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 = 12 $ we have:
$$ \sin( 60^o ) = \dfrac{ 12 }{ c } $$ $$ \frac{\sqrt{ 3 }}{ 2 } = \dfrac{ 12 }{ c } $$ $$ c = \dfrac{ 12 }{ \frac{\sqrt{ 3 }}{ 2 } } $$ $$ c = 8 \sqrt{ 3 } $$STEP 2: find side $ x $
To find side $ x $ use Pythagorean Theorem:
$$ h^2 + x^2 = c^2 $$After substituting $ h = 12 $ and $ c = 8 \sqrt{ 3 } $ we have:
$$ 12^2 + x^2 = \left(8 \sqrt{ 3 }\right)^2 $$ $$ x^2 = \left(8 \sqrt{ 3 }\right)^2 - 12^2 $$ $$ x^2 = 192 - 144 $$ $$ x^2 = 48 $$ $$ x = \sqrt{ 48 } $$$$ x = 4 \sqrt{ 3 } $$STEP 3: find long base $ a $
To find long base $ a $ use formula:
$$ x = \frac{ a - b } { 2 } $$After substituting $ x = 4 \sqrt{ 3 } $ and $ b = 16 $ we have:
$$ 4 \sqrt{ 3 } = \frac{ a - 16 } { 2 } $$ $$ 4 \sqrt{ 3 } \cdot 2 = a - 16 $$ $$ a - 16 = 8 \sqrt{ 3 } $$ $$ a = 8 \sqrt{ 3 } + 16 $$ $$ a = 29.8564 $$