To find the measure of ∠A in triangle ABC using the Law of Cosines, we can apply the following formula:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Given:
- \( a = 13 , \text{m} \)
- \( b = 6 , \text{m} \)
- \( c = 9 , \text{m} \)
Plugging in the values:
\[ \cos A = \frac{6^2 + 9^2 - 13^2}{2 \cdot 6 \cdot 9} \]
Calculating the squares:
\[ 6^2 = 36, \quad 9^2 = 81, \quad 13^2 = 169 \]
Now substituting these values into the equation:
\[ \cos A = \frac{36 + 81 - 169}{2 \cdot 6 \cdot 9} \] \[ \cos A = \frac{117 - 169}{108} \] \[ \cos A = \frac{-52}{108} \] \[ \cos A = -\frac{13}{27} \]
Now we need to find the angle A by taking the arccosine:
\[ A = \cos^{-1}\left(-\frac{13}{27}\right) \]
Calculating this value using a calculator:
\[ A \approx 119^\circ \]
Thus, the measure of ∠A is approximately 119 degrees.
The response to choose is: 119 degrees.