Question

determine the number of sides of regular polygon whose exterior and interior angles are in the ratio 2:7 at a vertex.​

Answer 1

$Let \: the\: number \: of\:sides \: of \: regular$

$polygon = n$

/* We know that , */

$The \: measure \: of \: each \: interior \: angle$

$is \: (n-2)(\frac{180}{2})$

$and \: the\: measure \: of \: each \: exterior$

$angle \: \frac{360}{n}$

$\frac{2}{7} = \frac{360}{n}\div (n-2)( \frac{180}{n})$

$\implies \frac{2}{7} = \frac{360}{n} \times \frac{n}{(n-2)\times 180}$

$\implies \frac{2}{7} = \frac{2}{n-2}$

$\implies 2(n-2) = 7 \times 2$

$\implies n-2 = 7$

$\implies n= 7 +2$

$\implies n= 9$

Therefore.,

$\red{The\: number \: of\:sides \: of \: regular }$

$\red{polygon } \green { = 9 }$

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