Question

a 30°-60°-90° triangles longer leg is 57 inches long. how long is the shorter leg?
write your answer in simplest radical form.
inches
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Explanation:

Step1: Recall 30-60-90 triangle ratios

In a \(30^\circ - 60^\circ - 90^\circ\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\), 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 longer leg

Given the longer leg is \(57\) inches, so \(x\sqrt{3}=57\), where \(x\) is the shorter leg.

Step3: Solve for \(x\)

To find \(x\), divide both sides by \(\sqrt{3}\): \(x = \frac{57}{\sqrt{3}}\). Rationalize the denominator by multiplying numerator and denominator by \(\sqrt{3}\): \(x=\frac{57\sqrt{3}}{3}=19\sqrt{3}\).

Answer:

\(19\sqrt{3}\)