STEP 1: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $\alpha = 53^o$ and $d_2 = 7\, \text{cm}$ we have:

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

STEP 2: find side $ a $

To find side $ a $ use formula:

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

After substituting $d_1 = 15.6881\, \text{cm}$ and $d_2 = 7\, \text{cm}$ we have:

$$ \left( 15.6881\, \text{cm} \right)^{2} + \left( 7\, \text{cm} \right)^{2} = 4 \cdot a^2 $$ $$ 246.1168\, \text{cm}^2 + 49\, \text{cm}^2 = 4 \cdot a^2 $$ $$ 4 \cdot a^2 = 295.1168\, \text{cm}^2 $$ $$ a^2 = \frac{ 295.1168\, \text{cm}^2 }{ 4 } $$ $$ a^2 = 73.7792\, \text{cm}^2 $$ $$ a = \sqrt{ 73.7792\, \text{cm}^2 } $$$$ a = 8.5895\, \text{cm} $$

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