Find the area $ A $ of a rhombus if perimeter $P = 58$ and diagonal $d_1 = 10$.

$A = 5 \sqrt{ 741 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ P = 4 \cdot a $$

After substituting $ P = 58 $ we have:

$$ 58 = 4 \cdot a $$ $$ a = \dfrac{ 58 }{ 4 } $$ $$ a = \frac{ 29 }{ 2 } $$

STEP 2: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ d_1 = 10 $ and $ a = \frac{ 29 }{ 2 } $ we have:

$$ 10^2 + d_2^2 = 4 \cdot \left(\frac{ 29 }{ 2 }\right)^2 $$ $$ 100 + d_2^2 = 841 $$ $$ d_2^2 = 841 - 100 $$ $$ d_2^2 = 741 $$ $$ d_2 = \sqrt{ 741 } $$

STEP 3: find area $ A $

To find area $ A $ use formula:

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

After substituting $ d_1 = 10 $ and $ d_2 = \sqrt{ 741 } $ we have:

$$ A = \dfrac{ 10 \cdot \sqrt{ 741 } }{ 2 }$$$$ A = \dfrac{ 10 \sqrt{ 741 } }{ 2 } $$$$ A = 5 \sqrt{ 741 } $$

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