STEP 1: find angle $ \gamma $
To find angle $ \gamma $ use Thw Law of Sines:
$$ \dfrac{ \sin( \alpha )} { a } = \dfrac{ \sin( \gamma )} { c } $$After substituting $a = \frac{ 709 }{ 10 }$, $\alpha = \frac{ 69 }{ 2 }^o$ and $c = \frac{ 174 }{ 5 }$ we have:
$$ \dfrac{ \sin( \frac{ 69 }{ 2 }^o )} { \frac{ 709 }{ 10 } } = \dfrac{ \sin( \gamma )} { \frac{ 174 }{ 5 } } $$ $$ \dfrac{ 0.5664 } { \frac{ 709 }{ 10 } } = \dfrac{ \sin( \gamma ) } { \frac{ 174 }{ 5 } } $$ $$ \sin( \gamma ) \cdot \frac{ 709 }{ 10 } = 0.5664 \cdot \frac{ 174 }{ 5 } $$ $$ \sin( \gamma ) \cdot \frac{ 709 }{ 10 } = 19.7109 $$ $$ \sin( \gamma ) = \dfrac{ 19.7109 }{ \frac{ 709 }{ 10 } } $$ $$ \sin( \gamma ) = 0.278 $$ $$ \gamma = \arcsin{ 0.278 } $$ $$ \gamma \approx 16.1415^o $$STEP 2: find angle $ \beta $
To find angle $ \beta $ use formula:
$$ \alpha + \beta + \gamma = 180^o $$After substituting $ \alpha = \frac{ 69 }{ 2 }^o $ and $ \gamma = 16.1415^o $ we have:
$$ \frac{ 69 }{ 2 }^o + \beta + 16.1415^o = 180^o $$ $$ \beta + 50.6415^o = 180^o $$ $$ \beta = 180^o - 50.6415^o $$ $$ \beta = 129.3585^o $$STEP 3: find side $ b $
To find side $ b $ use Law of Cosines:
$$ b^2 = a^2 + c^2 - 2 \cdot a \cdot c \cdot \cos( \beta ) $$After substituting $a = \frac{ 709 }{ 10 }$, $c = \frac{ 174 }{ 5 }$ and $\beta = 129.3585^o$ we have:
$$ b^2 = \left(\frac{ 709 }{ 10 }\right)^2 + \left(\frac{ 174 }{ 5 }\right)^2 - 2 \cdot \frac{ 709 }{ 10 } \cdot \frac{ 174 }{ 5 } \cdot \cos( 129.3585^o ) $$ $$ b^2 = \frac{ 502681 }{ 100 } + \frac{ 30276 }{ 25 } - 2 \cdot \frac{ 709 }{ 10 } \cdot \frac{ 174 }{ 5 } \cdot \cos( 129.3585^o ) $$ $$ b^2 = \frac{ 124757 }{ 20 } - 2 \cdot \frac{ 61683 }{ 25 } \cdot \cos( 129.3585^o ) $$ $$ b^2 = \frac{ 124757 }{ 20 } - \frac{ 123366 }{ 25 } \cdot \left(-0.6342\right) $$ $$ b^2 = \frac{ 124757 }{ 20 } - \left(-3129.4042\right) $$ $$ b^2 = 9367.2542 $$ $$ b = \sqrt{ 9367.2542 } $$$$ b \approx 96.7846 $$