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.(
To find the ratio of the side opposite ∠A to the hypotenuse of triangle ABC, we use the trigonometric function sine: sin(∠A) = (side opposite ∠A) / (hypotenuse). This gives us sin(∠A) = AB/CA = 6/10 = 3/5. Using this ratio, we can determine the side opposite ∠A in the second right triangle by multiplying it with the hypotenuse, giving us (3/5) * 32 = 19.2. Therefore, the side opposite ∠A in the second right triangle has a length of 19.2.
