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)
Responses

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

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

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

The side opposite the second triangle’s 25°angle has length 821−−√.

Answer 1

To find the length of the side opposite the 25° angle in the second right triangle, where the hypotenuse length is 10, we can use the sine function, which is defined for a right triangle as follows:

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

In this case:

- \(\theta = 25^\circ\)
- hypotenuse = 10
- opposite side = let’s denote it as \(x\).

Using the sine function for the 25° angle, we have:

\[
\sin(25^\circ) = \frac{x}{10}
\]

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

\[
x = 10 \cdot \sin(25^\circ)
\]

Now, we need to calculate \(10 \cdot \sin(25^\circ)\). Using a calculator or sine table, we find:

\[
\sin(25^\circ) \approx 0.4226
\]

Now calculate \(x\):

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

Thus, rounding to the nearest whole number, we have:

\[
x \approx 4
\]

Therefore, the length of the side opposite the 25° angle in the second triangle is approximately 4.

The correct response is:

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