Question

Question
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 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.
The ratio of the opposite side to the hypotenuse is 1.25, 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 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.
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Answer 1

To find the ratio of the side opposite ∠A to the hypotenuse of triangle ABC, we start by identifying the sides of triangle ABC.

We have:
- Side AB = 9 (opposite ∠C)
- Side BC = 12 (opposite ∠A)
- Hypotenuse AC = 15

The side opposite ∠A is BC, which has a length of 12. The hypotenuse AC has a length of 15. Thus, the ratio of the side opposite ∠A to the hypotenuse of triangle ABC is:

\[
\text{Ratio} = \frac{\text{Opposite Side}}{\text{Hypotenuse}} = \frac{BC}{AC} = \frac{12}{15} = \frac{4}{5} = 0.8
\]

Now, we will use this ratio to determine the location of point A' in the second triangle A'B'C'. In triangle A'B'C', the side lengths are:
- A'B' = 30
- B'C' = 24
- A'C' = 18

To identify point A', we need the side opposite to point A (which corresponds to the side BC in triangle ABC). The ratio (0.8) applies to the sides of triangle A'B'C' as follows:

Since point A' in triangle A'B'C' corresponds to the point opposite to side B'C', we can calculate the length of this side:

\[
\text{Opposite Side in A'B'C'} = 0.8 \times \text{Hypotenuse in A'B'C'} = 0.8 \times 30 = 24
\]

Since B'C' is 24, which is indeed the opposite side corresponding to A', this confirms its location.

Thus, we conclude:
**The ratio of the opposite side to the hypotenuse is 0.8, and point A' is opposite the side that has length 24.**