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? After completing this Concept, you'll be able to use the 30-60-90 Theorem to solve problems like this one.

### Watch This

CK-12 Foundation: Special Right Triangle: 30-60-90

Watch the first half of this video.

James Sousa: Trigonometric Function Values of Special Angles

Now watch the first half of this video.

James Sousa: Solving Special Right Triangles

### Guidance

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*}

The shorter leg is always \begin{align*}x\end{align*}

#### Example A

Find the length of the missing side.

We are given the shorter leg. If \begin{align*}x=5\end{align*}

#### Example B

Find the length of the missing side.

We are given the hypotenuse. \begin{align*}2x=20\end{align*}

#### Example C

A rectangle has sides 4 and \begin{align*}4 \sqrt{3}\end{align*}

If you are not given a picture, draw one.

The two lengths are \begin{align*}x, x \sqrt{3}\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*}

CK-12 Foundation: Special Right Triangle: 30-60-90

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### Guided Practice

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

1.

2.

3. \begin{align*}x\end{align*}

**Answers:**

1. 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*}

2. 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*}

3. We are given the shorter leg.

\begin{align*}& x=2(6)\\ & x = 12\\ & \text{The longer leg is}\\ & y= 6 \cdot \sqrt{3} = 6 \sqrt{3}\end{align*}

### Explore More

- 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.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 8.6.