### 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 \begin{align*}30^\circ, 60^\circ\end{align*} and \begin{align*}90^\circ\end{align*}, then the sides are in the ratio \begin{align*}x:x \sqrt{3}:2x\end{align*}.

The shorter leg is always \begin{align*}x\end{align*}, the longer leg is always \begin{align*}x \sqrt{3}\end{align*}, and the hypotenuse is always \begin{align*}2x\end{align*}. 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 \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

We are given the longer leg.

\begin{align*}& x \sqrt{3} = 12\\ & x = \frac{12}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{12 \sqrt{3}}{3} = 4 \sqrt{3}\\ & \text{The hypotenuse is}\\ & y = 2(4 \sqrt{3}) = 8 \sqrt{3}\end{align*}

#### Example 2

Find the value of \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

We are given the hypotenuse.

\begin{align*}& 2x =16\\ & x = 8\\ & \text{The longer leg is}\\ & y = 8 \cdot \sqrt{3} = 8 \sqrt{3}\end{align*}

#### Example 3

Find the length of the missing sides.

We are given the shorter leg. If \begin{align*}x=5\end{align*}, then the longer leg, \begin{align*}b=5 \sqrt{3}\end{align*}, and the hypotenuse, \begin{align*}c=2(5)=10\end{align*}.

#### Example 4

Find the length of the missing sides.

We are given the hypotenuse. \begin{align*}2x=20\end{align*}, so the shorter leg, \begin{align*}f = \frac{20}{2} = 10\end{align*}, and the longer leg, \begin{align*}g=10 \sqrt{3}\end{align*}.

#### Example 5

A rectangle has sides 4 and \begin{align*}4 \sqrt{3}\end{align*}. What is the length of the diagonal?

If you are not given a picture, draw one.

The two lengths are \begin{align*}x, x \sqrt{3}\end{align*}, so the diagonal would be \begin{align*}2x\end{align*}, or \begin{align*}2(4) = 8\end{align*}.

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

\begin{align*}4^2 + \left( 4 \sqrt{3} \right )^2 &= d^2\\ 16 + 48 &= d^2\\ d &= \sqrt{64} = 8\end{align*}

### 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 \begin{align*}x\end{align*}, then the longer leg is __________ and the hypotenuse is ___________.
- A rectangle has sides of length 6 and \begin{align*}6 \sqrt{3}\end{align*}. 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.