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

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 is defined as:

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

For the second triangle, we know:

• The angle \( \theta = 25° \)
• The hypotenuse = 10

Let \( x \) be the length of the side opposite the 25° angle. We can set up the equation as follows:

\[ \sin(25°) = \frac{x}{10} \]

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

\[ x = 10 \cdot \sin(25°) \]

Now we calculate \( \sin(25°) \). Using a calculator:

\[ \sin(25°) \approx 0.4226 \]

Therefore,

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

Rounding to the nearest whole number, we find \( x \approx 4 \).

Thus, the length of the side opposite the second triangle’s 25° angle is approximately 4.

So the answer is:

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