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 length of the shorter leg (\(a\)) 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 length of the longer leg (\(b\)) in a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle is \(b = a\sqrt{3}\). Substituting \(a = 5\) inches, we get \(b=5\sqrt{3}\approx5\times1.732 = 8.66\) inches.

Step3: Calculate the perimeter

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

Answer:

23.7 in.