Question

In a 30°-60°-90° right triangle, what is the length of the short leg if the hypotenuse has a length of 69?

In a 30°-60°-90° right triangle, what is the length of the short leg if the hypotenuse has a length of 69?

Answer 1

In a 30°-60°-90° right triangle, the sides are in the ratio of \(1 : \sqrt{3} : 2\). This means:

- The length of the short leg (opposite the 30° angle) is half the length of the hypotenuse.
- The length of the longer leg (opposite the 60° angle) is \( \sqrt{3} \) times the length of the short leg.

Given that the hypotenuse has a length of 69, we can find the length of the short leg as follows:

\[
\text{Length of the short leg} = \frac{1}{2} \times \text{hypotenuse}
\]

Substituting the given value for the hypotenuse:

\[
\text{Length of the short leg} = \frac{1}{2} \times 69 = 34.5
\]

Thus, the length of the short leg is \( \boxed{34.5} \).