Find the perimeter $ P $ of a rectangle if side $a = 2$ and diagonal $d = \frac{ 5 }{ 2 }$.

$P = 7$

STEP 1: find side $ b $

To find side $ b $ use Pythagorean Theorem:

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

After substituting $ a = 2 $ and $ d = \frac{ 5 }{ 2 } $ we have:

$$ 2^2 + b^2 = \left(\frac{ 5 }{ 2 }\right)^2 $$ $$ b^2 = \left(\frac{ 5 }{ 2 }\right)^2 - 2^2 $$ $$ b^2 = \frac{ 25 }{ 4 } - 4 $$ $$ b^2 = \frac{ 9 }{ 4 } $$ $$ b = \sqrt{ \frac{ 9 }{ 4 } } $$$$ b = \frac{ 3 }{ 2 } $$

STEP 2: find perimeter $ P $

To find perimeter $ P $ use formula:

$$ P = 2 a + 2 b $$

After substituting $ a = 2 $ and $ b = \frac{ 3 }{ 2 } $ we have:

$$ P = 2 \cdot 2 + 2 \cdot \frac{ 3 }{ 2 } $$ $$ P = 4 + 3 $$ $$ P = 7 $$

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