Question

Right triangle ABC has side lengths AB=6 , BC=8 , and CA=10 . A second right triangle has corresponding vertices of A′ , B′ , and C′ , with side lengths of 32, 40, and 24. In 3–5 sentences, describe how to 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.(4 points)

Answer 1

To find the ratio of the side opposite ∠A to the hypotenuse of triangle ABC, we first identify the side opposite angle A, which is side BC with a length of 8, and the hypotenuse CA with a length of 10. The ratio is calculated by dividing the length of the opposite side by the hypotenuse, giving us \( \frac{BC}{CA} = \frac{8}{10} = \frac{4}{5} \).

Next, we apply this ratio to triangle A′B′C′. The hypotenuse A′C′ is given as 40, so we find the corresponding length of the side opposite angle A, which is A′B′. Using the same ratio, we calculate \( A′B′ = \frac{4}{5} \times 40 = 32 \). This means that point A′ should be located on the segment opposite to the hypotenuse of triangle A′B′C′, maintaining the same ratio as in triangle ABC.