### 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 and , then the sides are in the ratio .

The shorter leg is always , the longer leg is always , and the hypotenuse is always . If you ever forget these theorems, you can still use the Pythagorean Theorem.

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 and .

We are given the longer leg.

#### Example 2

Find the value of and .

We are given the hypotenuse.

#### Example 3

Find the length of the missing sides.

We are given the shorter leg. If , then the longer leg, , and the hypotenuse, .

#### Example 4

Find the length of the missing sides.

We are given the hypotenuse. , so the shorter leg, , and the longer leg, .

#### Example 5

A rectangle has sides 4 and . What is the length of the diagonal?

If you are not given a picture, draw one.

The two lengths are , so the diagonal would be , or .

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 , then the longer leg is __________ and the hypotenuse is ___________.
- A rectangle has sides of length 6 and . 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.