,y) is anywhere inside the blue part of the circle (see image below).
Given
The radius, the center $(x, y)$, point $A(x,y)$, point $B(x,y)$. The point $P(x,y)$ is also given and I need to know if it's inside or outside the Circle's segment.
You can figure out if it is in the segment by finding out what angles bound the box, what the radius of the circle is, and the equation of the line between A and B. If the angle from $O(x,y)$ to $P(x,y)$ is between angle OA and OB, and the radius is between $y_{AB}$ and R, then the point is in the segment. | 677.169 | 1 |
Congruent vs. Equal
What's the Difference?
Congruent and equal are two terms used in mathematics to describe the relationship between geometric figures or numerical values. While they may seem similar, there is a subtle difference between the two. Congruent refers to figures or shapes that have the same shape and size, but they may be positioned differently. On the other hand, equal refers to numerical values that are exactly the same. In other words, congruent focuses on the similarity of shapes, while equal emphasizes the exactness of values.
Comparison
Attribute
Congruent
Equal
Definition
Figures that have the same shape and size.
Values or quantities that are the same.
Symbol
≅
=
Geometric Figures
Triangles, rectangles, circles, etc.
N/A
Measurements
Angles, sides, lengths, etc.
Numbers, quantities, values, etc.
Properties
Corresponding angles and sides are congruent.
Values or quantities are exactly the same.
Usage
Primarily used in geometry.
Used in various mathematical contexts.
Further Detail
Introduction
When it comes to geometry and mathematics, two terms that often come up are "congruent" and "equal." While they may seem similar at first glance, they have distinct meanings and applications. In this article, we will explore the attributes of congruent and equal, highlighting their differences and similarities.
Congruent
Congruent is a term used to describe geometric figures that have the same shape and size. When two figures are congruent, it means that all corresponding sides and angles are equal. This concept is fundamental in geometry, as it allows us to compare and analyze different shapes. Congruent figures can be translated, rotated, or reflected, but their size and shape remain the same.
For example, if we have two triangles, ABC and DEF, and we know that angle A is equal to angle D, side AB is equal to side DE, and side BC is equal to side EF, we can conclude that the triangles are congruent. This means that all other corresponding angles and sides will also be equal.
Congruence is denoted by the symbol ≅, which is placed between the names of the congruent figures. For instance, if triangle ABC is congruent to triangle DEF, we write it as ΔABC ≅ ΔDEF.
It is important to note that congruence applies to two-dimensional figures, such as triangles, quadrilaterals, and circles, but not to three-dimensional objects like spheres or cubes.
In summary, congruent figures have the same shape and size, with all corresponding angles and sides being equal.
Equal
Equal, on the other hand, is a broader term that can be applied to various mathematical concepts, not limited to geometry. When two quantities, values, or expressions are equal, it means they are exactly the same. Equality is a fundamental concept in mathematics, forming the basis for equations, identities, and mathematical operations.
For example, in arithmetic, if we have the equation 2 + 3 = 5, it means that the sum of 2 and 3 is equal to 5. In algebra, if we have the equation x + 4 = 9, we can solve for x and find that x is equal to 5. In both cases, equality is used to establish a relationship between different quantities.
Equality can also be applied to other mathematical concepts, such as inequalities, where we compare the relative size of two values. For instance, if we have the inequality 3< 5, it means that 3 is less than 5, but they are not equal.
Unlike congruence, which is specific to geometry, equality is a more general concept that can be applied across different branches of mathematics.
In summary, equal refers to two quantities or expressions that are exactly the same, establishing a relationship of equivalence.
Comparison
Now that we have explored the individual attributes of congruent and equal, let's compare them to understand their similarities and differences.
1. Geometric vs. General Concept
The most significant distinction between congruent and equal lies in their applications. Congruent is a term used exclusively in geometry to describe the equality of corresponding sides and angles in two-dimensional figures. On the other hand, equal is a broader concept that can be applied to various mathematical fields, including arithmetic, algebra, and calculus.
2. Shape and Size vs. Exactness
Congruent focuses on the shape and size of geometric figures. When two figures are congruent, it means they have the same shape and size, with all corresponding angles and sides being equal. In contrast, equal emphasizes exactness. Two quantities or expressions are equal when they are exactly the same, without considering their shape or size.
