### 30-60-90 Right Triangles

One of the two special right triangles is called a 30-60-90 triangle, after its three angles.

**30-60-90 Theorem:** If a triangle has angle measures

The shorter leg is always

What if you were given a 30-60-90 right triangle and the length of one of its side? How could you figure out the lengths of its other sides?

### Examples

#### Example 1

Find the value of

We are given the longer leg.

#### Example 2

Find the value of

We are given the hypotenuse.

#### Example 3

Find the length of the missing sides.

We are given the shorter leg. If

#### Example 4

Find the length of the missing sides.

We are given the hypotenuse.

#### Example 5

A rectangle has sides 4 and

If you are not given a picture, draw one.

The two lengths are

If you did not recognize this is a 30-60-90 triangle, you can use the Pythagorean Theorem too.

### Review

- In a 30-60-90 triangle, if the shorter leg is 5, then the longer leg is __________ and the hypotenuse is ___________.
- In a 30-60-90 triangle, if the shorter leg is
x , then the longer leg is __________ and the hypotenuse is ___________. - A rectangle has sides of length 6 and
63√ . What is the length of the diagonal? - Two (opposite) sides of a rectangle are 10 and the diagonal is 20. What is the length of the other two sides?

For questions 5-12, 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 8.6.