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: Identify 30-60-90 triangle sides

In a 30-60-90 triangle, the side opposite $30^\circ$ is $\frac{1}{2}$ the hypotenuse, and the side opposite $60^\circ$ is $\frac{\sqrt{3}}{2}$ the hypotenuse.
Side opposite $30^\circ$: $\frac{1}{2} \times 10 = 5$ inches
Side opposite $60^\circ$: $\frac{\sqrt{3}}{2} \times 10 = 5\sqrt{3} \approx 8.660$ inches

Step2: Calculate perimeter sum

Add all three sides together.
$\text{Perimeter} = 10 + 5 + 5\sqrt{3} \approx 10 + 5 + 8.660$

Step3: Compute and round result

Sum the values and round to nearest tenth.
$\text{Perimeter} \approx 23.660 \approx 23.7$ inches

Answer:

23.7 in.