Find the side $ a $ of a rhombus if area $A = 14$ and diagonal $d_1 = 7$.

$a = \frac{\sqrt{ 65 }}{ 2 }$

STEP 1: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ A = 14 $ and $ d_1 = 7 $ we have:

$$ 14 = \dfrac{ 7 \cdot d_2 }{ 2 } $$$$ 14 \cdot 2 = 7 \cdot d_2 $$$$ 28 = 7 \cdot d_2 $$$$ d_2 = \dfrac{ 28 }{ 7 } $$$$ d_2 = 4 $$

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 = 7 $ and $ d_2 = 4 $ we have:

$$ 7^2 + 4^2 = 4 \cdot a^2 $$ $$ 49 + 16 = 4 \cdot a^2 $$ $$ 4 \cdot a^2 = 65 $$ $$ a^2 = \frac{ 65 }{ 4 } $$ $$ a^2 = \frac{ 65 }{ 4 } $$ $$ a = \sqrt{ \frac{ 65 }{ 4 } } $$$$ a = \frac{\sqrt{ 65 }}{ 2 } $$

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