Chapter 5 unit 9
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The document discusses the Pythagorean theorem and distance formula. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The distance formula calculates the distance between two points and involves taking the difference of the x-coordinates squared and the difference of the y-coordinates squared and adding those values together. Examples of using both the Pythagorean theorem and distance formula are provided.
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Chapter 5 unit 9
The Pythagorean theorem is used to calculate the length of the hypotenuse of a right triangle using the lengths of the other two sides. It states that the sum of the squares of the two legs equals the square of the hypotenuse. The distance formula calculates the distance between two points and involves taking the difference of the x-coordinates squared and the difference of the y-coordinates squared and adding those values together. It is used to find distances in applications like navigation and physics.
Laverne thomas
The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. It is represented by the equation A^2 + B^2 = C^2, where C is the hypotenuse and A and B are the legs. The distance formula calculates the distance between two points in a plane by taking the square root of the sum of the squared differences of their x- and y-coordinates. It is written as √(x2 - x1)2 + (y2 - y1)2. An example of each is provided to demonstrate their use.
Powerpoint
The document explains the Pythagorean theorem and distance formula. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. An example calculates the length of the hypotenuse using the theorem. The distance formula calculates the distance between two points by taking the difference of their x- and y-coordinates and plugging them into the formula. An example finds the distance between two points using the formula.
Pythagorean theorem & distance formula ppt
The Pythagorean theorem states that in any right triangle, the sum of the squares of the two shorter sides equals the square of the longest side. It is used for right triangles, which have one 90 degree angle. The distance formula calculates the distance between two points by taking the difference of the x-coordinates squared and the difference of the y-coordinates squared, adding those values, and taking the square root of the sum. An example is worked out step-by-step to demonstrate using the distance formula.
Pythagorean theorem and distance formula powerpoint
The document discusses the Pythagorean theorem and distance formula. The Pythagorean theorem is used to find the length of the unknown side of a right triangle given two other side lengths using the formula a2 + b2 = c2. The distance formula calculates the distance between any two points in a plane by taking the square root of the sum of the squared differences between their x- and y-coordinates. An example demonstrates labeling points and plugging them into the distance formula to solve for the distance.
Pythagorean theorem and distance formula
The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The distance formula provides an algebraic expression to calculate the distance between two points by taking the difference of their x-coordinates squared plus the difference of their y-coordinates squared, and finding the square root of the total. Examples demonstrate using the distance formula to solve for the length of one side of a right triangle given the other two sides.
Capriesha gray
The document discusses the Pythagorean theorem and distance formula. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. The distance formula uses the Pythagorean theorem to calculate the distance between two points by finding the difference of their x-coordinates squared and y-coordinates squared and taking the square root of the sum. An example problem demonstrates applying the distance formula to find the distance between two points.
Pythagorean Theorem
The Pythagorean theorem states that the sum of the squares of the two shorter sides of a right triangle equals the square of the longest side or hypotenuse. It is used to calculate the length of the hypotenuse given the lengths of the other two sides. Two examples demonstrate using the Pythagorean theorem to find the length of unknown sides of right triangles.
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Chapter 5 unit 9
The Pythagorean theorem is used to calculate the length of the hypotenuse of a right triangle using the lengths of the other two sides. It states that the sum of the squares of the two legs equals the square of the hypotenuse. The distance formula calculates the distance between two points and involves taking the difference of the x-coordinates squared and the difference of the y-coordinates squared and adding those values together. It is used to find distances in applications like navigation and physics.
Laverne thomas
The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. It is represented by the equation A^2 + B^2 = C^2, where C is the hypotenuse and A and B are the legs. The distance formula calculates the distance between two points in a plane by taking the square root of the sum of the squared differences of their x- and y-coordinates. It is written as √(x2 - x1)2 + (y2 - y1)2. An example of each is provided to demonstrate their use.
Powerpoint
The document explains the Pythagorean theorem and distance formula. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. An example calculates the length of the hypotenuse using the theorem. The distance formula calculates the distance between two points by taking the difference of their x- and y-coordinates and plugging them into the formula. An example finds the distance between two points using the formula.
Pythagorean theorem & distance formula ppt
The Pythagorean theorem states that in any right triangle, the sum of the squares of the two shorter sides equals the square of the longest side. It is used for right triangles, which have one 90 degree angle. The distance formula calculates the distance between two points by taking the difference of the x-coordinates squared and the difference of the y-coordinates squared, adding those values, and taking the square root of the sum. An example is worked out step-by-step to demonstrate using the distance formula.
