Question

For △ ABC , find the measure of ∠ C to the nearest degree when side a=46m , side b=30m , and side c=20m. (1 point) 28 degrees 133 degrees 24 degrees 19 degrees

Answer 1

To find the measure of angle \( C \) in triangle \( ABC \), we can use the Law of Cosines, which states:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

where:
- \( a = 46 \, m \)
- \( b = 30 \, m \)
- \( c = 20 \, m \)

We can rearrange the formula to solve for \( \cos(C) \):

\[
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
\]

Now we can substitute the values for \( a \), \( b \), and \( c \):

1. Calculate \( a^2 \), \( b^2 \), and \( c^2 \):
- \( a^2 = 46^2 = 2116 \)
- \( b^2 = 30^2 = 900 \)
- \( c^2 = 20^2 = 400 \)

2. Substitute into the formula:
\[
\cos(C) = \frac{2116 + 900 - 400}{2 \cdot 46 \cdot 30}
\]
\[
\cos(C) = \frac{2616}{2760}
\]

3. Simplify:
\[
\cos(C) = 0.9489
\]

4. Now, find the angle \( C \) by taking the arccosine:
\[
C = \cos^{-1}(0.9489)
\]

Calculating this using a calculator:

\[
C \approx 18.2^\circ
\]

Rounding to the nearest degree, we get:

\[
C \approx 19^\circ
\]

Thus, the measure of \( \angle C \) to the nearest degree is \( \boxed{19} \) degrees.