Find the long diagonal $ d $ of a hexagon if incircle radius $r = \frac{ 85 }{ 2 }$.

$d = \frac{ 170 \sqrt{ 3}}{ 3 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ r = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$

After substituting $ r = \frac{ 85 }{ 2 } $ we have:

$$ \frac{ 85 }{ 2 } = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$ $$ \sqrt{ 3 } \cdot a = \frac{ 85 }{ 2 } \cdot 2 $$ $$ \sqrt{ 3 } \cdot a = 85 $$ $$ a = \dfrac{ 85 }{ \sqrt{ 3 } } $$ $$ a = \frac{ 85 \sqrt{ 3}}{ 3 } $$

STEP 2: find diagonal $ d $

To find diagonal $ d $ use formula:

$$ d = 2 \cdot a $$

After substituting $ a = \frac{ 85 \sqrt{ 3}}{ 3 } $ we have:

$$ d = 2 \cdot \frac{ 85 \sqrt{ 3}}{ 3 } $$ $$ d = \frac{ 170 \sqrt{ 3}}{ 3 } $$

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