Find the side $ a $ of a rhombus if diagonal $d_2 = 12$ and angle $\alpha = 30^o$.

$a = 23.9461$

STEP 1: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ \alpha = 15^o $ and $ d_2 = 12 $ we have:

$$ \sin \left( \frac{ 30^o }{ 2 } \right) = \dfrac{ d_2 }{ } $$ $$ \sin( 15^o ) = \dfrac{ 12 }{ d_1 } $$ $$ 0.2588 = \dfrac{ 12 }{ d_1 } $$ $$ d_1 = \dfrac{ 12 }{ 0.2588 } $$ $$ d_1 = 46.3644 $$

STEP 2: find side $ a $

To find side $ a $ use formula:

$$ d_1^2 + d_2^2 = 4 \cdot a^2 $$

After substituting $ d_1 = 46.3644 $ and $ d_2 = 12 $ we have:

$$ 46.3644^2 + 12^2 = 4 \cdot a^2 $$ $$ 2149.6613 + 144 = 4 \cdot a^2 $$ $$ 4 \cdot a^2 = 2293.6613 $$ $$ a^2 = \frac{ 2293.6613 }{ 4 } $$ $$ a^2 = 573.4153 $$ $$ a = \sqrt{ 573.4153 } $$$$ a = 23.9461 $$

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