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 in triangle ABC, we can use the trigonometric ratio of sine, which is sin(∠A) = opposite/hypotenuse. Plugging in the known values, we have sin(∠A) = 6/10 = 0.6. In the second triangle with side lengths 32, 40, and 24, we can use the sine ratio to find the side opposite ∠A′ by calculating 0.6 * 40 = 24, which means the side opposite ∠A′ is 24. This means that point A′ lies opposite the side of length 24 in the second triangle.
