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 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 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.
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 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 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.
Answers
Answered by
GPT-4o mini
To find the ratio of the side opposite ∠A to the hypotenuse of triangle ABC, we start by identifying the sides.
In triangle ABC:
- Side AB = 9 (opposite side to ∠C)
- Side BC = 12 (opposite side to ∠A)
- Hypotenuse AC = 15
The side opposite ∠A (which is BC = 12) and the hypotenuse (AC = 15) allows us to calculate the ratio:
\[
\text{Ratio} = \frac{\text{opposite side (BC)}}{\text{hypotenuse (AC)}} = \frac{12}{15} = 0.8
\]
Next, we will identify the location of point A' in triangle A'B'C'. The side lengths of triangle A'B'C' are:
- A'B' = 30
- B'C' = 24
- Hypotenuse A'C' = 18
To find where point A′ is located, we need to match triangles ABC and A′B′C′ considering their similarity due to both being right triangles.
In triangle A′B′C′, the side opposite the angle corresponding to angle A (which is BC in triangle ABC) must correspond to the side opposite to the angle A′. Since angle A corresponds to the angle B′, we need to match the sides:
- The length opposite A (which is BC = 12 in triangle ABC) corresponds to the length opposite A′ in triangle A′B′C′, which, according to our ratio of side lengths, aligns with the side of length 24 (since \(\frac{24}{30} = 0.8\)).
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.
In triangle ABC:
- Side AB = 9 (opposite side to ∠C)
- Side BC = 12 (opposite side to ∠A)
- Hypotenuse AC = 15
The side opposite ∠A (which is BC = 12) and the hypotenuse (AC = 15) allows us to calculate the ratio:
\[
\text{Ratio} = \frac{\text{opposite side (BC)}}{\text{hypotenuse (AC)}} = \frac{12}{15} = 0.8
\]
Next, we will identify the location of point A' in triangle A'B'C'. The side lengths of triangle A'B'C' are:
- A'B' = 30
- B'C' = 24
- Hypotenuse A'C' = 18
To find where point A′ is located, we need to match triangles ABC and A′B′C′ considering their similarity due to both being right triangles.
In triangle A′B′C′, the side opposite the angle corresponding to angle A (which is BC in triangle ABC) must correspond to the side opposite to the angle A′. Since angle A corresponds to the angle B′, we need to match the sides:
- The length opposite A (which is BC = 12 in triangle ABC) corresponds to the length opposite A′ in triangle A′B′C′, which, according to our ratio of side lengths, aligns with the side of length 24 (since \(\frac{24}{30} = 0.8\)).
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.
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