Question

in the 30 - 60 - 90 triangle below, side s has a length of __ and side r has a length of __. a. 12\sqrt{3}, 12\sqrt{3} b. 6\sqrt{2}, 6\sqrt{2} c. 6, 6\sqrt{3} d. 24\sqrt{3}, 6 e. 12\sqrt{3}, 24 f. 6, 12\sqrt{3}

Explanation:

Step1: Recall 30 - 60 - 90 triangle ratio

In a 30 - 60 - 90 triangle, if the hypotenuse $c$ is given, and the sides are in the ratio $1:\sqrt{3}:2$ where the side opposite the 30 - degree angle is the shortest side $a$, the side opposite the 60 - degree angle is $a\sqrt{3}$, and the hypotenuse $c = 2a$.

Step2: Find side $s$ (opposite 30 - degree angle)

The hypotenuse is 12. Since the side opposite the 30 - degree angle $s$ is half of the hypotenuse, $s=\frac{12}{2}=6$.

Step3: Find side $r$ (opposite 60 - degree angle)

Since the side opposite the 60 - degree angle $r$ is $\sqrt{3}$ times the side opposite the 30 - degree angle, $r = 6\sqrt{3}$.

Answer:

C. 6, $6\sqrt{3}$