Find the diagonal $ d_1 $ of a rhombus if side $a = 10$ and area $A = 80$.

$d_1 = 17.8885$

STEP 1: find height $ h $

To find height $ h $ use formula:

$$ A = a \cdot h $$

After substituting $ A = 80 $ and $ a = 10 $ we have:

$$ 80 = 10 \cdot h $$$$ h = \dfrac{ 80 }{ 10 } $$$$ h = 8 $$

STEP 2: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ h = 8 $ and $ a = 10 $ we have:

$$ \sin \left( \alpha \right) = \dfrac{ 8 }{ 10 } $$ $$ \sin \left( \alpha \right) = \frac{ 4 }{ 5 } $$ $$ \alpha = \arcsin\left( \frac{ 4 }{ 5 } \right) $$ $$ \alpha = 53.1301^o $$

STEP 3: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ \alpha = 26.5651 $ and $ h = 8 $ we have:

$$ \sin \left( \frac{ 53.1301^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( 26.5651 ) = \dfrac{ 8 }{ d_1 } $$ $$ 0.4472 = \dfrac{ 8 }{ d_1 } $$ $$ d_1 = \dfrac{ 8 }{ 0.4472 } $$ $$ d_1 = 17.8885 $$

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