Find the area $ A $ of a rhombus if perimeter $P = 100$ and diagonal $d_2 = 48$.

$A = 336$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ P = 4 \cdot a $$

After substituting $ P = 100 $ we have:

$$ 100 = 4 \cdot a $$ $$ a = \dfrac{ 100 }{ 4 } $$ $$ a = 25 $$

STEP 2: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ d_2 = 48 $ and $ a = 25 $ we have:

$$ d_1 ^ {\,2} + 48^2 = 4 \cdot 25^2 $$ $$ d_1 ^ {\,2} + 2304 = 2500 $$ $$ d_1 ^ {\,2} = 2500 - 2304 $$ $$ d_1 ^ {\,2} = 196 $$ $$ d_1 = \sqrt{ 196 } $$$$ d_1 = 14 $$

STEP 3: find area $ A $

To find area $ A $ use formula:

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

After substituting $ d_1 = 14 $ and $ d_2 = 48 $ we have:

$$ A = \dfrac{ 14 \cdot 48 }{ 2 }$$$$ A = \dfrac{ 672 }{ 2 } $$$$ A = 336 $$

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