Find the diagonal $ d_1 $ of a rhombus if area $A = 442$ and perimeter $P = 85$.

$d_1 = 32.9848$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ P = 4 \cdot a $$

After substituting $ P = 85 $ we have:

$$ 85 = 4 \cdot a $$ $$ a = \dfrac{ 85 }{ 4 } $$ $$ a = \frac{ 85 }{ 4 } $$

STEP 2: find height $ h $

To find height $ h $ use formula:

$$ A = a \cdot h $$

After substituting $ A = 442 $ and $ a = \frac{ 85 }{ 4 } $ we have:

$$ 442 = \frac{ 85 }{ 4 } \cdot h $$$$ h = \dfrac{ 442 }{ \frac{ 85 }{ 4 } } $$$$ h = \frac{ 104 }{ 5 } $$

STEP 3: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ h = \frac{ 104 }{ 5 } $ and $ a = \frac{ 85 }{ 4 } $ we have:

$$ \sin \left( \alpha \right) = \dfrac{ \frac{ 104 }{ 5 } }{ \frac{ 85 }{ 4 } } $$ $$ \sin \left( \alpha \right) = \frac{ 416 }{ 425 } $$ $$ \alpha = \arcsin\left( \frac{ 416 }{ 425 } \right) $$ $$ \alpha = 78.1877^o $$

STEP 4: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ \alpha = 39.0939 $ and $ h = \frac{ 104 }{ 5 } $ we have:

$$ \sin \left( \frac{ 78.1877^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( 39.0939 ) = \dfrac{ \frac{ 104 }{ 5 } }{ d_1 } $$ $$ 0.6306 = \dfrac{ \frac{ 104 }{ 5 } }{ d_1 } $$ $$ d_1 = \dfrac{ \frac{ 104 }{ 5 } }{ 0.6306 } $$ $$ d_1 = 32.9848 $$

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