Find the diagonal $ d_1 $ of a rhombus if side $a = 120$ and area $A = 10800$.

$d_1 = 218.7451$

STEP 1: find height $ h $

To find height $ h $ use formula:

$$ A = a \cdot h $$

After substituting $ A = 10800 $ and $ a = 120 $ we have:

$$ 10800 = 120 \cdot h $$$$ h = \dfrac{ 10800 }{ 120 } $$$$ h = 90 $$

STEP 2: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ h = 90 $ and $ a = 120 $ we have:

$$ \sin \left( \alpha \right) = \dfrac{ 90 }{ 120 } $$ $$ \sin \left( \alpha \right) = \frac{ 3 }{ 4 } $$ $$ \alpha = \arcsin\left( \frac{ 3 }{ 4 } \right) $$ $$ \alpha = 48.5904^o $$

STEP 3: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ \alpha = 24.2952 $ and $ h = 90 $ we have:

$$ \sin \left( \frac{ 48.5904^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( 24.2952 ) = \dfrac{ 90 }{ d_1 } $$ $$ 0.4114 = \dfrac{ 90 }{ d_1 } $$ $$ d_1 = \dfrac{ 90 }{ 0.4114 } $$ $$ d_1 = 218.7451 $$

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