Find the diagonal $ d_2 $ of a rhombus if side $a = 120$ and height $h = 90$.

$d_2 = 254.5584$

STEP 1: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ h = 90 $ and $ a = 120 $ we have:

$$ \sin \left( \alpha \right) = \dfrac{ 90 }{ 120 } $$ $$ \sin \left( \alpha \right) = \frac{ 3 }{ 4 } $$ $$ \alpha = \arcsin\left( \frac{ 3 }{ 4 } \right) $$ $$ \alpha = 48.5904^o $$

STEP 2: find angle $ \beta $

To find angle $ \beta $ use formula:

$$ \alpha + \beta = 90^o $$

After substituting $ \alpha = 48.5904^o $ we have:

$$ 48.5904^o + \beta = 90^o $$ $$ \beta = 90^o - 48.5904^o $$ $$ \beta = 41.4096^o $$

STEP 3: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

$$ \sin \left( \frac{ \beta }{ 2 } \right) = \dfrac{ h }{ d_2 } $$

After substituting $ \beta = 20.7048 $ and $ h = 90 $ we have:

$$ \sin \left( \frac{ 41.4096^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( 20.7048 ) = \dfrac{ 90 }{ d_2 } $$ $$ 0.3536 = \dfrac{ 90 }{ d_2 } $$ $$ d_2 = \dfrac{ 90 }{ 0.3536 } $$ $$ d_2 = 254.5584 $$

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