Question

what is the length of the hypotenuse of a 30-60-90 triangle if the side opposite the 60-degree angle is 6√3?

Answer 1

In a 30-60-90 triangle, the sides have a specific ratio:

- The side opposite the 30-degree angle is \( x \).
- The side opposite the 60-degree angle is \( x\sqrt{3} \).
- The hypotenuse is \( 2x \).

Given that the side opposite the 60-degree angle is \( 6\sqrt{3} \), we can set this equal to \( x\sqrt{3} \):

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

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

\[
x = 6
\]

Now, using this value of \( x \), we can find the hypotenuse, which is \( 2x \):

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

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