Find the perimeter $ P $ of a rhombus if diagonal $d_1 = \frac{ 707 }{ 50 }$ and diagonal $d_2 = \frac{ 283 }{ 50 }$.

$P = \frac{\sqrt{ 579938 }}{ 25 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ d_1 = \frac{ 707 }{ 50 } $ and $ d_2 = \frac{ 283 }{ 50 } $ we have:

$$ \left(\frac{ 707 }{ 50 }\right)^2 + \left(\frac{ 283 }{ 50 }\right)^2 = 4 \cdot a^2 $$ $$ \frac{ 499849 }{ 2500 } + \frac{ 80089 }{ 2500 } = 4 \cdot a^2 $$ $$ 4 \cdot a^2 = \frac{ 289969 }{ 1250 } $$ $$ a^2 = \frac{ \frac{ 289969 }{ 1250 } }{ 4 } $$ $$ a^2 = \frac{ 289969 }{ 5000 } $$ $$ a = \sqrt{ \frac{ 289969 }{ 5000 } } $$$$ a = \frac{\sqrt{ 579938 }}{ 100 } $$

STEP 2: find perimeter $ P $

To find perimeter $ P $ use formula:

$$ P = 4 \cdot a $$

After substituting $ a = \frac{\sqrt{ 579938 }}{ 100 } $ we have:

$$ P = 4 \cdot \frac{\sqrt{ 579938 }}{ 100 } $$ $$ P = \frac{\sqrt{ 579938 }}{ 25 } $$

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