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 where and are the legs of a right triangle and is the hypotenuse. Also, the side opposite the angle is lower case and the angle is upper case. For example, angle is opposite side .

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:
**
, we need to find the hypotenuse.

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.

#### Example C

Use the Pythagorean Theorem to find the missing side.

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Solution:
**
Use the Pythagorean Theorem to find the missing leg.

### Vocabulary

**
Pythagorean Theorem:
**
The
**
Pythagorean Theorem
**
is a mathematical relationship between the sides of a right triangle, given by
, where
,
, and
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. , we need to find the hypotenuse.

2. , we need to find the length of side .

3. 2. , we need to find the length of side .

### 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:

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 , 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 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 and the hypotenuse is , then the other leg is _____________.
- If the legs of a right triangle are and , 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 .
- Find the sum of the areas of the square with sides and the right triangles with legs and .
- 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.