Right triangle

Question

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
LA to the hypotenuse of triangle
ABC. Then use this ratio to identity the location of point
A' in the second right triangle.
(4 points)

Answer 1

To find the ratio of the side opposite angle A to the hypotenuse of triangle ABC, we first identify the lengths of the sides: side AB (which is opposite angle A) measures 6, and the hypotenuse CA measures 10. The ratio is thus calculated as \( \frac{AB}{CA} = \frac{6}{10} = \frac{3}{5} \).

Using this ratio, we can now identify the corresponding location of point A' in the second triangle with side lengths of 32 (opposite angle A'), 40 (opposite angle B'), and 24 (hypotenuse). Since the hypotenuse of the second triangle (24) corresponds to the same angle, we apply the same ratio \( \frac{3}{5} \) to the hypotenuse: \( A' = \frac{3}{5} \times 24 = \frac{72}{5} = 14.4 \). Therefore, point A' should be located at a position such that it is 14.4 units away from point B' in the second right triangle.