Find the area $ A $ of a rhombus if side $a = 2 \sqrt{ 5 }$ and diagonal $d_1 = 4 \sqrt{ 3 }$.

$A = 8 \sqrt{ 6 }$

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 } $ and $ a = 2 \sqrt{ 5 } $ we have:

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

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 } $ and $ d_2 = 4 \sqrt{ 2 } $ we have:

$$ A = \dfrac{ 4 \sqrt{ 3 } \cdot 4 \sqrt{ 2 } }{ 2 }$$$$ A = \dfrac{ 16 \sqrt{ 6 } }{ 2 } $$$$ A = 8 \sqrt{ 6 } $$

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