To find the length of side \( a \) in the oblique triangle, we can use the Law of Cosines. The Law states the following for a triangle with sides \( a \), \( b \), and \( c \) opposite to angles \( A \), \( B \), and \( C \), respectively:
\[ a^2 = b^2 + c^2 - 2bc \cdot \cos(A) \]
In this case, we are given:
- \( b = 12 , \text{mm} \)
- \( c = 14 , \text{mm} \)
- \( A = 35^\circ \)
Now, we can substitute these values into the Law of Cosines formula:
\[ a^2 = 12^2 + 14^2 - 2 \cdot 12 \cdot 14 \cdot \cos(35^\circ) \]
Calculating each term step-by-step:
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Calculate \( 12^2 \): \[ 12^2 = 144 \]
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Calculate \( 14^2 \): \[ 14^2 = 196 \]
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Now calculate \( 2 \cdot 12 \cdot 14 \): \[ 2 \cdot 12 \cdot 14 = 336 \]
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Find \( \cos(35^\circ) \). Using a calculator, we get: \[ \cos(35^\circ) \approx 0.8192 \]
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Substitute all values into the equation: \[ a^2 = 144 + 196 - 336 \cdot 0.8192 \]
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Calculate \( 336 \cdot 0.8192 \): \[ 336 \cdot 0.8192 \approx 275.3867 \]
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Putting it all together: \[ a^2 = 144 + 196 - 275.3867 \approx 340 - 275.3867 \approx 64.6133 \]
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Finally, take the square root to find \( a \): \[ a \approx \sqrt{64.6133} \approx 8.04 , \text{mm} \]
Rounding to the nearest millimeter, we find: \[ a \approx 8 , \text{mm} \]
Thus, the measure of side \( a \) is approximately 8 mm.