Question

solving for side lengths of right triangles 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. 23.1 in. 20.0 in. 27.4 in. 23.7 in.

Explanation:

Step1: Identify the type of triangle

This is a 30-60-90 right triangle. In a 30-60-90 triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\), where the side opposite 30° is the shortest (let's call it \(x\)), the side opposite 60° is \(x\sqrt{3}\), and the hypotenuse (opposite 90°) is \(2x\).

Step2: Find the length of the shorter leg

Given the hypotenuse is 10 inches. Since hypotenuse \(= 2x = 10\), we solve for \(x\):
\(x=\frac{10}{2}=5\) inches. This is the side opposite 30°.

Step3: Find the length of the longer leg

The longer leg (opposite 60°) is \(x\sqrt{3}\). Substituting \(x = 5\):
\(x\sqrt{3}=5\sqrt{3}\approx5\times1.732 = 8.66\) inches.

Step4: Calculate the perimeter

Perimeter \(= \) hypotenuse \(+\) shorter leg \(+\) longer leg \(= 10 + 5 + 8.66 = 23.66\) inches. Rounding to the nearest tenth, we look at the options. The closest option to 23.66 (or considering possible rounding during steps) is 23.1? Wait, no, wait. Wait, maybe I made a mistake. Wait, 5√3 is approximately 8.66, so 10 + 5 + 8.66 = 23.66. Wait, the options are 23.1, 20.0, 27.4, 23.7. Wait, 23.7 is closer to 23.66. Wait, maybe my calculation of the longer leg: wait, 5√3 is about 8.66, so 10 + 5 + 8.66 = 23.66, which is approximately 23.7 when rounded to the nearest tenth? Wait, no, 23.66 rounded to the nearest tenth is 23.7? Wait, 23.66: the tenths place is 6, hundredths is 6, so round up the tenths place: 23.7. Wait, but let's check again.

Wait, maybe the problem is that in a 30-60-90 triangle, hypotenuse is 2x, so x = 5 (shorter leg), longer leg is x√3 ≈ 8.66. Then perimeter is 10 + 5 + 8.66 = 23.66, which is approximately 23.7 when rounded to the nearest tenth. So the best estimate is 23.7 in. Wait, but let's check the options again. The options are 23.1, 20.0, 27.4, 23.7. So 23.7 is the closest to 23.66.

Wait, maybe I miscalculated the longer leg. Wait, 5√3 is approximately 8.66, so 10 + 5 + 8.66 = 23.66, which is 23.7 when rounded to the nearest tenth. So the answer should be 23.7 in.

Answer:

23.7 in.