Find the side $ a $ of a rhombus if diagonal $d_2 = 4$ and angle $\alpha = 110^o$.

$a = 3.1561$

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 = 55^o $ and $ d_2 = 4 $ we have:

$$ \sin \left( \frac{ 110^o }{ 2 } \right) = \dfrac{ d_2 }{ } $$ $$ \sin( 55^o ) = \dfrac{ 4 }{ d_1 } $$ $$ 0.8192 = \dfrac{ 4 }{ d_1 } $$ $$ d_1 = \dfrac{ 4 }{ 0.8192 } $$ $$ d_1 = 4.8831 $$

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 = 4.8831 $ and $ d_2 = 4 $ we have:

$$ 4.8831^2 + 4^2 = 4 \cdot a^2 $$ $$ 23.8446 + 16 = 4 \cdot a^2 $$ $$ 4 \cdot a^2 = 39.8446 $$ $$ a^2 = \frac{ 39.8446 }{ 4 } $$ $$ a^2 = 9.9612 $$ $$ a = \sqrt{ 9.9612 } $$$$ a = 3.1561 $$

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