Question

the hypotenuse of a 30°-60°-90° triangle measures $20\sqrt{2}$ inches. what is the measure of its shorter leg?
write your answer in simplest radical form.
\boxed{} inches

Explanation:

Step1: Recall 30-60-90 triangle ratios

In a \(30^\circ - 60^\circ - 90^\circ\) triangle, the ratios of the sides are \(1 : \sqrt{3} : 2\), where the shorter leg (opposite \(30^\circ\)) is \(x\), the longer leg (opposite \(60^\circ\)) is \(x\sqrt{3}\), and the hypotenuse is \(2x\).

Step2: Set up equation for hypotenuse

Given the hypotenuse is \(20\sqrt{2}\), and from the ratio, hypotenuse \(= 2x\). So we set \(2x = 20\sqrt{2}\).

Step3: Solve for \(x\) (shorter leg)

Divide both sides by 2: \(x=\frac{20\sqrt{2}}{2}=10\sqrt{2}\).

Answer:

\(10\sqrt{2}\)