To find the length of side \( a \) in an oblique triangle (non-right triangle), we can use the Law of Cosines, which states:
\[ a^2 = b^2 + c^2 - 2bc \cdot \cos(A) \]
Given:
- \( b = 12 \) mm
- \( c = 14 \) mm
- \( A = 35^\circ \)
First, we need to calculate \( \cos(35^\circ) \). Using a calculator, we find:
\[ \cos(35^\circ) \approx 0.8192 \]
Now we can substitute the values into the Law of Cosines formula:
\[ a^2 = 12^2 + 14^2 - 2 \cdot 12 \cdot 14 \cdot \cos(35^\circ) \]
Calculating \( 12^2 \) and \( 14^2 \):
\[ 12^2 = 144 \] \[ 14^2 = 196 \]
Now substituting:
\[ a^2 = 144 + 196 - 2 \cdot 12 \cdot 14 \cdot 0.8192 \]
Now calculating the product:
\[ 2 \cdot 12 \cdot 14 = 336 \] \[ 336 \cdot 0.8192 \approx 275.2 \]
Now substituting back into the equation for \( a^2 \):
\[ a^2 = 144 + 196 - 275.2 \] \[ a^2 = 340 - 275.2 \] \[ a^2 \approx 64.8 \]
Finally, taking the square root to find \( a \):
\[ a \approx \sqrt{64.8} \approx 8.06 \]
Rounding to the nearest millimeter:
\[ a \approx 8 \text{ mm} \]
Thus, the measure of side \( a \) is approximately 8 mm.