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)
• The ratio of the opposite side to the hypotenuse is 0.8, and point A' is opposite the side that has length 24.
O 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 0.8, and point A' 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.

Answer 1

To find the ratio of the side opposite angle A (which is side BC) to the hypotenuse (which is side AC) of triangle ABC, we use the given lengths.

1. **Identify the sides:**
- Opposite side BC = 12
- Hypotenuse AC = 15

2. **Calculate the ratio:**
\[
\text{Ratio} = \frac{\text{Opposite side}}{\text{Hypotenuse}} = \frac{BC}{AC} = \frac{12}{15} = \frac{4}{5} = 0.8
\]

Now we know the ratio of the side opposite angle A to the hypotenuse in triangle ABC is **0.8**.

Next, we need to identify the location of point A' in the second triangle A'B'C".

3. **Identify the sides of triangle A'B'C":**
- The sides of triangle A'B'C" are 30, 24, and 18. This triangle is also a right triangle.

4. **Determine which side corresponds to being opposite angle A:**
- The ratio found was for triangle ABC’s opposite side to hypotenuse: \(\frac{12}{15} = 0.8\).
- We need to find the corresponding angle in triangle A'B'C" such that the opposite side also has a ratio with the hypotenuse of 0.8.

Let's evaluate the sides of triangle A'B'C":
- **Hypotenuse (largest side)** = 30
- **Opposite side** = To determine which side corresponds, we consider which pairing maintains the same ratio of opposite to hypotenuse:
\[
\text{For opposite side 24:} \frac{24}{30} = 0.8 \quad \text{(matches the ratio)}
\]
\[
\text{For opposite side 18:} \frac{18}{30} = 0.6 \quad \text{(does not match the ratio)}
\]

Since **0.8** is the ratio when considering the side of length **24** as opposite, we conclude that point A' is opposite the side length **24** of triangle A'B'C".

Thus, the correct statements derived from the problem are:

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

So, the correct answer is:
**The ratio of the opposite side to the hypotenuse is 0.8, and point A' is opposite the side that has length 24.**