Find the side $ a $ of a rhombus if area $A = 16$ and diagonal $d_1 = 4$.

$a = 2 \sqrt{ 5 }$

STEP 1: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ A = 16 $ and $ d_1 = 4 $ we have:

$$ 16 = \dfrac{ 4 \cdot d_2 }{ 2 } $$$$ 16 \cdot 2 = 4 \cdot d_2 $$$$ 32 = 4 \cdot d_2 $$$$ d_2 = \dfrac{ 32 }{ 4 } $$$$ d_2 = 8 $$

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 $ and $ d_2 = 8 $ we have:

$$ 4^2 + 8^2 = 4 \cdot a^2 $$ $$ 16 + 64 = 4 \cdot a^2 $$ $$ 4 \cdot a^2 = 80 $$ $$ a^2 = \frac{ 80 }{ 4 } $$ $$ a^2 = 20 $$ $$ a = \sqrt{ 20 } $$$$ a = 2 \sqrt{ 5 } $$

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