Question

A right triangle has a hypotenuse of length 40 and an angle of 25 degrees with a side opposite this angle of length 16. A second right triangle also has an angle of 25 degree with a hypotenuse of length 10. Determine the length of the side opposite the 25 degree angle on the second triangle.

The side opposite the second triangle's 25 degree angle has a length 8.
The side opposite the second triangle's 25 degree angle has a length 8 square root 21.
The side opposite the second triangle's 25 degree angle has a length of 4.
The side opposite the second triangle's 25 degree angle has length 64

Answer 1

To determine the length of the side opposite the 25-degree angle in the second triangle, we can use the sine function. The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

For the second right triangle:

- Angle = 25 degrees
- Hypotenuse = 10

The sine of 25 degrees gives us the ratio of the opposite side (let's call it \( x \)) to the hypotenuse. Thus, we can write the equation:

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

To solve for \( x \), we can rearrange the equation:

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

Using a calculator:

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

Substituting this value in:

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

Approximately, \( x \approx 4 \).

Therefore, the length of the side opposite the 25-degree angle in the second triangle is approximately:

\[
\boxed{4}
\]