Two sides of a regular polygon with n sides, when extended, meet at an angle of 28^ . What is the smallest possible value of n ?

Correct answer: 45 Let the sides extended pass through the vertices which on joining the centre makes angle (k(2 π))/(n) aligned & 28+((n-2)/(n)) π+(2 k π)/(n)=360^ & ((2 k-2) π)/(n)=152^ & ((k-1) π)/(n)=76^ & n=((k-1) 45)/(19) aligned For minimum k=20 and n=45

Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Properties of Triangles

Olympiad workbook · IOQM · Properties of Triangles

Two sides of a regular polygon with n sides, when extended, meet at...

Two sides of a regular polygon with

n

sides, when extended, meet at an angle of

28^{\circ}

. What is the smallest possible value of

n

Correct answer

Step-by-step solution

Let the sides extended pass through the vertices which on joining the centre makes angle

\frac{k(2 \pi)}{n}

\begin{aligned} & \therefore \quad 28+\left(\frac{n-2}{n}\right) \pi+\frac{2 k \pi}{n}=360^{\circ} \\ & \Rightarrow \quad \frac{(2 k-2) \pi}{n}=152^{\circ} \\ & \Rightarrow \frac{(k-1) \pi}{n}=76^{\circ} \\ & \Rightarrow n=\frac{(k-1) 45}{19} \end{aligned}

For minimum

k=20

and

n=45

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