To find the measure of angle \(A\) in triangle \(ABC\) using the Law of Cosines, we can use the following formula:
\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \]
Given:
- \(a = 13\) m
- \(b = 6\) m
- \(c = 9\) m
First, we will calculate \(b^2 + c^2 - a^2\):
\[ b^2 = 6^2 = 36 \] \[ c^2 = 9^2 = 81 \] \[ a^2 = 13^2 = 169 \]
Now substitute into the equation:
\[ b^2 + c^2 - a^2 = 36 + 81 - 169 = 117 - 169 = -52 \]
Now calculate \(2bc\):
\[ 2bc = 2 \times 6 \times 9 = 108 \]
Now plug these values into the Law of Cosines formula:
\[ \cos(A) = \frac{-52}{108} \]
Calculating this gives:
\[ \cos(A) = -\frac{52}{108} \approx -0.4815 \]
Now, to find \(A\), take the arccosine:
\[ A \approx \cos^{-1}(-0.4815) \]
Using a calculator:
\[ A \approx 118.93^\circ \]
Rounding to the nearest degree, we find:
\[ A \approx 119^\circ \]
Thus, the measure of \(\angle A\) is \(119\) degrees. The correct response is:
119 degrees