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 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 0.8, 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.
Answers
GPT-4o mini
In triangle ABC, we can identify the sides based on the given information. The side lengths are:
- AB = 9 (this side is opposite angle C)
- BC = 12 (this side is opposite angle A)
- AC = 15 (this side is the hypotenuse)
To find the ratio of the side opposite to angle A (which is side BC) to the hypotenuse (which is side AC), we proceed as follows:
\[
\text{Ratio} = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{12}{15}
\]
Calculating this gives us:
\[
\text{Ratio} = \frac{12}{15} = \frac{4}{5} = 0.8
\]
Now we need to find out where point \( A' \) is located in triangle \( A'B'C' \). The lengths of the sides of triangle \( A'B'C' \) are given as 30, 24, and 18. We are looking for the side length opposite angle \( A' \).
In triangle \( A'B'C' \) (which we can map to triangle \( ABC \)), the sides can be configured as follows (based on the pattern known from triangle \( ABC \)):
- Side opposite \( A' = 24 \) (corresponding to BC)
- Side opposite \( B' = 18 \) (corresponding to AB)
- Hypotenuse (side opposite \( C' \)) = 30 (corresponding to AC)
Since we calculated the ratio of the side opposite angle \( A \) (which corresponds to \( A' \)) to the hypotenuse in triangle \( ABC \) and found it to be 0.8, we match it to triangle \( A'B'C' \):
The ratio of 0.8 corresponds to that of the longer side (24), thus the location of \( A' \) is opposite the side that has length 24.
Therefore, the correct statement is:
**The ratio of the opposite side to the hypotenuse is 0.8, and point \( A' \) is opposite the side that has length 24.**
- AB = 9 (this side is opposite angle C)
- BC = 12 (this side is opposite angle A)
- AC = 15 (this side is the hypotenuse)
To find the ratio of the side opposite to angle A (which is side BC) to the hypotenuse (which is side AC), we proceed as follows:
\[
\text{Ratio} = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{12}{15}
\]
Calculating this gives us:
\[
\text{Ratio} = \frac{12}{15} = \frac{4}{5} = 0.8
\]
Now we need to find out where point \( A' \) is located in triangle \( A'B'C' \). The lengths of the sides of triangle \( A'B'C' \) are given as 30, 24, and 18. We are looking for the side length opposite angle \( A' \).
In triangle \( A'B'C' \) (which we can map to triangle \( ABC \)), the sides can be configured as follows (based on the pattern known from triangle \( ABC \)):
- Side opposite \( A' = 24 \) (corresponding to BC)
- Side opposite \( B' = 18 \) (corresponding to AB)
- Hypotenuse (side opposite \( C' \)) = 30 (corresponding to AC)
Since we calculated the ratio of the side opposite angle \( A \) (which corresponds to \( A' \)) to the hypotenuse in triangle \( ABC \) and found it to be 0.8, we match it to triangle \( A'B'C' \):
The ratio of 0.8 corresponds to that of the longer side (24), thus the location of \( A' \) is opposite the side that has length 24.
Therefore, the correct statement is:
**The ratio of the opposite side to the hypotenuse is 0.8, and point \( A' \) is opposite the side that has length 24.**