Kaylee draws a right triangle ABC

𝐴
𝐵
𝐶
on a paper. The right angle is located at vertex A
𝐴
with side length BC
𝐵
𝐶
as the hypotenuse. The side length measurements are AB¯¯¯¯¯¯¯¯=9
𝐴
𝐵
¯
=
9
units, BC¯¯¯¯¯¯¯¯=41
𝐵
𝐶
¯
=
41
units, and CA¯¯¯¯¯¯¯¯=40
𝐶
𝐴
¯
=
40
units. What is the approximate angle measure at vertex C
𝐶
?(1 point)
Responses

55°
55°

29°
29°

77°
77°

13°

1 answer

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°.