Find the diagonal $ d_1 $ of a rhombus if side $a = 20$ and area $A = \frac{ 105 }{ 2 }$.

$d_1 = 39.9134$

STEP 1: find height $ h $

To find height $ h $ use formula:

$$ A = a \cdot h $$

After substituting $ A = \frac{ 105 }{ 2 } $ and $ a = 20 $ we have:

$$ \frac{ 105 }{ 2 } = 20 \cdot h $$$$ h = \dfrac{ \frac{ 105 }{ 2 } }{ 20 } $$$$ h = \frac{ 21 }{ 8 } $$

STEP 2: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ h = \frac{ 21 }{ 8 } $ and $ a = 20 $ we have:

$$ \sin \left( \alpha \right) = \dfrac{ \frac{ 21 }{ 8 } }{ 20 } $$ $$ \sin \left( \alpha \right) = \frac{ 21 }{ 160 } $$ $$ \alpha = \arcsin\left( \frac{ 21 }{ 160 } \right) $$ $$ \alpha = 7.5418^o $$

STEP 3: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ \alpha = 3.7709 $ and $ h = \frac{ 21 }{ 8 } $ we have:

$$ \sin \left( \frac{ 7.5418^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( 3.7709 ) = \dfrac{ \frac{ 21 }{ 8 } }{ d_1 } $$ $$ 0.0658 = \dfrac{ \frac{ 21 }{ 8 } }{ d_1 } $$ $$ d_1 = \dfrac{ \frac{ 21 }{ 8 } }{ 0.0658 } $$ $$ d_1 = 39.9134 $$

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