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.

A) The ratio of the opposite side to the hypotenuse is 1.04, and point A' is opposite the side that has length 48.

B) The ratio of the opposite side to the hypotenuse is 0.96, and point A' is opposite the side that has length 48.

C) The ratio of the opposite side to the hypotenuse is 0.28, and point A' is opposite the side that has length 14.

D) 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

In triangle ABC, the side lengths are \( AB = 7 \), \( BC = 24 \), and \( AC = 25 \). Since triangle ABC is a right triangle with \( AC \) as the hypotenuse, we can find the ratio of the side opposite \( \angle A \) to the hypotenuse.

The side opposite \( \angle A \) is \( BC = 24 \) and the hypotenuse \( AC = 25 \).

The ratio of the side opposite \( \angle A \) to the hypotenuse is calculated as follows:

\[
\text{Ratio} = \frac{\text{Opposite side}}{\text{Hypotenuse}} = \frac{BC}{AC} = \frac{24}{25} = 0.96
\]

Now, let’s analyze triangle \( A'B'C' \) with side lengths \( 50 \), \( 14 \), and \( 48 \).

To identify the location of \( A' \), we first recognize that the side opposite \( \angle A' \) corresponds to the side that is \( 14 \) units long, because the other two sides (\( 50 \) and \( 48 \)) could serve as the hypotenuse and adjacent side depending on which angle we consider. Given the dimensions, we identify the longest side \( 50 \) as the hypotenuse.

In a right triangle, the designation of the opposite side is relevant to which angle we're referring to. Since the ratio we've computed (0.96) corresponds to the height being 24 and the hypotenuse being 25 in triangle ABC, we are looking for a corresponding side in triangle \( A'B'C' \).

Since \( A'B'C' \) has \( 14 \) as its side opposite \( A' \):

- The hypotenuse is \( 50 \).
- Using our computed ratio, we can conclude that point \( A' \) in triangle \( A'B'C' \) is actually opposite the side of length \( 14 \).

Therefore, the correct choice is:

**D) The ratio of the opposite side to the hypotenuse is 0.96, and point A' is opposite the side that has length 14.**