To find the measure of angle \( A \) in triangle \( ABC \) using the Law of Cosines, we use the formula:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Given:
- Side \( a = 13 \) m
- Side \( b = 6 \) m
- Side \( c = 9 \) m
We will substitute these values into the formula.
Calculating \( b^2 \), \( c^2 \), and \( a^2 \):
\[ b^2 = 6^2 = 36 \]
\[ c^2 = 9^2 = 81 \]
\[ a^2 = 13^2 = 169 \]
Now we substitute these values into the cosine formula:
\[ \cos A = \frac{36 + 81 - 169}{2 \cdot 6 \cdot 9} \]
Calculating the numerator:
\[ 36 + 81 - 169 = 117 - 169 = -52 \]
Calculating the denominator:
\[ 2 \cdot 6 \cdot 9 = 108 \]
Now substituting back into the cosine equation:
\[ \cos A = \frac{-52}{108} \]
Calculating this gives:
\[ \cos A = -\frac{52}{108} \approx -0.4815 \]
Now, we need to find \( A \) by taking the inverse cosine:
\[ A = \cos^{-1}(-0.4815) \]
Using a calculator:
\[ A \approx 118.93^\circ \]
Rounding to the nearest whole number gives \( A \approx 119^\circ \).
Therefore, the measure of \( \angle A \) is approximately:
119 degrees.