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# 1.1: Lengths of Triangle Sides Using the Pythagorean Theorem

Difficulty Level: At Grade Created by: CK-12
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Practice Lengths of Triangle Sides Using the Pythagorean Theorem

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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?

### 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.

1. If the legs of a right triangle are 3 and 4, then the hypotenuse is _____________.
2. If the legs of a right triangle are 6 and 8, then the hypotenuse is _____________.
3. If the legs of a right triangle are 5 and 12, then the hypotenuse is _____________.
4. If the sides of a square are length 6, then the diagonal is _____________.
5. If the sides of a square are 9, then the diagonal is _____________.
6. If the sides of a square are , then the diagonal is _____________.
7. If the legs of a right triangle are 3 and 7, then the hypotenuse is _____________.
8. If the legs of a right triangle are and 6, then the hypotenuse is _____________.
9. If one leg of a right triangle is 4 and the hypotenuse is 8, then the other leg is _____________.
10. If one leg of a right triangle is 10 and the hypotenuse is 15, then the other leg is _____________.
11. If one leg of a right triangle is and the hypotenuse is , then the other leg is _____________.
12. 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.

1. Find the area of the square in the picture with sides .
2. Find the sum of the areas of the square with sides and the right triangles with legs and .
3. 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.

### Notes/Highlights Having trouble? Report an issue.

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### Vocabulary Language: English

Pythagorean Theorem

The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by $a^2 + b^2 = c^2$, where $a$ and $b$ are legs of the triangle and $c$ is the hypotenuse of the triangle.

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