### 45-45-90 Right Triangles

A right triangle with congruent legs and acute angles is an **Isosceles Right Triangle**. This triangle is also called a 45-45-90 triangle (named after the angle measures).

is a right triangle with , and .

**45-45-90 Theorem:** If a right triangle is isosceles, then its sides are in the ratio . For any isosceles right triangle, the legs are and the hypotenuse is always .

What if you were given an isosceles right triangle and the length of one of its sides? How could you figure out the lengths of its other sides?

### Examples

#### Example 1

Find the length of .

Use the ratio.

Here, we are given the hypotenuse. Solve for in the ratio.

#### Example 2

Find the length of , where is the hypotenuse of a 45-45-90 triangle with leg lengths of .

Use the ratio.

#### Example 3

Find the length of the missing side.

Use the ratio. because it is equal to . So, .

#### Example 4

Find the length of the missing side.

Use the ratio. because it is equal to . So, .

#### Example 5

A square has a diagonal with length 10, what are the lengths of the sides?

Draw a picture.

We know half of a square is a 45-45-90 triangle, so .

### Review

- In an isosceles right triangle, if a leg is 4, then the hypotenuse is __________.
- In an isosceles right triangle, if a leg is , then the hypotenuse is __________.
- A square has sides of length 15. What is the length of the diagonal?
- A square’s diagonal is 22. What is the length of each side?

For questions 5-11, find the lengths of the missing sides. Simplify all radicals.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 8.5.

### Resources