To find side $ c $ use Law of Cosines:
$$ c^2 = a^2 + b^2 - 2 \cdot a \cdot b \cdot \cos( \gamma ) $$After substituting $a = 10$, $b = \frac{ 661 }{ 50 }$ and $\gamma = \frac{ 419 }{ 20 }^o$ we have:
$$ c^2 = 10^2 + \left(\frac{ 661 }{ 50 }\right)^2 - 2 \cdot 10 \cdot \frac{ 661 }{ 50 } \cdot \cos( \frac{ 419 }{ 20 }^o ) $$ $$ c^2 = 100 + \frac{ 436921 }{ 2500 } - 2 \cdot 10 \cdot \frac{ 661 }{ 50 } \cdot \cos( \frac{ 419 }{ 20 }^o ) $$ $$ c^2 = \frac{ 686921 }{ 2500 } - 2 \cdot \frac{ 661 }{ 5 } \cdot \cos( \frac{ 419 }{ 20 }^o ) $$ $$ c^2 = \frac{ 686921 }{ 2500 } - \frac{ 1322 }{ 5 } \cdot 0.9339 $$ $$ c^2 = \frac{ 686921 }{ 2500 } - 246.9213 $$ $$ c^2 = 27.8471 $$ $$ c = \sqrt{ 27.8471 } $$$$ c \approx 5.277 $$