Pythagorean theorem and distance formula powerpoint
The document discusses the Pythagorean theorem and distance formula. The Pythagorean theorem is used to find the length of the unknown side of a right triangle given two other side lengths using the formula a2 + b2 = c2. The distance formula calculates the distance between any two points in a plane by taking the square root of the sum of the squared differences between their x- and y-coordinates. An example demonstrates labeling points and plugging them into the distance formula to solve for the distance.
Pythagorean theorem and distance formula
The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The distance formula provides an algebraic expression to calculate the distance between two points by taking the difference of their x-coordinates squared plus the difference of their y-coordinates squared, and finding the square root of the total. Examples demonstrate using the distance formula to solve for the length of one side of a right triangle given the other two sides.
Capriesha gray
The document discusses the Pythagorean theorem and distance formula. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. The distance formula uses the Pythagorean theorem to calculate the distance between two points by finding the difference of their x-coordinates squared and y-coordinates squared and taking the square root of the sum. An example problem demonstrates applying the distance formula to find the distance between two points.
Pythagorean Theorem
The Pythagorean theorem states that the sum of the squares of the two shorter sides of a right triangle equals the square of the longest side or hypotenuse. It is used to calculate the length of the hypotenuse given the lengths of the other two sides. Two examples demonstrate using the Pythagorean theorem to find the length of unknown sides of right triangles.
Presentation for chapter 9
The document explains the Pythagorean theorem and distance formula. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. Examples are provided to demonstrate calculating the length of one side of a right triangle given the other two sides. The distance formula is also explained as a way to calculate the length of one side of any triangle by squaring the differences between corresponding x- and y-coordinates of the triangle's vertices and summing the results. An example applies the distance formula to find the length between two points.
Powerpoint
The document discusses the Pythagorean theorem and distance formula. The Pythagorean theorem states that the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides. The distance formula calculates the distance between two points and is the square root of (x2 - x1)2 + (y2 - y1)2. An example problem applies the distance formula to find the distance between the points (-1,4) and (1,3).
Algebra pythagorean powerpoint
The document discusses the Pythagorean theorem and distance formula. The Pythagorean theorem states that for any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. The distance formula calculates the distance between two points by taking the difference of their x-coordinates squared plus the difference of their y-coordinates squared, which equals the distance. An example of each is provided to demonstrate their use.
Robin wilson
The document discusses the Pythagorean theorem and distance formula. It explains that the Pythagorean theorem states that the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides. It also provides an example of using the distance formula, which involves subtracting the x-coordinates and y-coordinates of two points and summing the squares of the differences.
PT2
The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It can be used to determine if a triangle is right, and to solve for missing sides. For example, if one side is 35m, another is 12m, and the hypotenuse is 37m, squaring and adding 35^2 + 12^2 equals 37^2, showing it is a right triangle. It can also solve for missing components, like finding a hypotenuse of 57.63m when the other two sides are 45m and 36m.
Baudhayan triples[1]
1) Bodhayana Triples describe right triangles using three numbers where the cosine, sine, and tangent of the angle can be expressed in terms of the numbers.
2) Triples can be added or subtracted to obtain the triple for the sum or difference of two angles. For example, the triple for 75 degrees is found by adding the triples for 30 and 45 degrees.
3) A point can be rotated by treating its coordinates as the adjacent and opposite sides of a right triangle with the triple corresponding to the angle of rotation. The new coordinates are obtained by adding the triples. For example, rotating (3,4) 30 degrees clockwise using triples yields the new coordinates of (3√3 - 4)/
Pythagorean theorem and distance formula
The document discusses the Pythagorean theorem and distance formula. The Pythagorean theorem states that for any right triangle, the sum of the squares of the two sides equals the square of the hypotenuse. The distance formula allows you to calculate the distance between two points in the xy-plane by taking the square root of the sum of the squared differences between the x-coordinates and y-coordinates. An example using points (2,4) and (7,13) demonstrates applying the distance formula to get a distance of 10.3 units between the two points.
