Question

1. choose the best answer. apply the 30°-60°-90° triangle theorem. what is the length of the shorter leg, a, and the longer leg, b, if the length of the hypotenuse, c, is 14 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 \(a\), the longer leg (opposite \(60^\circ\)) is \(b = a\sqrt{3}\), and the hypotenuse \(c = 2a\).

Step2: Solve for Shorter Leg (\(a\))

Given \(c = 14\) inches, and \(c = 2a\), we solve for \(a\):
\(2a = 14\)
Divide both sides by 2:
\(a = \frac{14}{2} = 7\) inches.

Step3: Solve for Longer Leg (\(b\))

Using \(b = a\sqrt{3}\) and \(a = 7\):
\(b = 7\sqrt{3}\) inches (or approximately \(12.12\) inches, but exact form is \(7\sqrt{3}\)).

Answer:

\(a = 7\) inches, \(b = 7\sqrt{3}\) inches (or \(b \approx 12.12\) inches)