# 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 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.

#### Using the Pythagorean Theorem

1.

, 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.

2.

Use the Pythagorean Theorem to find the missing leg.

3. Use the Pythagorean Theorem to find the missing leg in the triangle above.

### 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:

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:

, we need to find the hypotenuse.

#### Example 3

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

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

#### Example 4

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

** **

, we need to find the length of side .

### 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 , 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.

### Review (Answers)

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

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### Image Attributions

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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