### Pythagorean Theorem

The two shorter sides of a right triangle (the sides that form the right angle) are the **legs** and the longer side (the side opposite the right angle) is the **hypotenuse**. For the Pythagorean Theorem, the legs are “” and “” and the hypotenuse is “”.

**Pythagorean Theorem:** Given a right triangle with legs of lengths and and a hypotenuse of length , .

The converse of the Pythagorean Theorem is also true. It allows you to prove that a triangle is a right triangle even if you do not know its angle measures.

**Pythagorean Theorem Converse:** If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

If , then is a right triangle.

##### Pythagorean Triples

A combination of three numbers that makes the Pythagorean Theorem true is called a **Pythagorean triple.** Each set of numbers below is a Pythagorean triple.

Any multiple of a Pythagorean triple is also considered a Pythagorean triple. Multiplying 3, 4, 5 by 2 gives 6, 8, 10, which is another triple. To see if a set of numbers makes a Pythagorean triple, plug them into the Pythagorean Theorem.

What if you were told that a triangle had side lengths of 5, 12, and 13? How could you determine if the triangle were a right one?

### Examples

#### Example 1

What is the diagonal of a rectangle with sides 10 and 16?

For any square and rectangle, you can use the Pythagorean Theorem to find the length of a diagonal. Plug in the sides to find .

#### Example 2

Do 6, 7, and 8 make the sides of a right triangle?

Plug the three numbers into the Pythagorean Theorem. Remember that the largest length will always be the hypotenuse, . If , then they are the sides of a right triangle.

#### Example 3

Find the length of the hypotenuse.

Use the Pythagorean Theorem. Set and . Solve for .

#### Example 4

Is 20, 21, 29 a Pythagorean triple?

If , then the set is a Pythagorean triple.

Therefore, 20, 21, and 29 is a Pythagorean triple.

#### Example 5

Determine if the triangles below are right triangles.

Check to see if the three lengths satisfy the Pythagorean Theorem. Let the longest side represent .

### Review

Find the length of the missing side. Simplify all radicals.

- If the legs of a right triangle are 10 and 24, then the hypotenuse is __________.
- If the sides of a rectangle are 12 and 15, then the diagonal is _____________.
- If the sides of a square are 16, then the diagonal is ____________.
- If the sides of a square are 9, then the diagonal is _____________.

Determine if the following sets of numbers are Pythagorean Triples.

- 12, 35, 37
- 9, 17, 18
- 10, 15, 21
- 11, 60, 61
- 15, 20, 25
- 18, 73, 75

Determine if the following lengths make a right triangle.

- 7, 24, 25
- 15, 20, 25
- 20, 25, 30

### Review (Answers)

To see the Review answers, open this PDF file and look for section 8.2.

### Resources