### Pythagorean Theorem

The two shorter sides of a right triangle (the sides that form the right angle) are the **legs** and the longer side (the side opposite the right angle) is the **hypotenuse**. For the Pythagorean Theorem, the legs are “\begin{align*}a\end{align*}” and “\begin{align*}b\end{align*}” and the hypotenuse is “\begin{align*}c\end{align*}”.

**Pythagorean Theorem:** Given a right triangle with legs of lengths \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*}.

The converse of the Pythagorean Theorem is also true. It allows you to prove that a triangle is a right triangle even if you do not know its angle measures.

**Pythagorean Theorem Converse:** If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

If \begin{align*}a^2 + b^2 = c^2\end{align*}, then \begin{align*}\triangle ABC\end{align*} is a right triangle.

##### Pythagorean Triples

A combination of three numbers that makes the Pythagorean Theorem true is called a **Pythagorean triple.** Each set of numbers below is a Pythagorean triple.

\begin{align*}3, 4, 5 && 5, 12, 13 && 7, 24, 25 && 8, 15, 17 && 9, 12, 15 && 10, 24, 26\end{align*}

Any multiple of a Pythagorean triple is also considered a Pythagorean triple. Multiplying 3, 4, 5 by 2 gives 6, 8, 10, which is another triple. To see if a set of numbers makes a Pythagorean triple, plug them into the Pythagorean Theorem.

What if you were told that a triangle had side lengths of 5, 12, and 13? How could you determine if the triangle were a right one?

### Examples

#### Example 1

What is the diagonal of a rectangle with sides 10 and 16?

For any square and rectangle, you can use the Pythagorean Theorem to find the length of a diagonal. Plug in the sides to find \begin{align*}d\end{align*}.

\begin{align*}10^2 + 16^2 &= d^2\\ 100 + 256 &= d^2\\ 356 &= d^2\\ d &= \sqrt{356} = 2 \sqrt{89} \approx 18.87\end{align*}

#### Example 2

Do 6, 7, and 8 make the sides of a right triangle?

Plug the three numbers into the Pythagorean Theorem. Remember that the largest length will always be the hypotenuse, \begin{align*}c\end{align*}. If \begin{align*}6^2+7^2=8^2\end{align*}, then they are the sides of a right triangle.

\begin{align*}6^2 + 7^2 &= 36 + 49=85\\ 8^2 &= 64 && 85 \neq 64, \ \text{so the lengths are not the sides of a right triangle.}\end{align*}

#### Example 3

Find the length of the hypotenuse.

Use the Pythagorean Theorem. Set \begin{align*}a = 8\end{align*} and \begin{align*}b = 15\end{align*}. Solve for \begin{align*}c\end{align*}.

\begin{align*}8^2 + 15^2 &= c^2\\ 64 + 225 &= c^2\\ 289 &= c^2 && Take \ the \ square \ root \ of \ both \ sides.\\ 17 &= c\end{align*}

#### Example 4

Is 20, 21, 29 a Pythagorean triple?

If \begin{align*}20^2 + 21^2 = 29^2\end{align*}, then the set is a Pythagorean triple.

\begin{align*}20^2 + 21^2 &= 400+441=841\\ 29^2 &= 841\end{align*}

Therefore, 20, 21, and 29 is a Pythagorean triple.

#### Example 5

Determine if the triangles below are right triangles.

Check to see if the three lengths satisfy the Pythagorean Theorem. Let the longest side represent \begin{align*}c\end{align*}.

\begin{align*}a^2 + b^2 & = c^2\\ 8^2 + 16^2 & \overset{?} = \left( 8 \sqrt{5} \right )^2\\ 64 + 256 & \overset{?} = 64 \cdot 5\\ 320 &= 320 \qquad \text{Yes}\end{align*}

\begin{align*}a^2+b^2 & = c^2\\ 22^2 + 24^2 & \overset{?}= 26^2\\ 484 + 576 & \overset{?}= 676\\ 1060 & \neq 676 \qquad \text{No}\end{align*}

### Review

Find the length of the missing side. Simplify all radicals.

- If the legs of a right triangle are 10 and 24, then the hypotenuse is __________.
- If the sides of a rectangle are 12 and 15, then the diagonal is _____________.
- If the sides of a square are 16, then the diagonal is ____________.
- If the sides of a square are 9, then the diagonal is _____________.

Determine if the following sets of numbers are Pythagorean Triples.

- 12, 35, 37
- 9, 17, 18
- 10, 15, 21
- 11, 60, 61
- 15, 20, 25
- 18, 73, 75

Determine if the following lengths make a right triangle.

- 7, 24, 25
- \begin{align*}\sqrt{5}, 2 \sqrt{10}, 3 \sqrt{5}\end{align*}
- \begin{align*}2 \sqrt{3}, \sqrt{6}, 8\end{align*}
- 15, 20, 25
- 20, 25, 30
- \begin{align*}8 \sqrt{3}, 6, 2 \sqrt{39}\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 8.2.