STEP 1: find side $ x $
To find side $ x $ use formula:
$$ x = \frac{ a - b } { 2 } $$After substituting $ a = \frac{ 53 }{ 8 } $ and $ b = \frac{ 5 }{ 2 } $ we have:
$$ x = \frac{ \frac{ 53 }{ 8 } - \frac{ 5 }{ 2 } } { 2 } $$ $$ x = \frac{ \frac{ 33 }{ 8 } } { 2 } $$ $$ x = \frac{ 33 }{ 16 } $$STEP 2: find height $ h $
To find height $ h $ use Pythagorean Theorem:
$$ h^2 + x^2 = c^2 $$After substituting $ x = \frac{ 33 }{ 16 } $ and $ c = 24 $ we have:
$$ h ^ {\,2} + \left(\frac{ 33 }{ 16 }\right)^2 = 24^2 $$ $$ h ^ {\,2} = 24^2 - \left(\frac{ 33 }{ 16 }\right)^2 $$ $$ h ^ {\,2} = 576 - \frac{ 1089 }{ 256 } $$ $$ h ^ {\,2} = \frac{ 146367 }{ 256 } $$ $$ h = \sqrt{ \frac{ 146367 }{ 256 } } $$$$ h = \frac{ 9 \sqrt{ 1807}}{ 16 } $$STEP 3: find angle $ \alpha $
To find angle $ \alpha $ use formula:
$$ \sin \left( \alpha \right) = \dfrac{ h }{ c } $$After substituting $ h = \frac{ 9 \sqrt{ 1807}}{ 16 } $ and $ c = 24 $ we have:
$$ \sin \left( \alpha \right) = \dfrac{ \frac{ 9 \sqrt{ 1807}}{ 16 } }{ 24 } $$ $$ \sin \left( \alpha \right) = \frac{ 3 \sqrt{ 1807}}{ 128 } $$ $$ \alpha = \arcsin\left( \frac{ 3 \sqrt{ 1807}}{ 128 } \right) $$ $$ \alpha = 85.0701^o $$