Find the side $ a $ of a rhombus if diagonal $d_1 = 32$ and height $h = 24$.

$a = 24.1897$

STEP 1: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ h = 24 $ and $ d_1 = 32 $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 24 }{ 32 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 3 }{ 4 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 3 }{ 4 } \right) $$ $$ \frac{ \alpha }{ 2 } = 48.5904^o $$$$ \alpha = 48.5904^o \cdot 2 $$$$ \alpha = 97.1808^o $$

STEP 2: find side $ a $

To find side $ a $ use formula:

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

After substituting $ \alpha = 97.1808^o $ and $ h = 24 $ we have:

$$ \sin( 97.1808^o ) = \dfrac{ 24 }{ a } $$ $$ 0.9922 = \dfrac{ 24 }{ a } $$ $$ a = \dfrac{ 24 }{ 0.9922 } $$ $$ a = 24.1897 $$

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