You have a right triangle where the sides are given as follows:
- \( AB = 12 \) (opposite side to angle \( A \))
- \( BC = 5 \) (adjacent side to angle \( A \))
- \( AC = 13 \) (hypotenuse)
Since \( AC \) is the hypotenuse and the triangle follows the Pythagorean theorem, let's verify:
\[ AB^2 + BC^2 = AC^2 \] \[ 12^2 + 5^2 = 13^2 \] \[ 144 + 25 = 169 \] \[ 169 = 169 \]
The triangle is indeed right-angled.
To find angle \( A \), we can use the sine, cosine, or tangent functions. Let's use the tangent function, which relates the angle \( A \) to the opposite and adjacent sides:
\[ \tan(A) = \frac{AB}{BC} = \frac{12}{5} \]
To find angle \( A \):
\[ A = \tan^{-1}\left(\frac{12}{5}\right) \]
Now, you can calculate \( A \):
Using a calculator:
\[ A \approx \tan^{-1}(2.4) \approx 67.38^\circ \]
Thus, the measure of angle \( A \) is approximately \( 67.38^\circ \).