To find the ratio of the side opposite ∠A to the hypotenuse of triangle ABC, we need to identify the lengths of the sides relative to angle A.
In triangle ABC:
- The hypotenuse AC = 15
- The side opposite ∠A (which is side BC) = 12
Now, we can calculate the ratio:
\[ \text{Ratio} = \frac{\text{opposite side (BC)}}{\text{hypotenuse (AC)}} = \frac{12}{15} = \frac{4}{5} = 0.8 \]
Next, we need to identify point A′ in triangle A′B′C′, which has side lengths 30, 24, and 18. Since we have determined that the ratio of the opposite side to the hypotenuse is 0.8, we will find the side lengths that correspond to this ratio.
Let's identify the hypotenuse in triangle A′B′C′, which is the longest side:
- The hypotenuse (opposite the right angle) is 30.
- The side lengths are 30 (hypotenuse), 24, and 18.
Now calculate the ratio of the sides:
\[ \text{Ratio} = \frac{\text{opposite side}}{30} \]
We need this ratio to be 0.8. Let's solve for the opposite side:
\[ 0.8 = \frac{\text{opposite side}}{30} \]
Thus,
\[ \text{opposite side} = 0.8 \times 30 = 24 \]
Therefore, point A′ is opposite the side that has length 24.
The final conclusion is: The ratio of the opposite side to the hypotenuse is 0.8, and point A′ is opposite the side that has length 24.