To find side $ a $ use Law of Cosines:

$$ a^2 = b^2 + c^2 - 2 \cdot b \cdot c \cdot \cos( \alpha ) $$

After substituting $b = 10\, \text{cm}$, $c = 12\, \text{cm}$ and $\alpha = 50^o$ we have:

$$ a^2 = 10^2 + 12^2 - 2 \cdot 10 \cdot 12 \cdot \cos( 50^o ) $$ $$ a^2 = 100\, \text{cm}^2 + 144\, \text{cm}^2 - 2 \cdot 10 \cdot 12 \cdot \cos( 50^o ) $$ $$ a^2 = 244\, \text{cm}^2 - 2 \cdot 120\, \text{cm}^2 \cdot \cos( 50^o ) $$ $$ a^2 = 244\, \text{cm}^2 - 240\, \text{cm}^2 \cdot 0.6428 $$ $$ a^2 = 244\, \text{cm}^2 - 154.269\, \text{cm}^2 $$ $$ a^2 = 89.731\, \text{cm}^2 $$ $$ a = \sqrt{ 89.731\, \text{cm}^2 } $$$$ a \approx 9.4726\, \text{cm} $$

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