3. Symbolic Representation
Both congruent and equal have specific symbols to represent their concepts. Congruence is denoted by the symbol ≅, which is placed between the names of the congruent figures. On the other hand, equality is represented by the symbol =, which is used in equations and mathematical expressions.
4. Dimensionality
Congruence is limited to two-dimensional figures, such as triangles, quadrilaterals, and circles. It does not apply to three-dimensional objects like spheres or cubes. In contrast, equality has no such limitations and can be applied to any mathematical concept, regardless of its dimensionality.
5. Relationships and Operations
While congruence establishes a relationship between corresponding sides and angles of geometric figures, equality is used to establish relationships between different quantities or expressions. Equality is also fundamental in mathematical operations, allowing us to perform addition, subtraction, multiplication, and division.
Conclusion
In conclusion, congruent and equal are two terms used in mathematics with distinct meanings and applications. Congruent is specific to geometry, describing the equality of corresponding sides and angles in two-dimensional figures. On the other hand, equal is a broader concept that can be applied across various mathematical fields, emphasizing exactness and establishing relationships between different quantities or expressions. While they have some similarities, such as the use of symbols to represent their concepts, their differences lie in their focus, dimensionality, and applications. Understanding the attributes of congruent and equal is essential for a solid foundation in mathematics and geometry. | 677.169 | 1 |
Elementary Trigonometry
8. Now suppose the measures of the sides of a right-angled triangle to be p, q, r respectively, the right angle being subtended by that side whose measure is r.
Then since the geometrical property of such a triangle, established by Euclid 1. 47, may be extended to the case in which the sides are represented by numbers or symbols standing for numbers,
p2 + q2 = r2.
If any two of the numerical quantities involved in this equation are given, we can determine the third.
For example, if r = 5 and q = 3,
p2 + 9 = 25,
.. p2 = 16,
.. p=4.
EXAMPLES.-II.
(1) The hypothenuse being 51 yards, and one of the sides containing the right angle 24 yards, find the other side.
(2) The sides containing the right angle being 8 feet and 6 feet, find the hypothenuse.
(3) A rectangular field measures 225 yards in length, and 120 yards in breadth; what will be the length of a diagonal path across it?
(4) A rectangular field is 300 yards long and 200 yards broad; find the distance from corner to corner.
(5) A rectangular plantation, whose width is 88 yards, contains 2 acres; find the distance from corner to corner across the plantation.
(6) The sides of a right-angled triangle are in Arithmetical Progression and the hypothenuse is 20 feet; find the other sides.
(7) The sides of a right-angled triangle are in Arithmetical Progression; shew that they are proportional to 3, 4, 5.
(8) A ladder, whose foot rests in a given position, just reaches a window on one side of a street, and when turned about its foot, just reaches a window on the other side. If the two positions of the ladder be at right angles to each other, and the heights of the windows be 36 and 27 feet respectively, find the width of the street and the length of the ladder.
(9) In a right-angled isosceles triangle the hypothenuse is 12 feet, find the length of each of the other sides.
(10) What is the length of the diagonal of a square, whose side is 5 inches?
(11) The area of a square is 390625 square feet; what is the diagonal?
(12) Each side of an equilateral triangle is 13; find the length of the perpendicular dropped from one of the angles on the opposite side.
(13) If ABC be an equilateral triangle and the length of AD, a perpendicular on BC, be 15; find the length of AB.
(14) The radius of a circle is 37 inches; a chord is drawn in the circle: if the length of this chord be 70 inches, find its distance from the centre.
(15) The distance of a chord in a circle from the centre is 180 inches; the diameter of the circle is 362 inches: find the length of the chord.
(16) The length of a chord in a circle is 150 feet, and its distance from the centre is 308 feet; find the diameter of the circle.
9. We now proceed to treat of the ratio of the circumference of a circle to the diameter of that circle.
10. It is evident that a straight line can be compared as to its length with a circular arc, and that consequently the ratio between such lines can be represented in the form of a fraction.
11. We must assume as an axiom that an arc is greater than the chord subtending it: that is, if ABD be part of the circumference of a circle cut off by the straight line AD, the length of ABD is greater than the length of AD.
B
D
12. A figure enclosed by any number of straight lines is called a polygon.
13. A regular polygon is one in which all the sides and angles are equal.
14. The perimeter of a polygon is the sum of the sides. Hence if AB be one of the sides of a regular polygon of n sides the perimeter of the polygon will be n. AB.
