Find the perimeter $ P $ of a rhombus if area $A = 84$ and diagonal $d_1 = 14$.

$P = 4 \sqrt{ 85 }$

STEP 1: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ A = 84 $ and $ d_1 = 14 $ we have:

$$ 84 = \dfrac{ 14 \cdot d_2 }{ 2 } $$$$ 84 \cdot 2 = 14 \cdot d_2 $$$$ 168 = 14 \cdot d_2 $$$$ d_2 = \dfrac{ 168 }{ 14 } $$$$ d_2 = 12 $$

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 = 14 $ and $ d_2 = 12 $ we have:

$$ 14^2 + 12^2 = 4 \cdot a^2 $$ $$ 196 + 144 = 4 \cdot a^2 $$ $$ 4 \cdot a^2 = 340 $$ $$ a^2 = \frac{ 340 }{ 4 } $$ $$ a^2 = 85 $$ $$ a = \sqrt{ 85 } $$

STEP 3: find perimeter $ P $

To find perimeter $ P $ use formula:

$$ P = 4 \cdot a $$

After substituting $ a = \sqrt{ 85 } $ we have:

$$ P = 4 \cdot \sqrt{ 85 } $$ $$ P = 4 \sqrt{ 85 } $$

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