Pythagorean theorem and distance formula
The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. An example calculates the length of the hypotenuse of a triangle with sides of lengths 10 and 6 units. The distance formula calculates the distance between two points in the xy-plane by taking the square root of the sum of the squared differences of the x- and y-coordinates, as shown in an example finding the distance between points (1,4) and (10,16).
1..3 distance formula
This document discusses the Pythagorean theorem and how it relates to the distance formula. It states that in a right triangle, the hypotenuse squared is equal to the sum of the legs squared. It then shows how to derive the distance formula from the Pythagorean theorem, allowing you to calculate the distance between two points by taking the difference of their x-coordinates squared plus the difference of their y-coordinates squared.
Untitled 1
The document discusses the Pythagorean theorem and distance formula. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. It provides an example of using the theorem to solve for the length of one side. The distance formula calculates the distance between two points in the xy-plane by taking the square root of the sum of the squares of the differences between the x- and y-coordinates. An example of using the distance formula to find the distance between two points is also given.
The Pythagorean Theorem
The Pythagorean Theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It can be used to determine if a triangle is right, and to calculate unknown sides of a right triangle if two sides are known. Examples show how to use the theorem to find a missing hypotenuse or other side by setting up and solving an equation based on the formula a^2 + b^2 = c^2.
Welcome to International Journal of Engineering Research and Development (IJERD)
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals
Distance formula powerpoint
The document explains the Pythagorean theorem and distance formula. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The distance formula calculates the distance between two points by taking the square root of the sum of the squared differences between their x- and y-coordinates. An example is shown of each formula being used to solve a problem.
Pythagorean theorem and distance formula
The document discusses the Pythagorean theorem and distance formula. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. An example problem demonstrates using the theorem to solve for the length of one side. The distance formula calculates the distance between two points in a plane by taking the square root of the sum of the squared differences of their x- and y-coordinates. An example applies the formula to find the distance between two points.
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The document provides examples and step-by-step workings for evaluating algebraic expressions by substituting number values for variables. It demonstrates evaluating expressions including powers, integers, fractions, and evaluating expressions to find volumes. The examples show applying order of operations and substitution to simplify expressions down to a single numerical value.
1st semester geometry project
1. The document discusses geometry concepts such as parallel lines, segments and their measures, properties of parallel lines, triangles and angles, the exterior angle theorem, medians and altitudes of a triangle.
2. It provides examples using a swimming pool diagram to illustrate parallel pool dividers, finding segment lengths, applying angle theorems, and finding missing angle measures.
3. The final example uses the concurrency of medians theorem to find the length of a median and the distance from the centroid to a vertex in a triangle, given one side of the triangle and the distance from the centroid to that side.
5.13.3 Geometric Probability and Changing Dimensions
Students will
* Calculate geometric probabilities
* Use geometric probability to predict results in real-world situations
* Predict the effects of changing dimensions on the perimeter/circumference and area of a figure.
Powerpoint..
The document explains the Pythagorean theorem and distance formula. The Pythagorean theorem states that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse. It provides an example of using the theorem to solve for the missing length. The distance formula is also explained as taking the difference of the x-coordinates squared plus the difference of the y-coordinates squared to calculate the distance between any two points. An example using coordinates is provided.
On Triplet of Positive Integers Such That the Sum of Any Two of Them is a Per...
In this article we discussed determination of distinct positive integers a, b, c such that a + b, a + c, b + c are perfect squares. We can determine infinitely many such triplets. There are such four tuples and from them eliminating any one number we obtain triplets with the specific property. We can also obtain infinitely many such triplets from a single triplet.
Obj. 24 Special Right Triangles
Identify when a triangle is a 45-45-90 or 30-60-90 triangle
Use special right triangle relationships to solve problems
4.11.2 Special Right Triangles
This document discusses special right triangles and their properties. It defines 45-45-90 and 30-60-90 triangles, and provides the key relationships between their sides: for 45-45-90 triangles, the hypotenuse is √2 times the leg; for 30-60-90 triangles, the hypotenuse is 2 times the shorter leg and √3 times the longer leg. It provides examples of using these relationships to solve for missing side lengths.
Pythagorean theorem power point !
The document discusses the Pythagorean theorem and distance formula. It explains that the Pythagorean theorem uses the variables a, b, and c to calculate the hypotenuse of a right triangle according to the equation a^2 + b^2 = c^2. It also defines the distance formula as (X2 - X1)2 + (Y2 - Y1)2, which is used to calculate the distance between two points on a coordinate plane. An example of using the distance formula to find the distance between the points (5, -2) and (-3, 4) is shown.
