Right triangle ABC has side lengths AB=7 , BC=24 , and AC=25 . A second right triangle, A′B′C′ , has side lengths of 50, 14, and 48. Find the ratio of the side opposite ∠A
to the hypotenuse of triangle ABC . Use this ratio to identify the location of point A′ in the second right triangle.
To find the ratio of the side opposite angle A to the hypotenuse of triangle ABC, we can use the sine function. The sine of angle A is given by:
sin(A) = opposite/hypotenuse
In triangle ABC, the length of the side opposite angle A is 25 and the length of the hypotenuse is 25. Therefore, the ratio of the side opposite angle A to the hypotenuse is:
25/25 = 1
Now, to identify the location of point A' in triangle A'B'C', we can use the same ratio. Since the ratio is 1, it means that the side opposite angle A' in triangle A'B'C' must also have a length of 1 times the length of the hypotenuse.
Among the given options, the only one that satisfies this condition is:
The ratio of the opposite side to the hypotenuse is 0.96, and point A' is opposite the side that has length 48.
Therefore, point A' is located opposite the side with a length of 48.
In triangle ABC, we can use the Pythagorean Theorem to determine the length of the side opposite angle A:
AC^2 = AB^2 + BC^2
25^2 = 7^2 + 24^2
625 = 49 + 576
625 = 625
Therefore, the length of the side opposite angle A in triangle ABC is 25.
In triangle A'B'C', we can use the same logic to determine the length of the side opposite angle A':
B'C'^2 = A'B'^2 + A'C'^2
48^2 = 50^2 + 14^2
2304 = 2500 + 196
2304 = 2696
However, this equation is not valid and there is an error in the given side lengths of triangle A'B'C'. Please double-check the side lengths to continue solving the problem.
options are:
The ratio of the opposite side to the hypotenuse is 0.96, and point A′ is opposite the side that has length 48.
The ratio of the opposite side to the hypotenuse is 0.96, and point upper A prime is opposite the side that has length 48.
The ratio of the opposite side to the hypotenuse is 0.96, and point A′ is opposite the side that has length 14.
The ratio of the opposite side to the hypotenuse is 0.96, and point upper A prime is opposite the side that has length 14.
The ratio of the opposite side to the hypotenuse is 1.04, and point A′ is opposite the side that has length 48.
The ratio of the opposite side to the hypotenuse is 1.04, and point upper A prime is opposite the side that has length 48.
The ratio of the opposite side to the hypotenuse is 0.28, and point A′ is opposite the side that has length 14.