Question

right triangle ABC has side lengths AB = 9, BC = 12, and AC = 15. a second right triangle, A'B'C', has side lengths of 30, 24, and 18. find the ratio of the side opposite <A to the hypotenuse of triangle ABC. then use this ratio to identify the location of point A' in the second right triangle.
a. the ratio of the opposite side to the hypotenuse is 0.8, and point A' is opposite the side that has length 18
b. the ratio of the opposite side to the hypotenuse is 1.25, and point A' is opposite the side that has length 24
c. the ratio of the opposite side to the hypotenuse is 0.6, and point A' is opposite the side that has length 18
d. the ratio of the opposite side to the hypotenuse is 0.8, and point A' is opposite the side that has length 24

Answer 1

To find the ratio of the side opposite angle \( A \) to the hypotenuse of triangle \( ABC \):

1. **Identify the sides of triangle \( ABC \)**:
- \( AB = 9 \) (side opposite angle \( C \))
- \( AC = 12 \) (side opposite angle \( B \))
- \( BC = 15 \) (hypotenuse)

2. **Calculate the ratio**:
The side opposite angle \( A \) in triangle \( ABC \) is \( BC = 15 \).
- The hypotenuse is \( AC = 12 \).
The opposite side of angle \( A \) is \( AB = 9 \).

Thus, the ratio of the length of side opposite angle \( A \) (which is \( 9 \)) to the hypotenuse (which is \( 15 \)) is:

\[
\text{Ratio} = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{AB}{BC} = \frac{9}{15} = \frac{3}{5} = 0.6
\]

Next, we look at triangle \( A'B'C' \) to identify where point \( A' \) is located regarding the side lengths given.

The sides of triangle \( A'B'C' \) are:
- \( A'B' = 30 \)
- \( B'C' = 24 \)
- \( A'C' = 18 \)

Since we established that \( A \) corresponds to the side opposite which in triangle \( A'B'C' \) gives a ratio of \( 0.6 \), we check which side this corresponds to:
- The hypotenuse here is \( A'B' = 30 \).
- The opposite side that would form a \( 0.6 \) ratio with \( 30 \) would be:

\[
\text{Opposite side} = 0.6 \times 30 = 18
\]

Thus, point \( A' \) must be opposite the side that is \( 18 \).

Finally, based on our calculations, the correct answer is:
**c. the ratio of the opposite side to the hypotenuse is 0.6, and point A' is opposite the side that has length 18.**