Question

right triangle $abc$ has side lengths $ab = 6$, $bc = 6\sqrt{3}$, and $ac = 12$. a second right triangle, $abc$, has side lengths of $8\sqrt{3}$, $8$, and $16$. find the ratio of the side opposite $\angle c$ to the hypotenuse of triangle $abc$. then use this ratio to identify the location of point $c$ in the second right triangle. (1 point)

the ratio of the opposite side to the hypotenuse is $\frac{1}{2}$, and point $c$ is opposite the side that has $8\sqrt{3}$.
the ratio of the opposite side to the hypotenuse is $\frac{1}{2}$, and point $c$ is opposite the side that has length 8.
the ratio of the opposite side to the hypotenuse is $\frac{2}{1}$, and point $c$ is opposite the side that has length 8.
the ratio of the opposite side to the hypotenuse is $\frac{\sqrt{3}}{2}$, and point $c$ is opposite the side that has $8\sqrt{3}$.

Explanation:

Step1: Identify hypotenuse of △ABC

In right triangle ABC, the hypotenuse is the longest side, so $AC = 12$.

Step2: Find side opposite ∠C

The side opposite ∠C is $AB = 6$.

Step3: Calculate the ratio

Ratio = $\frac{\text{opposite side}}{\text{hypotenuse}} = \frac{6}{12} = \frac{1}{2}$

Step4: Locate C' in △A'B'C'

The hypotenuse of △A'B'C' is 16. We need the side with length equal to $\frac{1}{2} \times 16 = 8$, so C' is opposite the side of length 8.

Answer:

The ratio of the opposite side to the hypotenuse is $\frac{1}{2}$, and point $C'$ is opposite the side that has length 8.