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 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 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 0.28, 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 14.

1 answer

In triangle \( ABC \):

  • Side lengths are \( AB = 7 \), \( BC = 24 \), and \( AC = 25 \).
  • Angle \( \angle A \) is opposite side \( BC \) with length \( 24 \) and the hypotenuse \( AC \) has length \( 25 \).

We need to calculate the ratio of the side opposite \( \angle A \) to the hypotenuse:

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

Next, let's analyze triangle \( A'B'C' \):

  • Side lengths are \( 50 \), \( 14 \), and \( 48 \).
  • In this triangle, \( A' \) is the angle opposite the side with length \( B'C' = 48 \) which is the longest side and hence the hypotenuse.

Therefore, the ratio of the opposite side to the hypotenuse in triangle \( ABC \) is \( 0.96 \), and point \( A' \) in triangle \( A'B'C' \) is opposite the side that has length \( 48 \).

So the correct answer is:

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