Question

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

Explanation:

Step1: Recall 30-60-90 triangle ratios

In a 30-60-90 triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\), where the side opposite \(30^\circ\) is the shortest (\(x\)), the side opposite \(60^\circ\) is \(x\sqrt{3}\), and the hypotenuse is \(2x\).
Given hypotenuse \(= 10\) inches, so \(2x = 10\).

Step2: Solve for \(x\)

From \(2x = 10\), we get \(x=\frac{10}{2}=5\) inches (this is the side opposite \(30^\circ\)).

Step3: Find the side opposite \(60^\circ\)

The side opposite \(60^\circ\) is \(x\sqrt{3}=5\sqrt{3}\) inches. Calculate \(5\sqrt{3}\approx5\times1.732 = 8.66\) inches.

Step4: Calculate the perimeter

Perimeter \(= x + x\sqrt{3}+ 2x=5 + 8.66+ 10\).
First, \(5 + 8.66 = 13.66\), then \(13.66+ 10 = 23.66\approx23.7\) inches (rounded to the nearest tenth).

Answer:

23.7 in.