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)

Question

Responses

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 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 821−−√.

Answer 1

To find the length of the side opposite the 25° angle in the second right triangle, we can use the sine function, which relates the angle to the lengths of the sides of the triangle.

The sine of an angle in a right triangle is given by:

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

In this case, we have the angle \( \theta = 25° \) and the hypotenuse of the second triangle is 10. Let \( x \) be the length of the side opposite the 25° angle. From the sine function:

\[ \sin(25°) = \frac{x}{10} \]

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

\[ x = 10 \cdot \sin(25°) \]

We already know from the first triangle that with a hypotenuse of 40, the side opposite the 25° angle was calculated as 16, so we have:

\[ \sin(25°) = \frac{16}{40} = \frac{2}{5} = 0.4 \]

Using this approximation for a quick calculation:

\[ x = 10 \cdot \sin(25°) = 10 \cdot 0.4 = 4 \]

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

The correct response is:

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