Find the area $ A $ of a rectangle if side $a = \frac{ 9 }{ 2 }$ and diagonal $d = \frac{ 41 }{ 2 }$.

$A = 90$

STEP 1: find side $ b $

To find side $ b $ use Pythagorean Theorem:

$$ a^2 + b^2 = d^2 $$

After substituting $ a = \frac{ 9 }{ 2 } $ and $ d = \frac{ 41 }{ 2 } $ we have:

$$ \left(\frac{ 9 }{ 2 }\right)^2 + b^2 = \left(\frac{ 41 }{ 2 }\right)^2 $$ $$ b^2 = \left(\frac{ 41 }{ 2 }\right)^2 - \left(\frac{ 9 }{ 2 }\right)^2 $$ $$ b^2 = \frac{ 1681 }{ 4 } - \frac{ 81 }{ 4 } $$ $$ b^2 = 400 $$ $$ b = \sqrt{ 400 } $$$$ b = 20 $$

STEP 2: find area $ A $

To find area $ A $ use formula:

$$ A = a \cdot b $$

After substituting $ a = \frac{ 9 }{ 2 } $ and $ b = 20 $ we have:

$$ A = \frac{ 9 }{ 2 } \cdot 20 $$$$ A = \frac{ 9 }{ 2 } \cdot 20 $$$$ A = 90 $$

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