STEP 1: find side $ c $
To find side $ c $ use formula:
$$ \sin \left( \alpha \right) = \dfrac{ h }{ c } $$After substituting $ \alpha = 60^o $ and $ h = \frac{ 18 }{ 5 } $ we have:
$$ \sin( 60^o ) = \dfrac{ \frac{ 18 }{ 5 } }{ c } $$ $$ \frac{\sqrt{ 3 }}{ 2 } = \dfrac{ \frac{ 18 }{ 5 } }{ c } $$ $$ c = \dfrac{ \frac{ 18 }{ 5 } }{ \frac{\sqrt{ 3 }}{ 2 } } $$ $$ c = \frac{ 12 \sqrt{ 3}}{ 5 } $$STEP 2: find side $ x $
To find side $ x $ use Pythagorean Theorem:
$$ h^2 + x^2 = c^2 $$After substituting $ h = \frac{ 18 }{ 5 } $ and $ c = \frac{ 12 \sqrt{ 3}}{ 5 } $ we have:
$$ \left(\frac{ 18 }{ 5 }\right)^2 + x^2 = \left( \frac{ 12 \sqrt{ 3}}{ 5 } \right)^2 $$ $$ x^2 = \left( \frac{ 12 \sqrt{ 3}}{ 5 } \right)^2 - \left(\frac{ 18 }{ 5 }\right)^2 $$ $$ x^2 = \frac{ 432 }{ 25 } - \frac{ 324 }{ 25 } $$ $$ x^2 = \frac{ 108 }{ 25 } $$ $$ x = \sqrt{ \frac{ 108 }{ 25 } } $$$$ x = \frac{ 6 \sqrt{ 3}}{ 5 } $$STEP 3: find long base $ a $
To find long base $ a $ use formula:
$$ x = \frac{ a - b } { 2 } $$After substituting $ x = \frac{ 6 \sqrt{ 3}}{ 5 } $ and $ b = \frac{ 119 }{ 10 } $ we have:
$$ \frac{ 6 \sqrt{ 3}}{ 5 } = \frac{ a - \frac{ 119 }{ 10 } } { 2 } $$ $$ \frac{ 6 \sqrt{ 3}}{ 5 } \cdot 2 = a - \frac{ 119 }{ 10 } $$ $$ a - \frac{ 119 }{ 10 } = \frac{ 12 \sqrt{ 3}}{ 5 } $$ $$ a = \frac{ 12 \sqrt{ 3}}{ 5 } + \frac{ 119 }{ 10 } $$ $$ a = 16.0569 $$