A right triangle ABC has side lengths AB=6, BC=8, and CA=10. A second right triangle has corresponding vertices of A1, B1, and C1, with side lengths of 32,40, and 24. In 3-5 sentences describe how to find the ratio of the side opposite angleA to the hypotenuse of triangle ABC. Then use this ratio to identify the location of point A1 in the second right triangle

To find the ratio of the side opposite angle A to the hypotenuse in triangle ABC, we first identify the side lengths. In triangle ABC, the side opposite angle A is side BC, which measures 8 units, and the hypotenuse is side CA, which measures 10 units. The ratio is calculated as the length of the opposite side divided by the hypotenuse: \( \frac{BC}{CA} = \frac{8}{10} = \frac{4}{5} \).

In triangle A1B1C1, the hypotenuse is side A1C1, measuring 40 units. To find the corresponding length of the side opposite angle A1 (which is side B1C1), we apply the same ratio. The length of side B1C1 can be found using the ratio \(\frac{4}{5}\) relative to the hypotenuse: \( B1C1 = \frac{4}{5} \times 40 = 32 \). Thus, point A1 will be positioned such that the length B1C1 aligns with this calculation, indicating that it is opposite angle A1.