To find the angle measure at vertex C in the right triangle ABC, we can use the sine, cosine, or tangent functions. Since we have the lengths of the sides, we can use the cosine function, which relates the adjacent side (CA) and the hypotenuse (BC) to the angle at vertex C.
Given:
- \( AB = 9 \)
- \( AC = 40 \)
- \( BC = 41 \)
We can find angle measure \( C \) using the cosine function:
\[ \cos(C) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{AC}{BC} = \frac{40}{41} \]
Now, calculate the angle \( C \):
\[ C = \cos^{-1}\left(\frac{40}{41}\right) \]
Calculating this gives:
\[ C \approx \cos^{-1}(0.9756) \approx 12.5° \]
Rounding to the nearest whole number, the angle at vertex C is approximately \( 13° \).
Therefore, the correct response is:
13°.