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 triangl
To find the ratio of the side opposite ∠A to the hypotenuse of triangle ABC, we can use the Pythagorean theorem which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Specifically, in triangle ABC, the side opposite ∠A is BC=8 and the hypotenuse is CA=10. Therefore, the ratio of the side opposite ∠A to the hypotenuse is 8/10 or 4/5.
Using this ratio, we can identify the location of point A′ in the second right triangle. Since the hypotenuse in the second triangle is 40 and the side opposite ∠A′ is 32, we can multiply the ratio 4/5 by the hypotenuse 40 to find the corresponding side opposite ∠A′. Thus, 4/5 * 40 = 32. This means that the length of the side opposite ∠A′ is 32, so point A′ is located at the vertex opposite the 32 side in the second right triangle.