STEP 1: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $A = 186.8\, \text{cm}$ and $d_2 = 12.8\, \text{cm}$ we have:

$$ 186.8\, \text{cm} = \dfrac{ d_1 \cdot \left( 12.8\, \text{cm} \right)^{4} }{ 2 } $$$$ 186.8\, \text{cm} \cdot 2 = d_1 \cdot \left( 12.8\, \text{cm} \right)^{4} $$$$ 373.6\, \text{cm} = 12.8\, \text{cm} \cdot d_1 $$$$ d_1 = \dfrac{ 373.6\, \text{cm}}{ 12.8\, \text{cm} } $$$$ d_1 \approx 9.2906 $$

STEP 2: find side $ a $

To find side $ a $ use formula:

$$ d_1^2 + d_2^2 = 4 \cdot a^2 $$

After substituting $d_1 = 9.2906\, \text{cm}^0$ and $d_2 = 12.8\, \text{cm}$ we have:

$$ 9.2906 + \left( 12.8\, \text{cm} \right)^{2} = 4 \cdot a^2 $$ $$ 86.3161 + 163.84\, \text{cm}^2 = 4 \cdot a^2 $$ $$ 4 \cdot a^2 = 250.1561 $$ $$ a^2 = \frac{ 250.1561 }{ 4 } $$ $$ a^2 = 62.539 $$ $$ a = \sqrt{ 62.539 } $$$$ a = 7.9082 $$

STEP 3: find perimeter $ P $

To find perimeter $ P $ use formula:

$$ P = 4 \cdot a $$

After substituting $a = 7.9082\, \text{cm}^0$ we have:

$$ P = 4 \cdot 7.9082 $$ $$ P \approx 31.6327 $$

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