Question

a triangle has angles that measure 30°, 60°, and 90°. the hypotenuse of the triangle measures 10 inches. which is the best estimate for the perimeter of the triangle? round to the nearest tenth. 20.0 in. 23.1 in. 23.7 in. 27.4 in.

Explanation:

Step1: Find the shorter leg

In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, the shorter leg \(a\) (opposite the \(30^{\circ}\) angle) is half of the hypotenuse \(c\). Given \(c = 10\) inches, so \(a=\frac{c}{2}=\frac{10}{2}=5\) inches.

Step2: Find the longer leg

The longer leg \(b\) (opposite the \(60^{\circ}\) angle) is \(a\sqrt{3}\). Substitute \(a = 5\) into the formula, we get \(b = 5\sqrt{3}\approx5\times1.732 = 8.66\) inches.

Step3: Calculate the perimeter

The perimeter \(P=a + b + c\). Substitute \(a = 5\), \(b\approx8.66\), and \(c = 10\) into the formula. \(P=5+8.66 + 10=23.66\approx23.7\) inches.

Answer:

23.7 in.