Find the side $ a $ of a rhombus if area $A = 90$ and diagonal $d_1 = 15$.

$a = \frac{ 3 \sqrt{ 41}}{ 2 }$

STEP 1: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ A = 90 $ and $ d_1 = 15 $ we have:

$$ 90 = \dfrac{ 15 \cdot d_2 }{ 2 } $$$$ 90 \cdot 2 = 15 \cdot d_2 $$$$ 180 = 15 \cdot d_2 $$$$ d_2 = \dfrac{ 180 }{ 15 } $$$$ d_2 = 12 $$

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 = 15 $ and $ d_2 = 12 $ we have:

$$ 15^2 + 12^2 = 4 \cdot a^2 $$ $$ 225 + 144 = 4 \cdot a^2 $$ $$ 4 \cdot a^2 = 369 $$ $$ a^2 = \frac{ 369 }{ 4 } $$ $$ a^2 = \frac{ 369 }{ 4 } $$ $$ a = \sqrt{ \frac{ 369 }{ 4 } } $$$$ a = \frac{ 3 \sqrt{ 41}}{ 2 } $$

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