To solve this problem, we first need to identify the relationship between the sides of triangle \(ABC\) and the corresponding triangles.
Step 1: Determine sides of triangle \(ABC\)
The sides of triangle \(ABC\) are:
- \(AB = 9\)
- \(BC = 12\)
- \(AC = 15\)
First, we identify the hypotenuse. In a right triangle, the hypotenuse is the longest side, so here it is:
- \(AC = 15\)
The side opposite angle \(A\) (let's denote it as \(a\)) is:
- \(BC = 12\)
Step 2: Calculate the ratio of the opposite side to the hypotenuse in triangle \(ABC\)
The ratio \(R\) of the side opposite angle \(A\) to the hypotenuse is calculated as: \[ R = \frac{BC}{AC} = \frac{12}{15} = 0.8 \]
Step 3: Analyze triangle \(A'B'C'\)
The second triangle \(A'B'C'\) has side lengths:
- \(A'B' = 30\)
- \(B'C' = 24\)
- \(A'C' = 18\)
Again, the hypotenuse is the longest side, which is:
- \(A'B' = 30\)
To find which side corresponds to point \(A'\), we check the ratio we calculated earlier. Since in triangle \(A'B'C'\), we have:
- The side opposite point \(A'\) is \(B'C' = 24\).
Conclusion
Thus, the statement about the ratio of the opposite side to the hypotenuse being \(0.8\) and that point \(A'\) is opposite the side that has length \(24\) is correct.
So, the final answer 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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