Question

What is the length of the hypotenuse of a 30-60-90 triangle if the side opposite the 60 degree angle is 6 square root 3?

12
6
2 square root 3
12 square root 3

Answer 1

In a 30-60-90 triangle, the lengths of the sides are in the ratio 1 : √3 : 2. This means:

- The side opposite the 30-degree angle is the shortest and is \(x\).
- The side opposite the 60-degree angle (which is the given length) is \(x\sqrt{3}\).
- The hypotenuse (opposite the 90-degree angle) is \(2x\).

Given that the side opposite the 60-degree angle is \(6\sqrt{3}\):

\[
x\sqrt{3} = 6\sqrt{3}
\]

To find \(x\), divide both sides by \(\sqrt{3}\):

\[
x = 6
\]

Now, we can find the hypotenuse:

\[
\text{Hypotenuse} = 2x = 2(6) = 12
\]

Therefore, the length of the hypotenuse is \(\boxed{12}\).