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The document explains the Pythagorean theorem and distance formula. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. Examples are provided to demonstrate calculating the length of one side of a right triangle given the other two sides. The distance formula is also explained as a way to calculate the length of one side of any triangle by squaring the differences between corresponding x- and y-coordinates of the triangle's vertices and summing the results. An example applies the distance formula to find the length between two points.
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The document discusses the Pythagorean theorem and distance formula. The Pythagorean theorem states that the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides. The distance formula calculates the distance between two points and is the square root of (x2 - x1)2 + (y2 - y1)2. An example problem applies the distance formula to find the distance between the points (-1,4) and (1,3).
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The document discusses the Pythagorean theorem and distance formula. The Pythagorean theorem states that for any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. The distance formula calculates the distance between two points by taking the difference of their x-coordinates squared plus the difference of their y-coordinates squared, which equals the distance. An example of each is provided to demonstrate their use.
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The document discusses the Pythagorean theorem and distance formula. It explains that the Pythagorean theorem states that the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides. It also provides an example of using the distance formula, which involves subtracting the x-coordinates and y-coordinates of two points and summing the squares of the differences.
PT2
The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It can be used to determine if a triangle is right, and to solve for missing sides. For example, if one side is 35m, another is 12m, and the hypotenuse is 37m, squaring and adding 35^2 + 12^2 equals 37^2, showing it is a right triangle. It can also solve for missing components, like finding a hypotenuse of 57.63m when the other two sides are 45m and 36m.
Baudhayan triples[1]
1) Bodhayana Triples describe right triangles using three numbers where the cosine, sine, and tangent of the angle can be expressed in terms of the numbers.
2) Triples can be added or subtracted to obtain the triple for the sum or difference of two angles. For example, the triple for 75 degrees is found by adding the triples for 30 and 45 degrees.
3) A point can be rotated by treating its coordinates as the adjacent and opposite sides of a right triangle with the triple corresponding to the angle of rotation. The new coordinates are obtained by adding the triples. For example, rotating (3,4) 30 degrees clockwise using triples yields the new coordinates of (3√3 - 4)/
Pythagorean theorem and distance formula
The document discusses the Pythagorean theorem and distance formula. The Pythagorean theorem states that for any right triangle, the sum of the squares of the two sides equals the square of the hypotenuse. The distance formula allows you to calculate the distance between two points in the xy-plane by taking the square root of the sum of the squared differences between the x-coordinates and y-coordinates. An example using points (2,4) and (7,13) demonstrates applying the distance formula to get a distance of 10.3 units between the two points.
Pythagorean theorem and distance formula
The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. An example calculates the length of the hypotenuse of a triangle with sides of lengths 10 and 6 units. The distance formula calculates the distance between two points in the xy-plane by taking the square root of the sum of the squared differences of the x- and y-coordinates, as shown in an example finding the distance between points (1,4) and (10,16).
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This document discusses the Pythagorean theorem and how it relates to the distance formula. It states that in a right triangle, the hypotenuse squared is equal to the sum of the legs squared. It then shows how to derive the distance formula from the Pythagorean theorem, allowing you to calculate the distance between two points by taking the difference of their x-coordinates squared plus the difference of their y-coordinates squared.
Untitled 1
The document discusses the Pythagorean theorem and distance formula. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. It provides an example of using the theorem to solve for the length of one side. The distance formula calculates the distance between two points in the xy-plane by taking the square root of the sum of the squares of the differences between the x- and y-coordinates. An example of using the distance formula to find the distance between two points is also given.
The Pythagorean Theorem
The Pythagorean Theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It can be used to determine if a triangle is right, and to calculate unknown sides of a right triangle if two sides are known. Examples show how to use the theorem to find a missing hypotenuse or other side by setting up and solving an equation based on the formula a^2 + b^2 = c^2.
Welcome to International Journal of Engineering Research and Development (IJERD)
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals
Distance formula powerpoint
The document explains the Pythagorean theorem and distance formula. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The distance formula calculates the distance between two points by taking the square root of the sum of the squared differences between their x- and y-coordinates. An example is shown of each formula being used to solve a problem.
