Find the diagonal $ d_2 $ of a rhombus if area $A = 120$ and perimeter $P = 72$.

$d_2 = 11.8818$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ P = 4 \cdot a $$

After substituting $ P = 72 $ we have:

$$ 72 = 4 \cdot a $$ $$ a = \dfrac{ 72 }{ 4 } $$ $$ a = 18 $$

STEP 2: find height $ h $

To find height $ h $ use formula:

$$ A = a \cdot h $$

After substituting $ A = 120 $ and $ a = 18 $ we have:

$$ 120 = 18 \cdot h $$$$ h = \dfrac{ 120 }{ 18 } $$$$ h = \frac{ 20 }{ 3 } $$

STEP 3: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ h = \frac{ 20 }{ 3 } $ and $ a = 18 $ we have:

$$ \sin \left( \alpha \right) = \dfrac{ \frac{ 20 }{ 3 } }{ 18 } $$ $$ \sin \left( \alpha \right) = \frac{ 10 }{ 27 } $$ $$ \alpha = \arcsin\left( \frac{ 10 }{ 27 } \right) $$ $$ \alpha = 21.7385^o $$

STEP 4: find angle $ \beta $

To find angle $ \beta $ use formula:

$$ \alpha + \beta = 90^o $$

After substituting $ \alpha = 21.7385^o $ we have:

$$ 21.7385^o + \beta = 90^o $$ $$ \beta = 90^o - 21.7385^o $$ $$ \beta = 68.2615^o $$

STEP 5: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

$$ \sin \left( \frac{ \beta }{ 2 } \right) = \dfrac{ h }{ d_2 } $$

After substituting $ \beta = 34.1308 $ and $ h = \frac{ 20 }{ 3 } $ we have:

$$ \sin \left( \frac{ 68.2615^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( 34.1308 ) = \dfrac{ \frac{ 20 }{ 3 } }{ d_2 } $$ $$ 0.5611 = \dfrac{ \frac{ 20 }{ 3 } }{ d_2 } $$ $$ d_2 = \dfrac{ \frac{ 20 }{ 3 } }{ 0.5611 } $$ $$ d_2 = 11.8818 $$

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