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: Recall side - length ratios in 30 - 60 - 90 triangle

In a 30 - 60 - 90 triangle, if the hypotenuse \(c = 10\) inches, the side opposite the 30 - degree angle \(a=\frac{c}{2}\), and the side opposite the 60 - degree angle \(b = a\sqrt{3}\).
Since \(c = 10\) inches, then \(a=\frac{10}{2}=5\) inches.

Step2: Calculate the side opposite the 60 - degree angle

\(b = a\sqrt{3}\), substituting \(a = 5\) inches, we get \(b = 5\sqrt{3}\approx5\times1.732 = 8.66\) inches.

Step3: Calculate the perimeter

The perimeter \(P=a + b + c\). Substituting \(a = 5\) inches, \(b\approx8.66\) inches, and \(c = 10\) inches, we have \(P=5+8.66 + 10=23.66\approx23.7\) inches.

Answer:

23.7 in.