Right triangle ABCABC has side lengths AB=6AB=6, BC=8BC=8, and CA=10CA=10. A second right triangle has corresponding vertices of A′A′, B′B′, and C′C′, with side lengths of 32, 40, and 24. In 3–5 sentences, describe how to find the ratio of the side opposite ∠A∠A to the hypotenuse of triangle ABCABC. Then use this ratio to identify the location of point A′A′ in the second right triangle.
To find the ratio of the side opposite ∠A∠A to the hypotenuse of triangle ABCABC, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. So, in triangle ABCABC, AC^2 = AB^2 + BC^2. We can substitute the given values to find the length of AC. Then, we can find the ratio of the side opposite ∠A∠A (which is AB) to the hypotenuse AC. Based on this ratio, we can determine the location of point A′A′ in the second right triangle.