Pythagorean theorem and distance formula
The document discusses the Pythagorean theorem and distance formula. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. An example problem demonstrates using the theorem to solve for the length of one side. The distance formula calculates the distance between two points in a plane by taking the square root of the sum of the squared differences of their x- and y-coordinates. An example applies the formula to find the distance between two points.
Alegebra Powers Substitution
The document provides examples and step-by-step workings for evaluating algebraic expressions by substituting number values for variables. It demonstrates evaluating expressions including powers, integers, fractions, and evaluating expressions to find volumes. The examples show applying order of operations and substitution to simplify expressions down to a single numerical value.
1st semester geometry project
1. The document discusses geometry concepts such as parallel lines, segments and their measures, properties of parallel lines, triangles and angles, the exterior angle theorem, medians and altitudes of a triangle.
2. It provides examples using a swimming pool diagram to illustrate parallel pool dividers, finding segment lengths, applying angle theorems, and finding missing angle measures.
3. The final example uses the concurrency of medians theorem to find the length of a median and the distance from the centroid to a vertex in a triangle, given one side of the triangle and the distance from the centroid to that side.
5.13.3 Geometric Probability and Changing Dimensions
Students will
* Calculate geometric probabilities
* Use geometric probability to predict results in real-world situations
* Predict the effects of changing dimensions on the perimeter/circumference and area of a figure.
Powerpoint..
The document explains the Pythagorean theorem and distance formula. The Pythagorean theorem states that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse. It provides an example of using the theorem to solve for the missing length. The distance formula is also explained as taking the difference of the x-coordinates squared plus the difference of the y-coordinates squared to calculate the distance between any two points. An example using coordinates is provided.
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Welcome to International Journal of Engineering Research and Development (IJERD)
Welcome to International Journal of Engineering Research and Development (IJERD)
5.13.3 Geometric Probability and Changing Dimensions
5.13.3 Geometric Probability and Changing Dimensions
Similar to Chapter 5 unit 9
On Triplet of Positive Integers Such That the Sum of Any Two of Them is a Per...
In this article we discussed determination of distinct positive integers a, b, c such that a + b, a + c, b + c are perfect squares. We can determine infinitely many such triplets. There are such four tuples and from them eliminating any one number we obtain triplets with the specific property. We can also obtain infinitely many such triplets from a single triplet.
Obj. 24 Special Right Triangles
Identify when a triangle is a 45-45-90 or 30-60-90 triangle
Use special right triangle relationships to solve problems
4.11.2 Special Right Triangles
This document discusses special right triangles and their properties. It defines 45-45-90 and 30-60-90 triangles, and provides the key relationships between their sides: for 45-45-90 triangles, the hypotenuse is √2 times the leg; for 30-60-90 triangles, the hypotenuse is 2 times the shorter leg and √3 times the longer leg. It provides examples of using these relationships to solve for missing side lengths.
Pythagorean theorem power point !
The document discusses the Pythagorean theorem and distance formula. It explains that the Pythagorean theorem uses the variables a, b, and c to calculate the hypotenuse of a right triangle according to the equation a^2 + b^2 = c^2. It also defines the distance formula as (X2 - X1)2 + (Y2 - Y1)2, which is used to calculate the distance between two points on a coordinate plane. An example of using the distance formula to find the distance between the points (5, -2) and (-3, 4) is shown.
4.11.2 Special Right Triangles
1) Special right triangles have properties that allow for determining side lengths without using the Pythagorean theorem. The 45-45-90 triangle has legs that are equal in length and a hypotenuse that is √2 times the leg length. The 30-60-90 triangle has a shorter leg that is half the hypotenuse, a longer leg that is √3 times the shorter leg, and a hypotenuse that is 2 times the shorter leg.
2) Examples demonstrate using the properties of special right triangles to find missing side lengths by setting up and solving equations based on the corresponding theorems. Rationalizing denominators may be required when finding a leg from a known hypotenuse.
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The document summarizes the Pythagorean theorem and distance formula. It states that the Pythagorean theorem equation is a^2 + b^2 = c^2, where a and b are the legs of a right triangle and c is the hypotenuse. An example problem demonstrates solving for the missing side of a right triangle. The distance formula is defined as the square root of (x2-x1)^2 + (y2-y1)^2, and a worked example finds the distance between two points.
