Question

right triangle abc has side lengths ab = 9, bc = 12, and ac = 15. a second right triangle, abc, has side lengths of 30, 24, and 18. 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. (1 point)
the ratio of the opposite side to the hypotenuse is 0.6, and point a is opposite the side that has length 18.
the ratio of the opposite side to the hypotenuse is 0.8, and point a is opposite the side that has length 18.
the ratio of the opposite side to the hypotenuse is 1.25, and point a is opposite the side that has length 24.
the ratio of the opposite side to the hypotenuse is 0.8, and point a is opposite the side that has length 24.

Explanation:

Step1: Identify opposite side and hypotenuse for ∠A in △ABC

In right - triangle ABC with sides AB = 9, BC = 12, and AC = 15, for ∠A, the opposite side is BC and the hypotenuse is AC.

Step2: Calculate the ratio

The ratio of the side opposite ∠A to the hypotenuse is $\frac{BC}{AC}=\frac{12}{15}=0.8$.

Step3: Analyze the second triangle

In right - triangle A'B'C' with side lengths 30, 24, and 18, we need to find the location of A'. Since the ratio of the opposite side to the hypotenuse in the first triangle is 0.8, we know that in the second triangle, if we consider the sides, the side opposite A' should be such that the ratio of the side opposite A' to the hypotenuse gives 0.8. If we assume the hypotenuse of △A'B'C' is 30, and we want a ratio of 0.8, the side opposite A' is $0.8\times30 = 24$. So point A' is opposite the side that has length 24.

Answer:

The ratio of the opposite side to the hypotenuse is 0.8, and point A' is opposite the side that has length 24.