Find the height $ h $ of a rhombus if diagonal $d_1 = 35$ and diagonal $d_2 = 20$.

$h = 20$

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 = 20 $ and $ d_1 = 35 $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 20 }{ 35 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 4 }{ 7 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 4 }{ 7 } \right) $$ $$ \frac{ \alpha }{ 2 } = 34.8499^o $$$$ \alpha = 34.8499^o \cdot 2 $$$$ \alpha = 69.6998^o $$

STEP 2: find height $ h $

To find height $ h $ use formula:

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

After substituting $ \alpha = 34.8499 $ and $ d_1 = 35 $ we have:

$$ \sin \left( \frac{ 69.6998^o }{ 2 } \right) = \dfrac{ h }{ 35 } $$ $$ \sin( 34.8499 ) = \dfrac{ h }{ 35 } $$ $$ 0.5714 = \dfrac{ h }{ 35 } $$$$ h = 0.5714 \cdot 35 $$$$ h = 20 $$

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