Find the diagonal $ d_1 $ of a rhombus if side $a = 16$ and area $A = 214$.

$d_1 = 28.1602$

STEP 1: find height $ h $

To find height $ h $ use formula:

$$ A = a \cdot h $$

After substituting $ A = 214 $ and $ a = 16 $ we have:

$$ 214 = 16 \cdot h $$$$ h = \dfrac{ 214 }{ 16 } $$$$ h = \frac{ 107 }{ 8 } $$

STEP 2: find angle $ \alpha $

To find angle $ \alpha $ use formula:

$$ \sin \left( \alpha \right) = \dfrac{ h }{ a } $$

After substituting $ h = \frac{ 107 }{ 8 } $ and $ a = 16 $ we have:

$$ \sin \left( \alpha \right) = \dfrac{ \frac{ 107 }{ 8 } }{ 16 } $$ $$ \sin \left( \alpha \right) = \frac{ 107 }{ 128 } $$ $$ \alpha = \arcsin\left( \frac{ 107 }{ 128 } \right) $$ $$ \alpha = 56.7136^o $$

STEP 3: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ h }{ d_1 } $$

After substituting $ \alpha = 28.3568 $ and $ h = \frac{ 107 }{ 8 } $ we have:

$$ \sin \left( \frac{ 56.7136^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( 28.3568 ) = \dfrac{ \frac{ 107 }{ 8 } }{ d_1 } $$ $$ 0.475 = \dfrac{ \frac{ 107 }{ 8 } }{ d_1 } $$ $$ d_1 = \dfrac{ \frac{ 107 }{ 8 } }{ 0.475 } $$ $$ d_1 = 28.1602 $$

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