Find the side $ a $ of a rhombus if area $A = 42$ and diagonal $d_1 = 8$.

$a = \frac{\sqrt{ 697 }}{ 4 }$

STEP 1: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ A = 42 $ and $ d_1 = 8 $ we have:

$$ 42 = \dfrac{ 8 \cdot d_2 }{ 2 } $$$$ 42 \cdot 2 = 8 \cdot d_2 $$$$ 84 = 8 \cdot d_2 $$$$ d_2 = \dfrac{ 84 }{ 8 } $$$$ d_2 = \frac{ 21 }{ 2 } $$

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 = 8 $ and $ d_2 = \frac{ 21 }{ 2 } $ we have:

$$ 8^2 + \left(\frac{ 21 }{ 2 }\right)^2 = 4 \cdot a^2 $$ $$ 64 + \frac{ 441 }{ 4 } = 4 \cdot a^2 $$ $$ 4 \cdot a^2 = \frac{ 697 }{ 4 } $$ $$ a^2 = \frac{ \frac{ 697 }{ 4 } }{ 4 } $$ $$ a^2 = \frac{ 697 }{ 16 } $$ $$ a = \sqrt{ \frac{ 697 }{ 16 } } $$$$ a = \frac{\sqrt{ 697 }}{ 4 } $$

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