Find the diagonal $ d_2 $ of a rhombus if area $A = 20$ and height $h = 3$.

$d_2 = 5.7208$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ A = a \cdot h $$

After substituting $ A = 20 $ and $ h = 3 $ we have:

$$ 20 = 3 \cdot a $$$$ a = \dfrac{ 20}{ 3 } $$$$ a = \frac{ 20 }{ 3 } $$

STEP 2: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

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

$$ \sin \left( \alpha \right) = \dfrac{ 3 }{ \frac{ 20 }{ 3 } } $$ $$ \sin \left( \alpha \right) = \frac{ 9 }{ 20 } $$ $$ \alpha = \arcsin\left( \frac{ 9 }{ 20 } \right) $$ $$ \alpha = 26.7437^o $$

STEP 3: find angle $ \beta $

To find angle $ \beta $ use formula:

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

After substituting $ \alpha = 26.7437^o $ we have:

$$ 26.7437^o + \beta = 90^o $$ $$ \beta = 90^o - 26.7437^o $$ $$ \beta = 63.2563^o $$

STEP 4: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ \beta = 31.6282 $ and $ h = 3 $ we have:

$$ \sin \left( \frac{ 63.2563^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( 31.6282 ) = \dfrac{ 3 }{ d_2 } $$ $$ 0.5244 = \dfrac{ 3 }{ d_2 } $$ $$ d_2 = \dfrac{ 3 }{ 0.5244 } $$ $$ d_2 = 5.7208 $$

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