# 3.6: Pythagorean Theorem, Part 1: Proof & Finding a Missing Side

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Identify and employ the Pythagorean Theorem when working with right triangles.

## Introduction to Pythagorean Theorem

If you are interested, here is a bit of history to begin our lesson:

Pythagoras of Samos (c. 570-c. 495 BC) was an Ionian Greek philosopher and founder of the religious movement called Pythagoreanism. Most of our information about Pythagoras was written down centuries after he lived, thus very little reliable information is known about him. He was born on the island of Samos, and may have travelled widely in his youth, visiting Egypt and other places seeking knowledge. Around 530 BC, he moved to Croton, a Greek colony in southern Italy, and there set up a religious sect. His followers pursued the religious rites and practices developed by Pythagoras, and studied his philosophical theories.

Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. He is often revered as a great mathematician, mystic and scientist, and he is best known for the Pythagorean Theorem which bears his name. It was said that he was the first man to call himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Plato, and through him, all of Western philosophy.

## The Pythagorean Theorem

The triangle below is a right triangle:

The sides labeled \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are called the legs of the triangle and meet at the right angle.

The third side, labeled \begin{align*}c\end{align*} is called the hypotenuse. The hypotenuse is the side opposite (or across from) the right angle. The hypotenuse of a right triangle is also the longest side of the triangle.

• The longest side of a right triangle is called the ______________________.
• The hypotenuse is across from the ___________________ angle in a right triangle.
• The two shorter sides of a right triangle are called __________________.
• The legs of a right triangle form a vertex at the _____________________ angle.

The Pythagorean Theorem states that there is a relationship between the three sides of ANY right triangle. When you square the length of the hypotenuse, it will equal the sum of the squares of the lengths of the two legs. In the triangle on the previous page, the sum of the squares of the legs is \begin{align*}a^2 + b^2\end{align*} and the square of the hypotenuse is \begin{align*}c^2\end{align*}.

The Pythagorean Theorem

Given a right triangle with legs whose lengths are \begin{align*}a\end{align*} and \begin{align*}b\end{align*} and a hypotenuse of length \begin{align*}c\end{align*},

\begin{align*}a^2 + b^2 = c^2\end{align*}

Be careful when using this theorem!

You must make sure that the legs are correctly labeled \begin{align*}a\end{align*} and \begin{align*}b\end{align*} and the hypotenuse is correctly labeled \begin{align*}c\end{align*} to use this equation. A more accurate way to write the Pythagorean Theorem is:

\begin{align*}(leg_1)^2+(leg_2)^2=hypotenuse^2\end{align*}

When using the Pythagorean Theorem:

• You must make sure that the hypotenuse is side _____________ in the triangle
• and the legs are sides ____________ and _____________.

The ________________________________ Theorem is only true for right triangles!

Example 1

Use the side lengths of the following triangle to test the Pythagorean Theorem.

The legs of the triangle above are 3 inches and 4 inches. The hypotenuse is 5 inches. So, using the Pythagorean Theorem, let’s make \begin{align*}a = 3, b = 4\end{align*}, and \begin{align*}c = 5\end{align*}. We can substitute these values into the formula for the Pythagorean Theorem to verify that the relationship works:

\begin{align*}a^2+b^2&=c^2\\ 3^2+4^2 &= 5^2\\ 9 + 16 &= 25\\ 25 &= 25\end{align*}

Since both sides of the equation equal 25, the equation is true. Therefore, the Pythagorean Theorem works on this right triangle.

1. True or false: The Pythagorean Theorem works for all triangles.

2. True or false: The hypotenuse of a right triangle is the shortest side of the triangle.

3. What are 2 characteristics of the hypotenuse of a triangle?

1) 2)

4. In the triangle to the right,

Which side is a leg? ________________________

Which side is the other leg? __________________

Which side is the hypotenuse? ________________

Example 2

What is the length of \begin{align*}b\end{align*} in the triangle below?

Use the Pythagorean Theorem to find the length of the missing leg, \begin{align*}b\end{align*}. Set up the equation \begin{align*}a^2 + b^2 = c^2\end{align*}, letting \begin{align*}a = 6\end{align*} and \begin{align*}c = 10\end{align*}:

\begin{align*}a^2+b^2 & \ = \ c^2\\ 6^2+b^2 & \ = \ 10^2\\ 36+b^2 & \ = \ 100\\ - 36 & \ \ \ -36\\ b^2 & \ = \ 64\\ \sqrt{b^2} & \ = \ \sqrt{64}\\ b & \ = \ \pm 8\\ b & \ = \ 8\end{align*}

In algebra you learned that \begin{align*}\sqrt{x^2} = \pm x\end{align*} because, for example, \begin{align*}(5)^2 = (-5)^2 = 25\end{align*}.

However, in this case (and in much of geometry), we are only interested in the positive solution to \begin{align*}b= \sqrt{64}\end{align*} because geometric lengths are positive (having a side with a negative length does not make sense.)

So in Example 2, we can disregard the solution \begin{align*}b = -8\end{align*} and our final answer is \begin{align*}b = 8\end{align*} inches.

In Example 2 above,

• The hypotenuse is side ____________, which is ____________ inches long.
• One leg is side ____________, which is ____________ inches long.
• The other leg is side ____________, which is ____________ inches long.

When using the Pythagorean Theorem and taking the square root of a number,

• We only care about the ___________________________ answer because lengths cannot be negative.

Example 3

Find the length of the missing side in the triangle below.

Use the Pythagorean Theorem to set up an equation and solve for the missing side, which in this problem is the hypotenuse.

Let \begin{align*}a = 5\end{align*} and \begin{align*}b = 12\end{align*}. We do not know \begin{align*}c\end{align*}:

\begin{align*}a^2+b^2 &= c^2\\ 5^2+ 12^2 &= c^2\\ 36+144 &= c^2\\ 169 &= c^2\\ \sqrt{169} &= \sqrt{c^2}\\ 13 &= c\end{align*}

So, the length of the missing side, the ______________________________, is 13 centimeters.

Find the length of the missing side in the triangle below.

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