Find the height $ h $ of a rhombus if diagonal $d_1 = \frac{ 56 }{ 5 }$ and diagonal $d_2 = \frac{ 42 }{ 5 }$.

$h = 8.4$

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{ 42 }{ 5 } $ and $ d_1 = \frac{ 56 }{ 5 } $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ \frac{ 42 }{ 5 } }{ \frac{ 56 }{ 5 } } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 3 }{ 4 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 3 }{ 4 } \right) $$ $$ \frac{ \alpha }{ 2 } = 48.5904^o $$$$ \alpha = 48.5904^o \cdot 2 $$$$ \alpha = 97.1808^o $$

STEP 2: find height $ h $

To find height $ h $ use formula:

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

After substituting $ \alpha = 48.5904 $ and $ d_1 = \frac{ 56 }{ 5 } $ we have:

$$ \sin \left( \frac{ 97.1808^o }{ 2 } \right) = \dfrac{ h }{ \frac{ 56 }{ 5 } } $$ $$ \sin( 48.5904 ) = \dfrac{ h }{ \frac{ 56 }{ 5 } } $$ $$ 0.75 = \dfrac{ h }{ \frac{ 56 }{ 5 } } $$$$ h = 0.75 \cdot \frac{ 56 }{ 5 } $$$$ h = 8.4 $$

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