You've just signed up to be an architect's assistant in a new office downtown. You're asked to draw a scale model of a sculpture for a business plaza. The sculpture has a large triangular piece where one of the angles between the sides is ninety degrees. This type of triangle is called a "right triangle." The architect you're working for comes into the room and tells you that the sides of the triangle that form the right angle are 9 feet and 12 feet. Can you tell how long the third side is? When you've completed this Concept, you'll be able to find the length of an unknown side of a right triangle by using the lengths of the other two sides.

### Watch This

James Sousa: The Pythagorean Theorem

### Guidance

From Geometry, recall that the Pythagorean Theorem is \begin{align*}a^2 + b^2 = c^2\end{align*} where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are the legs of a right triangle and \begin{align*}c\end{align*} is the hypotenuse. Also, the side opposite the angle is lower case and the angle is upper case. For example, angle \begin{align*}A\end{align*} is opposite side \begin{align*}a\end{align*} .

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

#### Example A

Use the Pythagorean Theorem to find the missing side.

**
Solution:
**
\begin{align*}a = 8, \ b = 15\end{align*}
, we need to find the hypotenuse.

\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.

#### Example B

Use the Pythagorean Theorem to find the missing side.

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Solution:
**
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*}

#### Example C

Use the Pythagorean Theorem to find the missing side.

**
Solution:
**
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*}

### Vocabulary

**
Pythagorean Theorem:
**
The
**
Pythagorean Theorem
**
is a mathematical relationship between the sides of a right triangle, given by
\begin{align*}a^2 + b^2 = c^2\end{align*}
, where
\begin{align*}a\end{align*}
,
\begin{align*}b\end{align*}
, and
\begin{align*}c\end{align*}
are lengths of the triangle.

### Guided Practice

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

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

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

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Solutions:
**

1. \begin{align*}a = 1, \ b = 8\end{align*} , we need to find the hypotenuse.

\begin{align*}1^2 + 8^2 & = c^2\\ 1 + 64 & = c^2\\ 65 & = c^2\\ \left ( \sqrt{65} \right ) & = c\end{align*}

2. \begin{align*}a = 3, \ c = 11\end{align*} , we need to find the length of side \begin{align*}b\end{align*} .

\begin{align*}3^2 + b^2 & = 11^2\\ 9 + b^2 & = 11^2\\ 121 - 9 & = b^2\\ 112 = b^2\\ \left ( \sqrt{112} \right ) & = b\end{align*}

3. 2. \begin{align*}a = 7, \ c = 18\end{align*} , we need to find the length of side \begin{align*}b\end{align*} .

\begin{align*}7^2 + b^2 & = 18^2\\ 49 + b^2 & = 18^2\\ 324 - 49 & = b^2\\ 275 = b^2\\ \left ( \sqrt{275} \right ) & = b\end{align*}

### Concept Problem Solution

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.

### Practice

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*} , 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*} 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*} and the hypotenuse is \begin{align*}10\sqrt{6}\end{align*} , then the other leg is _____________.
- If the legs of a right triangle are \begin{align*}x\end{align*} and \begin{align*}y\end{align*} , 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*} .
- Find the sum of the areas of the square with sides \begin{align*}c\end{align*} and the right triangles with legs \begin{align*}a\end{align*} and \begin{align*}b\end{align*} .
- 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.