To find the ratio of the side opposite angle \( A \) to the hypotenuse in triangle \( ABC \), we first note that triangle \( ABC \) is a right triangle with side lengths \( AB = 9 \), \( BC = 12 \), and \( AC = 15 \).
In triangle \( ABC \):
- The side opposite angle \( A \) (which is side \( BC \)) has a length of \( 12 \).
- The hypotenuse (which is side \( AC \)) has a length of \( 15 \).
Now we can calculate the ratio of the opposite side to the hypotenuse:
\[ \text{Ratio} = \frac{\text{Opposite side}}{\text{Hypotenuse}} = \frac{BC}{AC} = \frac{12}{15} = \frac{4}{5} = 0.8. \]
Next, we need to identify the location of point \( A' \) in triangle \( A'B'C' \) with side lengths \( 30 \), \( 24 \), and \( 18 \). In triangle \( A'B'C' \):
- The side opposite angle \( A' \) (which is side \( B'C' \)) has a length of \( 24 \).
- The hypotenuse (which is side \( A'B' \)) has a length of \( 30 \).
Since we previously calculated the ratio to be \( 0.8 \) (which corresponds to the opposite side being \( 24 \) and the hypotenuse being \( 30 \)), we find that point \( A' \) is opposite the side that has the length \( 24 \).
Thus, the correct choice is: The ratio of the opposite side to the hypotenuse is 0.8, and point A' is opposite the side that has length 24.
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