Find the perimeter $ P $ of a rhombus if area $A = \frac{ 934 }{ 5 }$ and diagonal $d_2 = \frac{ 64 }{ 5 }$.

$P = \frac{\sqrt{ 6500801 }}{ 40 }$

STEP 1: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ A = \frac{ 934 }{ 5 } $ and $ d_2 = \frac{ 64 }{ 5 } $ we have:

$$ \frac{ 934 }{ 5 } = \dfrac{ d_1 \cdot \frac{ 64 }{ 5 } }{ 2 } $$$$ \frac{ 934 }{ 5 } \cdot 2 = d_1 \cdot \frac{ 64 }{ 5 } $$$$ \frac{ 1868 }{ 5 } = \frac{ 64 }{ 5 } \cdot d_1 $$$$ d_1 = \dfrac{ \frac{ 1868 }{ 5 }}{ \frac{ 64 }{ 5 } } $$$$ d_1 = \frac{ 467 }{ 16 } $$

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 = \frac{ 467 }{ 16 } $ and $ d_2 = \frac{ 64 }{ 5 } $ we have:

$$ \left(\frac{ 467 }{ 16 }\right)^2 + \left(\frac{ 64 }{ 5 }\right)^2 = 4 \cdot a^2 $$ $$ \frac{ 218089 }{ 256 } + \frac{ 4096 }{ 25 } = 4 \cdot a^2 $$ $$ 4 \cdot a^2 = \frac{ 6500801 }{ 6400 } $$ $$ a^2 = \frac{ \frac{ 6500801 }{ 6400 } }{ 4 } $$ $$ a^2 = \frac{ 6500801 }{ 25600 } $$ $$ a = \sqrt{ \frac{ 6500801 }{ 25600 } } $$$$ a = \frac{\sqrt{ 6500801 }}{ 160 } $$

STEP 3: find perimeter $ P $

To find perimeter $ P $ use formula:

$$ P = 4 \cdot a $$

After substituting $ a = \frac{\sqrt{ 6500801 }}{ 160 } $ we have:

$$ P = 4 \cdot \frac{\sqrt{ 6500801 }}{ 160 } $$ $$ P = \frac{\sqrt{ 6500801 }}{ 40 } $$

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