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Combination Based on Geometry

A 1, A 2, …, ...

Question

$A_{1}, A_{2}, . . ., A_{18}$ are the vertices of a $18$ sided regular polygon. " $B$ " is an external point such that $A_{1} A_{2} B$ is an equilateral triangle. If $A_{18} A_{1}$ and $A_{1} B$ are the adjacent sides of another regular polygon of $n$ sides, then the value of $n$ is

6

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Solution

The correct option is B $9$

$2 π = ∠ A_{18} A_{1} A_{2} + ∠ A_{2} A_{1} B + ∠ A_{18} A_{1} B$
$= \frac{8 π}{9} + \frac{π}{3} + (1 - \frac{2}{n}) π$
$\Rightarrow n = 9$

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