Find the side $ a $ of a rhombus if area $A = 20$ and diagonal $d_2 = 5$.

$a = \frac{\sqrt{ 89 }}{ 2 }$

STEP 1: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ A = 20 $ and $ d_2 = 5 $ we have:

$$ 20 = \dfrac{ d_1 \cdot 5 }{ 2 } $$$$ 20 \cdot 2 = d_1 \cdot 5 $$$$ 40 = 5 \cdot d_1 $$$$ d_1 = \dfrac{ 40}{ 5 } $$$$ d_1 = 8 $$

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 = 8 $ and $ d_2 = 5 $ we have:

$$ 8^2 + 5^2 = 4 \cdot a^2 $$ $$ 64 + 25 = 4 \cdot a^2 $$ $$ 4 \cdot a^2 = 89 $$ $$ a^2 = \frac{ 89 }{ 4 } $$ $$ a^2 = \frac{ 89 }{ 4 } $$ $$ a = \sqrt{ \frac{ 89 }{ 4 } } $$$$ a = \frac{\sqrt{ 89 }}{ 2 } $$

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