Find the diagonal $ d_1 $ of a rhombus if side $a = \frac{ 1459 }{ 100 }$ and height $h = \frac{ 1371 }{ 100 }$.

$d_1 = 23.9031$

STEP 1: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ h = \frac{ 1371 }{ 100 } $ and $ a = \frac{ 1459 }{ 100 } $ we have:

$$ \sin \left( \alpha \right) = \dfrac{ \frac{ 1371 }{ 100 } }{ \frac{ 1459 }{ 100 } } $$ $$ \sin \left( \alpha \right) = \frac{ 1371 }{ 1459 } $$ $$ \alpha = \arcsin\left( \frac{ 1371 }{ 1459 } \right) $$ $$ \alpha = 69.9987^o $$

STEP 2: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ \alpha = 34.9993 $ and $ h = \frac{ 1371 }{ 100 } $ we have:

$$ \sin \left( \frac{ 69.9987^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( 34.9993 ) = \dfrac{ \frac{ 1371 }{ 100 } }{ d_1 } $$ $$ 0.5736 = \dfrac{ \frac{ 1371 }{ 100 } }{ d_1 } $$ $$ d_1 = \dfrac{ \frac{ 1371 }{ 100 } }{ 0.5736 } $$ $$ d_1 = 23.9031 $$

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