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 0.6, 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 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 1.25, and point A′
is opposite the side that has length 24.
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 0.6, 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 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 1.25, and point A′
is opposite the side that has length 24.
Answers
GPT-4o mini
To find the ratio of the side opposite \( \angle A \) to the hypotenuse of triangle \( ABC \):
1. In triangle \( ABC \):
- \( AB = 9 \) is opposite \( \angle C \)
- \( BC = 12 \) is opposite \( \angle A \)
- \( AC = 15 \) is the hypotenuse
The opposite side to \( \angle A \) is \( BC = 12 \).
2. The hypotenuse is \( AC = 15 \).
Now, we can find the ratio of the opposite side to the hypotenuse:
\[
\text{Ratio} = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{12}{15} = \frac{4}{5} = 0.8
\]
Next, we need to identify the location of point \( A' \) in triangle \( A'B'C' \):
In triangle \( A'B'C' \):
- The side lengths are 30, 24, and 18.
- The largest side is the hypotenuse, which is 30.
The side lengths:
- Let the side opposite \( A' \) be \( B'C' = 18 \) (as it's opposite to the angle \( A' \)).
- The side opposite \( B' \) is \( A'C' = 24 \).
Since we found that the ratio of the opposite side to the hypotenuse is \( 0.8 \), which corresponds to the side opposite \( A \) as \( BC = 12 \) and hypotenuse \( AC = 15 \) in triangle \( ABC \).
In triangle \( A'B'C' \), since point \( A' \) corresponds to the angle opposite side \( B'C' = 18 \), 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 18.**
1. In triangle \( ABC \):
- \( AB = 9 \) is opposite \( \angle C \)
- \( BC = 12 \) is opposite \( \angle A \)
- \( AC = 15 \) is the hypotenuse
The opposite side to \( \angle A \) is \( BC = 12 \).
2. The hypotenuse is \( AC = 15 \).
Now, we can find the ratio of the opposite side to the hypotenuse:
\[
\text{Ratio} = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{12}{15} = \frac{4}{5} = 0.8
\]
Next, we need to identify the location of point \( A' \) in triangle \( A'B'C' \):
In triangle \( A'B'C' \):
- The side lengths are 30, 24, and 18.
- The largest side is the hypotenuse, which is 30.
The side lengths:
- Let the side opposite \( A' \) be \( B'C' = 18 \) (as it's opposite to the angle \( A' \)).
- The side opposite \( B' \) is \( A'C' = 24 \).
Since we found that the ratio of the opposite side to the hypotenuse is \( 0.8 \), which corresponds to the side opposite \( A \) as \( BC = 12 \) and hypotenuse \( AC = 15 \) in triangle \( ABC \).
In triangle \( A'B'C' \), since point \( A' \) corresponds to the angle opposite side \( B'C' = 18 \), 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 18.**