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The document discusses the Pythagorean theorem and distance formula. It defines the Pythagorean theorem equation as a^2 + b^2 = c^2, where a and b are the legs of a right triangle and c is the hypotenuse. An example problem demonstrates using the theorem to solve for the missing side of a right triangle. The distance formula is defined as the square root of (x2-x1)^2 + (y2-y1)^2, and an example shows labeling points and plugging them into the formula to calculate the distance between two points.
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The document discusses the Pythagorean theorem and the distance formula. It explains that the Pythagorean theorem states that for any right triangle, a^2 + b^2 = c^2, where a and b are the lengths of the two sides and c is the length of the hypotenuse. It also provides an example problem. The document then explains that the distance formula calculates the distance between two points (x1, y1) and (x2, y2) by taking the difference of the x-coordinates squared and the difference of the y-coordinates squared, then summing and taking the square root of the result. It provides an example distance formula problem.
Pythagorean theorem
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. The theorem can be expressed as a2 + b2 = c2, where a and b are the legs and c is the hypotenuse.
The distance formula is used to calculate the distance between two points in a coordinate plane. It is calculated by taking the square root of the sum of the squares of the differences between the x-coordinates and y-coordinates of the two points.
For example, to find the distance between points (3,17) and (5,27), you would calculate √(5-3)2 +
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The document discusses the Pythagorean theorem and distance formula.
The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It is expressed as a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse.
The distance formula uses the same concept as the Pythagorean theorem to calculate the distance between two points in a coordinate plane. It is written as (x2 - x1)^2 + (y2 - y1)^2 = c^2, where c is the distance between the points (x1, y1) and
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(4) Parabola theory Module.pdf
A parabola is the locus of a point which moves in such a way that its distance from a fixed point is equal to its perpendicular distance from a fixed straight line.
1.1 Focus : The fixed point is called the focus of the Parabola.
1.2 Directrix : The fixed line is called the directrix of the Parabola.
(focus)
2.3 Vertex : The point of intersection of a parabola and its axis is called the vertex of the Parabola.
NOTE: The vertex is the middle point of the focus and the point of intersection of axis and directrix
2.4 Focal Length (Focal distance) : The distance of any point P (x, y) on the parabola from the focus is called the focal length. i.e.
The focal distance of P = the perpendicular distance of the point P from the directrix.
2.5 Double ordinate : The chord which is perpendicular to the axis of Parabola or parallel to Directrix is called double ordinate of the Parabola.
2.6 Focal chord : Any chord of the parabola passing through the focus is called Focal chord.
2.7 Latus Rectum : If a double ordinate passes through the focus of parabola then it is called as latus rectum.
2.7.1 Length of latus rectum :
The length of the latus rectum = 2 x perpendicular distance of focus from the directrix.
2.1 Eccentricity : If P be a point on the parabola and PM and PS are the distances from the directrix and focus S respectively then the ratio PS/PM is called the eccentricity of the Parabola which is denoted by e.
Note: By the definition for the parabola e = 1.
If e > 1 Hyperbola, e = 0 circle, e < 1
ellipse
2.2 Axis : A straight line passes through the focus and perpendicular to the directrix is called the axis of parabola.
If we take vertex as the origin, axis as x- axis and distance between vertex and focus as 'a' then equation of the parabola in the simplest form will be-
y2 = 4ax
3.1 Parameters of the Parabola y2 = 4ax
(i) Vertex A (0, 0)
(ii) Focus S (a, 0)
(iii) Directrix x + a = 0
(iv) Axis y = 0 or x– axis
(v) Equation of Latus Rectum x = a
(vi) Length of L.R. 4a
(vii) Ends of L.R. (a, 2a), (a, – 2a)
(viii) The focal distance sum of abscissa of the point and distance between vertex and L.R.
(ix) If length of any double ordinate of parabola
y2 = 4ax is 2 𝑙 then coordinates of end points of this Double ordinate are
𝑙2 𝑙2
, 𝑙
and
, 𝑙 .
