STEP 1: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $d_1 = 4 \sqrt{ 3 }\, \text{cm}$ and $a = 2 \sqrt{ 5 }\, \text{cm}$ we have:

$$ \left( 4 \sqrt{ 3 }\, \text{cm} \right)^{2} + d_2^2 = 4 \cdot \left( 2 \sqrt{ 5 }\, \text{cm} \right)^{2} $$ $$ 48\, \text{cm}^2 + d_2^2 = 80\, \text{cm}^2 $$ $$ d_2^2 = 80\, \text{cm}^2 - 48\, \text{cm}^2 $$ $$ d_2^2 = 32\, \text{cm}^2 $$ $$ d_2 = \sqrt{ 32\, \text{cm}^2 } $$$$ d_2 = 4 \sqrt{ 2 }\, \text{cm} $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $d_1 = 4 \sqrt{ 3 }\, \text{cm}$ and $d_2 = 4 \sqrt{ 2 }\, \text{cm}$ we have:

$$ A = \dfrac{ 4 \sqrt{ 3 }\, \text{cm} \cdot 4 \sqrt{ 2 }\, \text{cm} }{ 2 }$$$$ A = \dfrac{ 16 \sqrt{ 6 }\, \text{cm}^2 }{ 2 } $$$$ A = 8 \sqrt{ 6 }\, \text{cm}^2 $$

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