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*} 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.

#### Example A

Find the length of the missing side.

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 B

Find the length of the missing side.

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 C

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.

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

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

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### Guided Practice

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

1.

2.

3. @$\begin{align*}x\end{align*}@$ is the hypotenuse of a 30-60-90 triangle and @$\begin{align*}y\end{align*}@$ is the longer leg of the same triangle. The shorter leg has a length of @$\begin{align*}6\end{align*}@$.

**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.