Question

a rigth 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.
a. the side opposite the second triangle's 25* angle has length 8<21
b. the side opposite the second triangle's 25* angle has length 64
c. the side opposite the second triangle's 25* angle has length 4
d. the side opposite the second triangle's 25* angle has length 8

Answer 1

To find the length of the side opposite the 25° angle in the second right triangle, we can use the relationship of the sine function in right triangles.

The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

For the second triangle, we have:

- Hypotenuse \( = 10 \)
- Angle \( \theta = 25° \)

We can find the length of the side opposite the 25° angle (let's call it \( x \)) using the sine function:

\[
\sin(25°) = \frac{x}{10}
\]

To find \( x \), we rearrange this equation:

\[
x = 10 \cdot \sin(25°)
\]

Now, we need to calculate \( \sin(25°) \). Using a calculator or sine table:

\[
\sin(25°) \approx 0.4226
\]

Now substituting this value into the equation:

\[
x = 10 \cdot 0.4226
\]
\[
x \approx 4.226
\]

Now we can compare this value with the given options.

The closest match is:
- c. the side opposite the second triangle's 25° angle has length 4.

Therefore, the correct answer is **c. the side opposite the second triangle's 25° angle has length 4**.