4a
4a
3.2 Other standard Parabola :
Equation of Parabola Vertex Axis Focus Directrix Equation of Latus rectum Length of Latus rectum
y2 = 4ax (0, 0) y = 0 (a, 0) x = –a x = a 4a
y2 = – 4ax (0, 0) y = 0 (–a, 0) x = a x = –a 4a
x2 = 4ay (0, 0) x = 0 (0, a) y = a y = a 4a
x2 = – 4ay (0, 0) x = 0 (0, –a) y = a y = –a 4a
Standard form of an equation of Parabola
Ex.1 If focus of a parabola is (3,–4) and directrix is x + y – 2 = 0, then its vertex is (A) (4/15, – 4/13)
(B) (–13/4, –15/4)
(C) (15/2, – 13/2)
(D) (15/4, – 13/4)
Sol. First we find the equation of axis of parabola
Pythagorean theorem and distance formula
The Pythagorean theorem is used to calculate the length of the hypotenuse of a right triangle using the lengths of the other two sides. The distance formula calculates the distance between two points in a plane by taking the square root of the sum of the squared differences between their x- and y-coordinates. Examples are provided to demonstrate how to apply the Pythagorean theorem to solve for the length of one side of a right triangle and how to use the distance formula to calculate the distance between two points.
Pythagorean theorem and distance formula
The Pythagorean theorem is used to calculate the length of the hypotenuse of a right triangle using the lengths of the two legs. The distance formula calculates the distance between two points in a plane by taking the square root of the sum of the squared differences between their x- and y-coordinates. Examples are provided to demonstrate how to apply the Pythagorean theorem to solve for the length of the third side of a right triangle and how to use the distance formula to calculate the distance between two points.
2.1 Rectangular Coordinates
This document defines and explains ordered pairs, rectangular coordinate systems, and formulas for distance and midpoint between points in a coordinate plane. It begins by defining ordered pairs and how they represent relationships between elements of two sets. It then introduces the rectangular coordinate system using perpendicular x and y axes intersecting at the origin, and how points are located using ordered pair coordinates. It provides the distance formula for finding the distance between any two points using their x and y coordinates. Examples are given to demonstrate using the formula and interpreting results. The midpoint formula is also defined for finding the midpoint of a line segment given the endpoints. Examples are worked through for both formulas.
Pythagorean Theorem
Pythagoras was an ancient Greek thinker, but he was not the founder of the Pythagorean theorem. That honor goes to his followers, known as the Pythagorean Brotherhood, who established the theorem over 100 years after Pythagoras' death. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This relationship has many practical applications in fields like engineering, construction, physics, and astronomy that involve calculating distances.
Obj. 24 Special Right Triangles
Identify when a triangle is a 45-45-90 or 30-60-90 triangle
Use special right triangle relationships to solve problems
Geometry Section 1-3 1112
This document discusses distance and midpoints between points in a coordinate plane. It defines distance as the length of a segment between two points and the Pythagorean theorem. The midpoint of a segment is the point halfway between the two endpoints. Examples are provided to demonstrate calculating distance and midpoints using formulas like the distance formula and midpoint formula.
Core sub math_att_4pythagoreantheorem
The document provides information about the Pythagorean theorem:
1) It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
2) It gives examples of right triangles that satisfy the theorem, such as ones with sides of 3, 4, 5 or 5, 12, 13.
3) It includes an animated proof of the theorem showing how the area of the square on the hypotenuse equals the combined areas of the squares on the other two sides.
mathematics part-2.docx
The document contains solutions to several math problems involving arithmetic progressions (APs) and geometric progressions (GPs). It summarizes key information about various AP and GP sequences, including their first terms, common differences/ratios, number of terms, and calculated sums. It also shows sample calculations for finding terms, differences, and sums of numeric AP and GP sequences.
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Chapter 5 unit 9
- 1. Pythagorean Theorem & Distance Formula Laquanda Baker
- 3. A^2 + B^2 = C^2
- 4. We use a right angle triangle
- 5. The variables stand for the sides of the right triangle
- 6. A and B stands for the legs on the triangle
- 7. C stands for the hypotenuse which is the longest side of the 90* angle triangle
- 9. 4^2 + 12^2 = X^2
- 10. 16 + 144= 160
- 11. Square root of the total 4 12 X
- 13. Distance Formula= (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2
- 14. Ex: 1,4- 10,16
- 15. (10^2 – 1^2) + (16^2 – 4^2)
- 16. 9^2 + 12^2
- 17. 81 + 144 = 225
- 18. Square root of 225 is 15
- 20. Remember : (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2
- 21. (4 - 3)^2 + (-1 – 1)^2
- 22. 1^2 + -2^2
- 23. 1 + -4
- 24. No square root of -3