Question

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 0.96, 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 0.96, and point A′ is opposite the side that has length 14. The ratio of the opposite side to the hypotenuse is 0.96, and point upper A prime is opposite the side that has length 14. 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 0.28, 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 identify the relevant side lengths. The sides of triangle ABC are:

- AB = 7 (opposite ∠C)
- BC = 24 (opposite ∠A)
- AC = 25 (hypotenuse)

In this case, the side opposite ∠A (which is side BC) is 24, and the hypotenuse (side AC) is 25. Thus, we can calculate the ratio:

\[
\text{Ratio} = \frac{\text{Opposite side}}{\text{Hypotenuse}} = \frac{BC}{AC} = \frac{24}{25}
\]

Calculating that gives us:

\[
\text{Ratio} = 0.96
\]

Now, we need to identify the location of point A′ in the second triangle A′B′C′ with sides 50, 14, and 48.

In triangle A′B′C′, we determine which side is opposite angle A′ by using the same criterion:

The sides of triangle A′B′C′ are:

- A′B′ = 50 (hypotenuse)
- B′C′ = 14 (opposite angle A′)
- A′C′ = 48 (opposite angle B′)

Given this setup, the side that is opposite angle A′ is the side with length 14.

Since the ratio of the opposite side to the hypotenuse is a consistent property of similar triangles, and we found that ratio to be 0.96 for triangle ABC (which corresponds to A), point A′ in triangle A′B′C′ is opposite the side that has length 14.

So, the correct conclusion is:

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