STEP 1: find side $ a $

To find side $ a $ use Pythagorean Theorem:

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

After substituting $b = 4\, \text{cm}$ and $d = 55\, \text{cm}$ we have:

$$ a ^ {\,2} + \left( 4\, \text{cm} \right)^{2} = \left( 55\, \text{cm} \right)^{2} $$ $$ a ^ {\,2} = \left( 55\, \text{cm} \right)^{2} - \left( 4\, \text{cm} \right)^{2} $$ $$ a ^ {\,2} = 3025\, \text{cm}^2 - 16\, \text{cm}^2 $$ $$ a ^ {\,2} = 3009\, \text{cm}^2 $$ $$ a = \sqrt{ 3009\, \text{cm}^2 } $$$$ a = \sqrt{ 3009 }\, \text{cm} $$

STEP 2: find perimeter $ P $

To find perimeter $ P $ use formula:

$$ P = 2 a + 2 b $$

After substituting $a = \sqrt{ 3009 }\, \text{cm}$ and $b = 4\, \text{cm}$ we have:

$$ P = 2 \cdot \sqrt{ 3009 }\, \text{cm} + 2 \cdot 4\, \text{cm} $$ $$ P = 2 \sqrt{ 3009 }\, \text{cm} + 8\, \text{cm} $$ $$ P = 117.7087\, \text{cm} $$

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