Find the diagonal $ d_2 $ of a rhombus if side $a = 8$ and height $h = \frac{ 693 }{ 100 }$.

$d_2 = 26.7979$

STEP 1: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ h = \frac{ 693 }{ 100 } $ and $ a = 8 $ we have:

$$ \sin \left( \alpha \right) = \dfrac{ \frac{ 693 }{ 100 } }{ 8 } $$ $$ \sin \left( \alpha \right) = \frac{ 693 }{ 800 } $$ $$ \alpha = \arcsin\left( \frac{ 693 }{ 800 } \right) $$ $$ \alpha = 60.0257^o $$

STEP 2: find angle $ \beta $

To find angle $ \beta $ use formula:

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

After substituting $ \alpha = 60.0257^o $ we have:

$$ 60.0257^o + \beta = 90^o $$ $$ \beta = 90^o - 60.0257^o $$ $$ \beta = 29.9743^o $$

STEP 3: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ \beta = 14.9871 $ and $ h = \frac{ 693 }{ 100 } $ we have:

$$ \sin \left( \frac{ 29.9743^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( 14.9871 ) = \dfrac{ \frac{ 693 }{ 100 } }{ d_2 } $$ $$ 0.2586 = \dfrac{ \frac{ 693 }{ 100 } }{ d_2 } $$ $$ d_2 = \dfrac{ \frac{ 693 }{ 100 } }{ 0.2586 } $$ $$ d_2 = 26.7979 $$

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