# How do you find the remaining side of a 30^circ-60^circ-90^circ triangle if the longest side is 8?

Jan 4, 2018

The three sides are 4, $4 \sqrt{3}$, and 8.

#### Explanation:

The ratio of the sides in a 30-60-90 triangle is $x : x \sqrt{3} : 2 x$. That means if the side opposite ${30}^{\circ}$ is $x$, then the side opposite ${60}^{\circ}$ is $x \sqrt{3}$ and the side opposite ${90}^{\circ}$ is $2 x$.

Since the longest side is the hypotenuse, which is opposite ${90}^{\circ}$, we know that $2 x = 8 \setminus \rightarrow x = 4$. So the side opposite ${30}^{\circ}$ is 4.

If the side opposite ${30}^{\circ}$ is 4, we know that the side opposite ${60}^{\circ}$, $x \sqrt{3}$, is going to be $4 \sqrt{3}$.

So the sides are 4, $4 \sqrt{3}$, and 8.