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The Pythagorean Theorem: Practice and Application

  • 0:06 The Theorem and Its Purpose
  • 1:49 Pythagorean Theorem Basics
  • 4:22 Pythagorean Theorem in…
  • 6:56 Lesson Summary
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Lesson Transcript
Instructor: DaQuita Hester

DaQuita has taught high school mathematics for six years and has a master's degree in secondary mathematics education.

The Pythagorean theorem is one of the most famous geometric theorems. Written by the Greek mathematician Pythagoras, this theorem makes it possible to find a missing side length of a right triangle. Learn more about the famous theorem here and test your understanding with a quiz.

Introduction

Jessica has a square garden. To make watering her plants easier, she would like to create a diagonal pathway that connects the opposite corners. If one side of her garden is 13 feet long, what will be the length of her diagonal pathway?

This scenario presents a perfect example of geometry in real life. Since the garden is in the shape of a square, we will be able to use the Pythagorean theorem to figure out how long the diagonal pathway will be. Let's take a closer look at this theorem and how it's used so that we can calculate the solution to this problem.

The Theorem and Its Purpose

Written by the Greek mathematician Pythagoras, the Pythagorean theorem states that in right triangles, the sum of the squares of the two legs is equal to the square of the hypotenuse. In clearer terms, if you square both of the legs and then add these numbers together, you will end up with the same value as the hypotenuse squared. As a formula, this theorem is written as a^2 + b^2 = c^2, where a and b represent the two legs of the triangle, and c represents the hypotenuse.

When labeling any right triangle with these variables, it is important to know that a and b can be written on either leg. However, c must be written on the hypotenuse, which is the longest side and opposite the right angle.

The purpose of the Pythagorean theorem is to make it possible to find the length of any missing side of a right triangle. For the theorem to be usable, the value of the other two sides must be known.

Let's take a look at a few basic problems.

Pythagorean Theorem Basics

In our first triangle, one leg has a length of 4 centimeters, and the second leg has a length of 7 centimeters. Let's find the length of the hypotenuse. Since I have the value for the two legs, which are represented by a and b, we will let a = 4 and b = 7.

We are now ready to plug this information into the formula. Once we do, we get 4^2 + 7^2 = c^2. After simplifying, we end up with 16 + 49 = c^2, and when we combine our like terms, we have 65 = c^2. For our last step, we will take the square root of both sides to get a final answer of c = 8.06. Therefore, we can see that the length of the diagonal is 8.06 centimeters.

In our next triangle, the measure of one leg is 12 inches and the measure of the hypotenuse is 20 inches. For this triangle, we have the value of the hypotenuse, but must find the length of the missing leg. Therefore, we will let a = 12 and c = 20.

By substituting this information into the formula, we have 12^2 + b^2 = 20^2. When we simplify, we get 144 + b^2 = 400. At this point, we should subtract 144 from both sides to get b^2 = 256. Once we take the square root of both sides, we see that b, the missing leg in our triangle, has a value of 16 inches.

Did anything about the numbers in that last example stand out to you? Notice that all of the side lengths are whole numbers, not fractions or decimals. This means that the side lengths for this triangle are a Pythagorean triple, which is defined as a set of nonzero whole numbers that satisfy the Pythagorean theorem.

A few more examples of Pythagorean triples are 3, 4, 5, and 28, 45, 53. If you plug these numbers into the Pythagorean theorem, you will see that 3^2 + 4^2 = 5^2, and 28^2 + 45^2 = 53^2.

Pythagorean Theorem in Everyday Life

Now that we have an understanding of the Pythagorean theorem, let's apply it to a few everyday examples, beginning with Jessica's garden from the beginning of the lesson.

Let's review the problem again. Jessica has a square garden. To make watering her plants easier, she would like to create a diagonal pathway that connects opposite corners. If one side of her garden is 13 feet long, what will be the length of her diagonal pathway?

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