Find the side $ a $ of a rhombus if diagonal $d_2 = 7$ and angle $\alpha = 53^o$.

$a = 8.5895$

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 = \frac{ 53 }{ 2 }^o $ and $ d_2 = 7 $ we have:

$$ \sin \left( \frac{ 53^o }{ 2 } \right) = \dfrac{ d_2 }{ } $$ $$ \sin( \frac{ 53 }{ 2 }^o ) = \dfrac{ 7 }{ d_1 } $$ $$ 0.4462 = \dfrac{ 7 }{ d_1 } $$ $$ d_1 = \dfrac{ 7 }{ 0.4462 } $$ $$ d_1 = 15.6881 $$

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 = 15.6881 $ and $ d_2 = 7 $ we have:

$$ 15.6881^2 + 7^2 = 4 \cdot a^2 $$ $$ 246.1168 + 49 = 4 \cdot a^2 $$ $$ 4 \cdot a^2 = 295.1168 $$ $$ a^2 = \frac{ 295.1168 }{ 4 } $$ $$ a^2 = 73.7792 $$ $$ a = \sqrt{ 73.7792 } $$$$ a = 8.5895 $$

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