# 1.1: Lengths of Triangle Sides Using the Pythagorean Theorem

**At Grade**Created by: CK-12

**Practice**Lengths of Triangle Sides Using the Pythagorean Theorem

### Finding the Length of Triangle Sides Using Pythagorean Theorem

From Geometry, recall that the Pythagorean Theorem is \begin{align*}a^2 + b^2 = c^2\end{align*}

The Pythagorean Theorem is used to solve for the sides of a right triangle.

#### Use the Pythagorean Theorem to find the missing side.

\begin{align*}a = 8, \ b = 15\end{align*}

\begin{align*}8^2 + 15^2 = c^2\\
64 + 225 = c^2\\
289 = c^2\\
17 = c\end{align*}

Notice, we do not include -17 as a solution because a negative number cannot be a side of a triangle.

#### Use the Pythagorean Theorem to find the missing side.

Use the Pythagorean Theorem to find the missing leg.

\begin{align*}\left ( 5\sqrt{7} \right )^2 + x^2 = \left ( 5\sqrt{13} \right )^2\\
25 \cdot 7 + x^2 = 25 \cdot 13\\
175 + x^2 = 325\\
x^2 = 150\\
x = 5\sqrt{6}\end{align*}

#### Use the Pythagorean Theorem to find the missing side.

Use the Pythagorean Theorem to find the missing leg.

\begin{align*}10^2 + x^2 = \left ( 10\sqrt{2} \right )^2\\
100 + x^2 = 100 \cdot 2\\
100 + x^2 = 100\\
x^2 = 100\\
x = 10\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem asking you to draw a scale model of a sculpture for a business plaza.

With your knowledge of the Pythagorean Theorem, you can see that the triangle has sides with lengths 9 feet and 12 feet. You work to find the hypotenuse:

\begin{align*}a^2 + b^2 = c^2\\
9^2 + 12^2 = c^2\\
81 + 144 = c^2\\
225 = c^2\\
\left ( \sqrt{225} \right ) = 15 = c\end{align*}

With the knowledge that the length of the third side of the triangle is 15 feet, you are able to construct your scale model with ease.

#### Example 2

Use the Pythagorean Theorem to find the missing side of the following triangle:

\begin{align*}a = 1, \ b = 8\end{align*}

\begin{align*}1^2 + 8^2 = c^2\\
1 + 64 = c^2\\
65 = c^2\\
\sqrt{65} = c\end{align*}

#### Example 3

Use the Pythagorean Theorem to find the missing side of the following triangle:

\begin{align*}a = 3, \ b = 11\end{align*}

\begin{align*}3^2 + 11^2 = c^2\\
9 + 121 = c^2\\
130 = c^2\\
\sqrt{130} = c\end{align*}

#### Example 4

Find the missing side of the right triangle below. Leave the answer in simplest radical form.

** **

\begin{align*}a = 7, \ c = 18\end{align*}

\begin{align*}7^2 + b^2 = 18^2\\
49 + b^2 = 18^2\\
324 - 49 = b^2\\
275 = b^2\\
\sqrt{275} = b\end{align*}

### Review

Find the missing sides of the right triangles. Leave answers in simplest radical form.

- If the legs of a right triangle are 3 and 4, then the hypotenuse is _____________.
- If the legs of a right triangle are 6 and 8, then the hypotenuse is _____________.
- If the legs of a right triangle are 5 and 12, then the hypotenuse is _____________.
- If the sides of a square are length 6, then the diagonal is _____________.
- If the sides of a square are 9, then the diagonal is _____________.
- If the sides of a square are \begin{align*}x\end{align*}
x , then the diagonal is _____________. - If the legs of a right triangle are 3 and 7, then the hypotenuse is _____________.
- If the legs of a right triangle are \begin{align*}2\sqrt{5}\end{align*}
25√ and 6, then the hypotenuse is _____________. - If one leg of a right triangle is 4 and the hypotenuse is 8, then the other leg is _____________.
- If one leg of a right triangle is 10 and the hypotenuse is 15, then the other leg is _____________.
- If one leg of a right triangle is \begin{align*}4\sqrt{7}\end{align*}
47√ and the hypotenuse is \begin{align*}10\sqrt{6}\end{align*}106√ , then the other leg is _____________. - If the legs of a right triangle are \begin{align*}x\end{align*}
x and \begin{align*}y\end{align*}y , then the hypotenuse is ____________.

*Pythagorean Theorem Proof*

Use the picture below to answer the following questions.

- Find the area of the square in the picture with sides \begin{align*}(a + b)\end{align*}
(a+b) . - Find the sum of the areas of the square with sides \begin{align*}c\end{align*}
c and the right triangles with legs \begin{align*}a\end{align*}a and \begin{align*}b\end{align*}b . - Explain why the areas found in the previous two problems should be the same value. Then, set the expressions equal to each other and simplify to get the Pythagorean Theorem.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.1.

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Here you'll learn what the Pythagorean Theorem is and how to use it to find the length of an unknown side of a right triangle.

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