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: Identify the triangle type

This is a 30 - 60 - 90 triangle. 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 side (let's call it \(x\)), the side opposite \(60^{\circ}\) is \(x\sqrt{3}\), and the hypotenuse (opposite \(90^{\circ}\)) is \(2x\).

Step2: Find the length of the shortest side

We know the hypotenuse is 10 inches. Since the hypotenuse is \(2x\), we can set up the equation \(2x = 10\). Solving for \(x\), we divide both sides by 2: \(x=\frac{10}{2}=5\) inches. So the side opposite \(30^{\circ}\) is 5 inches.

Step3: Find the length of the other leg

The side opposite \(60^{\circ}\) is \(x\sqrt{3}\). Substituting \(x = 5\), we get \(5\sqrt{3}\) inches. Calculating the numerical value, \(5\sqrt{3}\approx5\times1.732 = 8.66\) inches.

Step4: Calculate the perimeter

The perimeter \(P\) of a triangle is the sum of its three sides. The sides are 5 inches, \(5\sqrt{3}\) inches (≈8.66 inches), and 10 inches. So \(P=5 + 5\sqrt{3}+10\). Combining like terms, we get \(P = 15+5\sqrt{3}\). Substituting the approximate value of \(5\sqrt{3}\) (≈8.66), we have \(P\approx15 + 8.66=23.66\) inches. Rounding to the nearest tenth, we get \(23.7\) inches.

Answer:

23.7 in.