Question

Right triangle ABC
has side lengths AB=7
, BC=24
, and AC=25
. A second right triangle, A′B′C′
, has side lengths of 50, 14, and 48. Find the ratio of the side opposite ∠A
to the hypotenuse of triangle ABC
. 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 0.28, and point A′
is opposite the side that has length 14.
The ratio of the opposite side to the hypotenuse is 0.28, and point upper A prime is opposite the side that has length 14.

The ratio of the opposite side to the hypotenuse is 0.96, and point A′
is opposite the side that has length 48.
The ratio of the opposite side to the hypotenuse is 0.96, and point upper A prime is opposite the side that has length 48.

The ratio of the opposite side to the hypotenuse is 1.04, and point A′
is opposite the side that has length 48.
The ratio of the opposite side to the hypotenuse is 1.04, and point upper A prime is opposite the side that has length 48.

The ratio of the opposite side to the hypotenuse is 0.96, and point A′
is opposite the side that has length 14.

Answer 1

To find the ratio of the side opposite ∠A to the hypotenuse of triangle ABC, we can use the given side lengths of triangle ABC:

- \( AB = 7 \) (opposite side to ∠A)
- \( AC = 25 \) (hypotenuse)

The ratio is calculated as follows:

\[
\text{Ratio} = \frac{\text{Opposite side}}{\text{Hypotenuse}} = \frac{AB}{AC} = \frac{7}{25}
\]

Calculating this gives:

\[
\frac{7}{25} = 0.28
\]

Next, we need to identify the corresponding point A' in triangle A'B'C', which has sides of lengths 50, 14, and 48. In a right triangle, the side opposite the angle corresponds to the same ratio of the opposite side to the hypotenuse.

The hypotenuse of triangle A'B'C' is the longest side, which is 50. We have found that the ratio of the opposite side (to the angle corresponding to A) to the hypotenuse in triangle ABC is 0.28. Therefore, we can find the equivalent opposite side in triangle A'B'C':

Using the ratio:
\[
0.28 = \frac{\text{Side opposite A'}}{50}
\]

Thus, the side opposite angle A' can be calculated:
\[
\text{Side opposite A'} = 0.28 \times 50 = 14
\]

Therefore, point A' is opposite the side that has length 14 in triangle A'B'C'.

The final answer is:

**The ratio of the opposite side to the hypotenuse is 0.28, and point A′ is opposite the side that has length 14.**