Find the height $ h $ of a rhombus if diagonal $d_1 = 81$ and diagonal $d_2 = \frac{ 75 }{ 2 }$.

$h = 37.5$

STEP 1: find angle $ \alpha $

To find angle $ \alpha $ use formula:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ d_2 }{ d_1 } $$

After substituting $ d_2 = \frac{ 75 }{ 2 } $ and $ d_1 = 81 $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ \frac{ 75 }{ 2 } }{ 81 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 25 }{ 54 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 25 }{ 54 } \right) $$ $$ \frac{ \alpha }{ 2 } = 27.5785^o $$$$ \alpha = 27.5785^o \cdot 2 $$$$ \alpha = 55.1569^o $$

STEP 2: find height $ h $

To find height $ h $ use formula:

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

After substituting $ \alpha = 27.5785 $ and $ d_1 = 81 $ we have:

$$ \sin \left( \frac{ 55.1569^o }{ 2 } \right) = \dfrac{ h }{ 81 } $$ $$ \sin( 27.5785 ) = \dfrac{ h }{ 81 } $$ $$ 0.463 = \dfrac{ h }{ 81 } $$$$ h = 0.463 \cdot 81 $$$$ h = 37.5 $$

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