15. The circumference of a circle is greater than the perimeter of any polygon which can be inscribed in the circle, but as the
number of sides of such a polygon is increased the perimeter of the polygon approaches nearer to the circumference of the circle, as will appear from the following illustration.
Let AB be the side of a regular hexagon ABDEFG inscribed in a circle.
Then AB is equal to the radius of the circle. EUCL. IV. 15.
Now the arc ACB is greater than AB,
and the circumference of the circle is therefore larger than the perimeter of the hexagon.
Hence
the circumference is greater than six times the radius,
Now
and greater than three times the diameter. suppose C to be the middle point of the arc AB. Join AC, CB.
These will be sides of a duodecagon or regular figure of 12 sides inscribed in the circle.
Now AC, CB are together greater than AB: but AC, CB are together less than the arc ACB.
Hence the perimeter of the duodecagon will be less than the circumference of the circle, but will approximate more nearly to the circumference of the circle than the perimeter of the hexagon.
So the larger the number of sides of a polygon inscribed in a circle, the more nearly does the perimeter of the polygon approach to the circumference of the circle; and when the number of sides is infinitely large, the perimeter of the polygon will become ultimately equal to the circumference of the circle.
16.
To shew that the ratio of the circumference of a circle to the radius is the same for all circles.
Let O and o be the centres of two circles.
Let AB, ab be sides of regular polygons of n sides inscribed in the circles, P, p the perimeters of the polygons, and C, c the circumferences of the circles.
Then OAB, oab are similar triangles.
.. OA: oa :: AB : ab
:: n.AB: n. ab
:: P: p.
Now when n is very large the perimeters of the polygons may be regarded as equal to the circumferences of the circles; | 677.169 | 1 |
Congruent Triangles
How is being "equal" different from being "congruent" in Geometry? When talking about shapes, we can say that one side is equal to another or the respective angles are equal: this term works well when talking about lengths or another numeric value. If you say, 'These two triangles are equal,' you're not providing the full story. The next questions would be, 'What is equal? The area? The lengths of the sides?' When two or more triangles have all equal areas, angles, and side lengths, we can more precisely describe them as congruent.
Congruent triangles: Meaning and examples
Congruent triangles have the same shape and size, with equal sides and angles, but they can be positioned differently from each other in space. When talking about congruent triangles, there must be two or more triangles in order to compare them with each other. You can't evaluate congruence on one triangle because it will always be congruent to itself! Let's look at an example that compares two triangles.
Imagine you have a right triangle, and your friend is sitting on the opposite side of a table from you with a copy of your triangle. You both put your triangles on the table with the right angle on the left side like this:
Right triangles turned 180º relative to one another, StudySmarter Original
These two triangles are congruent: they are the same size and shape. With a turn and a drag, the triangles can precisely overlap each other, like so:
When two or more triangles can overlap each other exactly, we know that they are congruent.
Given the example above, can you define congruent triangles? Try comparing your definition with the following one:
Congruent triangles are triangles of the same shape and size. However, they can be positioned differently in space.
Let's see a different example.
Three triangles are positioned differently from each other. One of them is also oriented differently; that is, it is rotated relative to the others. Just by looking at them, do you think these triangles are congruent? Take a look at the given triangles in the picture below.
Differently positioned triangles, StudySmarter Original
The first two triangles on top look congruent, right? They are the same shape and size as each other. The third triangle on the bottom may look a bit different from the first two because of the way it's oriented. If you were to rotate the third triangle 70° clockwise, you could more easily see that it is in fact congruent to the other two: it's the same shape and size. See the picture below.
Congruent triangles with the same orientation, StudySmarter Original
From this example, we see that triangles can still be congruent even if they are rotated or oriented differently in space. The same goes for congruent triangles that are flipped (reflected) or slid over (translated).
Notations for congruent triangles
We know that two triangles can be congruent, but now the question is: "How can we tell which side or angle from each triangle corresponds with which?" So, let's see how equal sides and angles are marked and identified.
Typically, we mark equal sides of congruent triangles with dash-like lines, while the angles have curved markings over them. We can see these notations in the figure of congruent triangles below. Notice that different corresponding sides and angles have their own matching notations to show which one matches with which. To avoid mixing up the separate sides and angles, different pairs have differing numbers of marks (i.e., one line, two lines, three lines).
