Question

a right triangle has a hypotenuse of length 40 and an angle of 25°, with a side opposite this angle of length 16. a second right triangle also has an angle of 25°, with a hypotenuse of length 10. determine the length of the side opposite the 25° angle on the second triangle. (1 point)

the side opposite the second triangle’s 25° angle has length 64.

the side opposite the second triangle’s 25° angle has length 4.

the side opposite the second triangle’s 25°angle has length 8√21.

the side opposite the second triangle’s 25° angle has length 8.

Explanation:

Step1: Use the sine - ratio property

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. For the first right - triangle with $\theta = 25^{\circ}$, hypotenuse $h_1 = 40$ and opposite side $a_1=16$, $\sin25^{\circ}=\frac{a_1}{h_1}=\frac{16}{40}$.

Step2: Apply the same ratio to the second triangle

Let the length of the side opposite the $25^{\circ}$ angle in the second triangle be $a_2$, and the hypotenuse of the second triangle $h_2 = 10$. Since $\sin25^{\circ}$ is the same for both right - triangles (because the angle is the same), and $\sin25^{\circ}=\frac{a_2}{h_2}$, and $\sin25^{\circ}=\frac{16}{40}$. We can set up the proportion $\frac{a_2}{10}=\frac{16}{40}$.

Step3: Solve the proportion for $a_2$

Cross - multiply: $40a_2=16\times10$. Then $a_2=\frac{16\times10}{40}$. Simplify the right - hand side: $a_2 = 4$.

Answer:

The side opposite the second triangle's $25^{\circ}$ angle has length 4.