Question

What is the length of the hypotenuse of a 30-60-90 triangle if the side opposite the 60-degree angle is 6square root symbol3?

Answer 1

In a 30-60-90 triangle, the sides are in a specific ratio. The lengths of the sides are as follows:

- The side opposite the 30-degree angle (let's call it \( a \)) is \( x \).
- The side opposite the 60-degree angle is \( x\sqrt{3} \).
- The hypotenuse (opposite the 90-degree angle) is \( 2x \).

In this case, we are given that the side opposite the 60-degree angle is \( 6\sqrt{3} \). This corresponds to \( x\sqrt{3} \) in our ratio. Setting this equal to \( 6\sqrt{3} \):

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

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

\[
x = 6.
\]

Now, we can find the length of the hypotenuse:

\[
\text{Hypotenuse} = 2x = 2 \times 6 = 12.
\]

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