Congruent triangles, StudySmarter Originals
In the above figure, sides AB and DE are equal, so it is marked by a single dash. Similarly, sides BC and EF are equal, and sides AC and DF are equal. Also, ∠A=∠D, which is shown by a single curved mark. Double and triple curved marks are used as a notation to show that ∠B=∠E and ∠C=∠F, respectively. Note that the ≅ sign is used to show that the triangles are congruent. For the above figure, we can say that △ABC≅△DEF; that is, triangle ABC is congruent to triangle DEF.
Congruent triangles versus non-congruent triangles
It's important to note that for congruent triangles to stay congruent, we can only perform transformations of translation (location change) or rotation on any one of them. If we need to transform the shape or size (or some angles or lengths) of a triangle to make them exactly overlap one another, then the triangles are non-congruent. Let's define non-congruent triangles.
Non-congruent triangles are triangles differing in shape and/or size relative to each other.
Let's look at an example.
Two triangles are given in the picture below. Do they look congruent?
Two non-congruent triangles, StudySmarter Originals
Solution: This case is pretty evident: the given triangles are different in shape and size, no matter how you move or rotate them. This means the given triangles are non-congruent.
Check whether the given triangles are congruent or not.
Two similar triangles, StudySmarter Originals
Solution: Based on the triangles' angle notations, we can see that ∠A=∠D,∠B=∠E,∠C=∠F. By looking at the figure, it may seem as though the triangles are congruent. However, the triangle ABC is bigger than the triangle DEF. Although the triangles have the same shape, they are not congruent. When we drag triangle DEF onto triangle ABC, they don't fit each other exactly. This is because their sides' lengths are not equal.
Similar triangles placed on each other, StudySmarter Originals
Triangle DEF fits inside triangle ABC. Hence, the triangles have the same shape but different sizes, making them non-congruent triangles. They are, however, similar!
Similar triangles are triangles that have the same shape (due to their equal corresponding angles) but are different in size.
Rules for determining congruent triangles
Now that we know the concept of congruent triangles, how can we determine if triangles are congruent? In the first two examples, we dragged and turned the triangles to see if they were congruent. You may be wondering if we can tell if triangles are congruent just by looking at them closely. However, relying on vision alone can be faulty and inaccurate. Rotating and dragging the triangles is also not the best method! There are more accurate methods for finding out if triangles are congruent.
As you might have guessed, we can determine congruency by measuring the triangles. Those with equal corresponding sides and angles are congruent. Let's see an example of this method.
Two triangles are positioned right next to each other. The second one is rotated. We need to find out whether these two are congruent.
Differently oriented triangles, StudySmarter Original
First, let's rotate the second triangle, so it's positioned in the same orientation as the first triangle. This step is not mandatory, but it helps to see the shapes more clearly and compare the right sides and angles.
Equally oriented triangles, StudySmarter Originals
Next, let's measure the angles and sides of both triangles. To understand our results more clearly, let's name the triangles ABC and DEF.
By comparing the triangles' measurements, we can see that they are all equal. This means the triangles are congruent. We denote this with the following symbol:
△ABC≅△DEF
It would take quite some time to measure every side and every angle whenever we wanted to determine triangle congruency. For this reason, we rely on triangle congruency theorems which help us reduce the number of measurements needed to evaluate congruency.
Congruent triangles types and theorems
Suppose you only know some information about two triangles' measurements, such as the measurements of two of their angles and the side length between those angles. If these specific measurements are equal between the two triangles, this limited amount of information is enough to prove that they are congruent. Why don't we need all of the triangles' angle measurements and side lengths to confirm it? This is because we can refer to the known triangle congruency theorem, Angle-Side-Angle (ASA), which states that our equal measurements (two angles and their shared side) are sufficient.
Depending on the type of information that we have about the triangles' measurements, we can select from the five triangle congruency theorems to help us evaluate congruency. The known theorems for triangle congruence are shown in the table below. We can also think of these theorems as shortcuts of measurement, each with their own specific rules and conditions.
Theorem Name
Statement
Figure
Side-Side-Side(SSS)
The triangles are congruent if... all three sides of one triangle are equal to all three sides of another triangle.
