Find the side $ a $ of a rhombus if area $A = 222$ and diagonal $d_2 = 4$.

$a = \frac{ 13 \sqrt{ 73}}{ 2 }$

STEP 1: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

$$ A = \dfrac{ d_1 \cdot d_2 }{ 2 } $$

After substituting $ A = 222 $ and $ d_2 = 4 $ we have:

$$ 222 = \dfrac{ d_1 \cdot 4 }{ 2 } $$$$ 222 \cdot 2 = d_1 \cdot 4 $$$$ 444 = 4 \cdot d_1 $$$$ d_1 = \dfrac{ 444}{ 4 } $$$$ d_1 = 111 $$

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 = 111 $ and $ d_2 = 4 $ we have:

$$ 111^2 + 4^2 = 4 \cdot a^2 $$ $$ 12321 + 16 = 4 \cdot a^2 $$ $$ 4 \cdot a^2 = 12337 $$ $$ a^2 = \frac{ 12337 }{ 4 } $$ $$ a^2 = \frac{ 12337 }{ 4 } $$ $$ a = \sqrt{ \frac{ 12337 }{ 4 } } $$$$ a = \frac{ 13 \sqrt{ 73}}{ 2 } $$

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