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.(1 point)
Responses

The ratio of the opposite side to the hypotenuse is 0.8, and point A′
is opposite the side that has length 18.
The ratio of the opposite side to the hypotenuse is 0.8, and point upper A prime is opposite the side that has length 18.

The ratio of the opposite side to the hypotenuse is 1.25, and point A′
is opposite the side that has length 24.
The ratio of the opposite side to the hypotenuse is 1.25, and point upper A prime is opposite the side that has length 24.

The ratio of the opposite side to the hypotenuse is 0.6, and point A′
is opposite the side that has length 18.
The ratio of the opposite side to the hypotenuse is 0.6, and point upper A prime is opposite the side that has length 18.

The ratio of the opposite side to the hypotenuse is 0.8, and point A′
is opposite the side that has length 24.
The ratio of the opposite side to the hypotenuse is 0.8, and point upper A prime is opposite the side that has length 24.
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Answer 1

To find the ratio of the side opposite ∠A to the hypotenuse in triangle ABC, we first identify the sides. In triangle ABC:

- The hypotenuse AC = 15
- The side opposite ∠A (which is side BC) = 12

Now we can calculate the ratio:

\[
\text{Ratio} = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{12}{15} = \frac{4}{5} = 0.8
\]

Next, for triangle A′B′C′, we need to locate point A′, which is opposite the corresponding side. The side lengths of triangle A′B′C′ are:

- Hypotenuse (side opposite the right angle): A′C′ = 30
- One side (corresponding to side BC of triangle ABC) = B′C′ = 24
- The other side (corresponding to side AB of triangle ABC) = A′B′ = 18

Since we just calculated that the ratio of the opposite side to the hypotenuse is 0.8, we need to find the corresponding side to side BC of triangle A′B′C′. The side opposite the corresponding angle A′ would be the one corresponding to length 24, because:

\[
\text{For triangle A′B′C′, the analogous hypotenuse is 30 and the side opposite to A′ (like BC in triangle ABC) is 24.}
\]

Thus, the correct identification is:

**The ratio of the opposite side to the hypotenuse is 0.8, and point A′ is opposite the side that has length 24.**