Question

For ∆ ABC, find the measure of ∠ C to the nearest degree when side a = 46 m, side b = 30 m, and side c = 20 m

a. 28 degrees
b. 19 degrees
c. 133 degrees
d. 24 degrees

Answer 1

To find the measure of angle \( C \) in triangle \( ABC \), where side lengths are given as \( a = 46 \, m \), \( b = 30 \, m \), and \( c = 20 \, m \), we can use the Law of Cosines:

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

Substituting the known values:

\[
20^2 = 46^2 + 30^2 - 2 \cdot 46 \cdot 30 \cdot \cos(C)
\]

Calculating the squares:

\[
400 = 2116 + 900 - 2760 \cdot \cos(C)
\]

Simplifying:

\[
400 = 3016 - 2760 \cdot \cos(C)
\]

Rearranging the equation to isolate \( \cos(C) \):

\[
2760 \cdot \cos(C) = 3016 - 400
\]
\[
2760 \cdot \cos(C) = 2616
\]
\[
\cos(C) = \frac{2616}{2760}
\]
\[
\cos(C) \approx 0.948
\]

Now, finding \( C \):

\[
C = \cos^{-1}(0.948)
\]

Calculating the angle:

\[
C \approx 18.9^\circ
\]

Rounding to the nearest degree, we find:

\[
C \approx 19^\circ
\]

Therefore, the measure of angle \( C \) is:

**b. 19 degrees**.