Find the diagonal $ d_2 $ of a rhombus if diagonal $d_1 = 12 \sqrt{ 3 }$ and angle $\beta = 30^o$.

$d_2 = 6 \sqrt{ 3 }$

STEP 1: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ \beta = 30^o $ we have:

$$ \alpha + 30^o = 90^o $$ $$ \alpha = 90^o - 30^o $$ $$ \alpha = 60^o $$

STEP 2: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ d_2 }{ d_1 } $$

After substituting $ \alpha = 30^o $ and $ d_1 = 12 \sqrt{ 3 } $ we have:

$$ \sin \left( \frac{ 60^o }{ 2 } \right) = \dfrac{ d_2 }{ 12 \sqrt{ 3 } } $$ $$ \sin( 30^o ) = \dfrac{ d_2 }{ 12 \sqrt{ 3 } } $$ $$ \frac{ 1 }{ 2 } = \dfrac{ d_2 }{ 12 \sqrt{ 3 } } $$$$ d_2 = \frac{ 1 }{ 2 } \cdot 12 \sqrt{ 3 } $$$$ d_2 = 6 \sqrt{ 3 } $$

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