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 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 0.8, 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 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.

Answer 1

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

- The side opposite angle A (∠A) is side BC, which has a length of 12.
- The hypotenuse AC has a length of 15.

Now, we calculate the ratio of the length of the side opposite ∠A to the hypotenuse:

\[
\text{Ratio} = \frac{\text{Opposite Side}}{\text{Hypotenuse}} = \frac{BC}{AC} = \frac{12}{15} = \frac{4}{5} = 0.8.
\]

Now, we’ll check the corresponding side in triangle A′B′C′. The sides of triangle A′B′C′ are:

- A′B′ (30)
- B′C′ (24)
- A′C′ (18)

Since the ratio calculated for triangle ABC is 0.8, we need to find the corresponding side in triangle A′B′C′ that holds this ratio concerning its hypotenuse. The hypotenuse A′C′ is 18, and the other side should relate as such:

\[
\text{Side opposite ∠A′ in triangle A′B′C′} = 0.8 \times \text{Hypotenuse} = 0.8 \times 18 = 14.4.
\]

However, from the available sides, we find:

- The side A′B′ is 30,
- The side B′C′ is 24,
- A′C′ is 18.

Since the side length of 14.4 does not exist, we consider only the available sides. We see that:

\[
\frac{B′C′}{A′C′} = \frac{24}{18} = \frac{4}{3}.
\]

This indicates that B′C′ can be corresponding to the side opposite to angle A′. Therefore, the side B′C′ is the length that corresponds to the previously calculated ratio of 0.8, since 24 is closer than the others.

Ultimately, we can conclude:

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

So, the correct response is:

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