Assignment 3 Pythagoreas\' Theorem PDF

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Waseet Kazmi...

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Pythagoras’ Theorem

February 2021

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Introduction

In this paper we will show that we use the Pythagorean Theorem to find the length of the third or missing side of a right triangle. In Section 2 we will introduce the concept of how a right triangle and hypotenuse play into the Pythagorean theorem. In Section 3 we will provide a proof of how the Pythagorean Theorem helps us find the length of the third side of a right triangle. Finally, Section 4 we will look through a few examples using the Pythagorean theorem.

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Preliminaries

First we define a right triangle. Definition 1. A right triangle is a triangle in which one of the interior angles is 90◦ . A right triangle consists of three sides which are the hypotenuse, the height, and the base. They are also referred to as as c, a and b respectively. Now we define a hypotenuse. Definition 2. A hypotenuse is the longest side of the right triangle which is also the side opposite of the right angle. And finally, we can see say the Pythagorean theorem states that in a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides (the base and height). It’s given by a 2 + b2 = c 2 1

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Source:[1]

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The Proof

We are finally ready to show our main result. Theorem 3. In a right triangle, the sum of the squares of the base and height is equal to the square of the hypotenuse. Proof. To show that a2 + b2 = c2 we can use algebra along with a diagram which depicts two squares and four triangles. We can look at the diagram below which has the ”abc” triangle in it four times (in the blue).

Source:[1] Each side of the square is length a + b so the total area is A = (a + b)(a + b). Now, we can add up the areas of the smaller pieces. The smaller, yellow square has an area equal to c2 and each of the four triangles have an area = 2ab. Then, . If we put all four of the triangles together we get 4ab of ab 2 2 adding up the yellow square and the four triangles gives A = c2 + 2ab. As you can tell from the diagram, the area of the large square is equal to area of the tilted square and the four triangles. Putting the two equations together we get (a + b)(a + b) = c2 + 2ab Now, we can rearrange the equation to get the Pythagorean theorem. First step is to expand (a + b)(a + b) by multiplication to get

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a2 + 2ab + b2 = c2 + 2ab Then, we can subtract 2ab from both sides which ultimately gives the Pythagorean theorem a 2 + b2 = c 2

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Pythagorean Theorem Examples

Below are a few examples of Pythagorean theorem being used to find the missing length of a side of a triangle. Example 4. Solve this triangle

Source: [2] Solution We start with the formula given for the Pythagorean theorem a2 + b2 = c2 . Then, we can identify from the figure that a = 5 and b = 12. Next, we plug these values into the formula which gives 52 + 122 = c2 . To simplify the expression we calculate both squares and add them together√resulting √ in 2 2 169 = c . Now, to solve for c we take the square root of each side c = 169 which finally gives c = 13. Example 5. Solve this triangle

Source: [2] Solution Like the above example, we start with the formula a2 + b2 = c2 and identify the sides a = 9 and c = 15. Now, we plug these values into the formula which gives 92 + b2 = 152 . To simplify the equation, we calculate both 3

squares first 81 + b2 = 225. Then, to isolate b further we subtract 81 from 2 both √ sides √ resulting in b = 144. Lastly, we take the square root of both sides b2 = 144 ultimatelt resulting in b = 12. As you can see, Pythagoreans theorem can be used to find the missing side of a right triangle.

References [1]

MathIsFun: Pythagorean Theorem https://www.mathsisfun.com/pythagoras.html

[2]

MathIsFun: Pythagorean Theorem Proof https://www.mathsisfun.com/geometry/ pythagorean-theorem-proof.html

[3]

Splash Learn: What is Right Triangle? https://www.splashlearn.com/math-vocabulary/ geometry/right-triangle

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