SSS congruent triangles, StudySmarter Originals
Side-Angle-Side(SAS)
The triangles are congruent if... two sides and the included angle of these sides of one triangle are equal to the two sides and included angle of another triangle.
SAS congruent triangles, StudySmarter Originals
Hypothenuse-Leg(HL)
The triangles are congruent if... the hypothenuse side and any one side of one right triangle are equal to the hypotenuse and another side of another right triangle.
HL congruent triangles, StudySmarter Originals
Angle-Side-Angle(ASA)
The triangles are congruent if... two angles and the included side of one triangle are equal to the two angles and included side of another triangle.
ASA congruent triangles, StudySmarter Originals
Angle-Angle-Side(AAS)
The triangles are congruent if... two angles and a non-included side of one triangle are equal to the two angles and a non-included side of another triangle.
AAS congruent triangles, StudySmarter Originals
Congruent triangles - Key takeaways
Triangles are congruent if their respective sides and angles are equal (they have the same shape and size).
Congruent triangles can be placed in different locations and rotated relative to each other.
Non-congruent triangles differ in shape and/or size.
There are rules or theorems that help determine whether triangles are congruent or not.
Frequently Asked Questions about Congruent Triangles
When talking about congruence, it usually involves two or more triangles. If two or more triangles are congruent, it means they have the same shape and size - equal sides and angles.
How to prove triangles are congruent?
To prove that two or more triangles are congruent, you need to know that the given triangles have equal respective sides and angles. You can also use one of the five theorems for proving triangle congruence (SSS, SAS, ASA, AAS and HL).
How to find congruent triangles?
To find congruent triangles, you need to look at the shape and size of the given triangles. The triangles that have equal respective sides and angles are congruent. You can also use one of the five theorems to find congruent triangles (SSS, SAS, ASA, AAS and HL).
What are the rules in congruent triangles?
The rules for congruent triangles are - they need to have equal respective sides and angles. These two rules are also encompassed in five theorems for proving triangle congruence - SSS, SAS, ASA, AAS and HL.
What is an example of congruent triangles?
An example of congruent triangles is two equilateral triangles with a side of length 6cm. Equilateral triangles have all sides of the same length, so this means two such triangles with a side of the same length are congruent - all of the sides and angles are equal between the two triangles.
Test your knowledge with multiple choice flashcards
Are triangles with equal areas congruent?
A. No
B. Yes
C. Only if they are right triangles
D. It depends on the shapes of the triangles
Which statements are true?
A. Triangles with equal areas are always congruent
B. Congruent triangles can be of different magnitudes but have to be of the same shape
C. Congruent triangles have equal respective angles and equal areas
D. Congruent triangles have equal respective sides and angles
How many triangles need to be given to talk about congruence?
A. One or none
B. Less than two
C. At least one
D. Two or more | 677.169 | 1 |
It is always true that one of the two sets of opposite sides of a trapezium has the same length
Asked on 5/31/2023, 93 pageviews
2 Answers
as the other set of opposite sides.
This is not always true. In a trapezium, opposite sides are not necessarily congruent unless it is an isosceles trapezium. An isosceles trapezium has two parallel sides of equal length, which means that the opposite sides are also of equal length. However, in a non-isosceles trapezium, the two pairs of opposite sides can have different lengths.
by gpt-3.5-turbo
Second Answer
No, it is not always true that one of the two sets of opposite sides of a trapezium has the same length. In a trapezium, the opposite sides are not necessarily equal in length, unless it is a special case of an isosceles trapezium, where the non-parallel sides are equal in length. In a general trapezium, the lengths of the opposite sides can be different. | 677.169 | 1 |
Congruent Corresponding Angles to Start? (Quick Investigation)
How it works ?
In the applet below, the purple angle's measure can be changed by adjusting the slider. In addition, the BIG WHITE POINTS can be moved anywhere you'd like | 677.169 | 1 |
Find the measure of all the angles of a parallelogram, if one angle is 24∘
less than twice the smallest angle.
Video Solution
Text Solution
Verified by Experts
Let the smallest angle of the parallelogram be x. According to the question: The largest angle=(2x−24∘) Sum of the adjacent angles of the parallelogram=180∘ x+2x−24∘=180∘ 3x=180∘+24∘ 3x=204∘ x=68∘ ∴ The largest angle=2(68∘−24∘)=112∘ | 677.169 | 1 |
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