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

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 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 1.25, and point A′
is opposite the side that has length 24.

1 answer

To find the ratio of the side opposite ∠A to the hypotenuse of triangle ABC, we first identify the sides of triangle ABC:

  • Side lengths:
    • \( AB = 9 \) (opposite to vertex C)
    • \( BC = 12 \) (opposite to vertex A)
    • \( AC = 15 \) (the hypotenuse)

The side opposite ∠A in triangle ABC is BC, which measures 12. The hypotenuse is AC, which measures 15. Now, we can compute the ratio:

\[ \text{Ratio} = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{12}{15} = \frac{4}{5} = 0.8 \]

Now, we can move to triangle A′B′C′, which has the following side lengths:

  • Side lengths:
    • \( A'B' = 30 \)
    • \( B'C' = 24 \)
    • \( A'C' = 18 \) (hypotenuse)

The side opposite corresponding vertex A′ in triangle A′B′C′ would be the side B′C′, which has a length of 24.

Thus, the information concludes as follows:

  • The ratio of the opposite side to the hypotenuse in triangle ABC is \( 0.8 \).
  • Point A′ in triangle A′B′C′ is opposite the side that has length \( 24 \).

Therefore, 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.