Question

in a 30° - 60° - 90° triangle, if the hypotenuse is 9 inches, find the side opposite the 60° angle. round your answer to the tenths place.

Explanation:

Step1: Recall the side - length ratios in a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle

In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, if the hypotenuse \(c\) is given, and the sides are \(a\) (opposite \(30^{\circ}\)), \(b\) (opposite \(60^{\circ}\)), and \(c\) (hypotenuse). The ratios are \(a:b:c = 1:\sqrt{3}:2\). So, \(b=\frac{\sqrt{3}}{2}c\).

Step2: Substitute the value of the hypotenuse

Given \(c = 9\) inches. Then \(b=\frac{\sqrt{3}}{2}\times9\).

$$b=\frac{9\sqrt{3}}{2}$$

Since \(\sqrt{3}\approx1.732\), then \(b=\frac{9\times1.732}{2}\).

$$b=\frac{15.588}{2}=7.794$$

Answer:

\(7.8\)