Answer:
The polygon has 20 sides (Icosagon)
Step-by-step explanation:
As the smallest angle of rotation for a regular polygon is 18°, so a 18° rotation yields the same figure.
According to the attachment, the side AB subtends an angle of 18° at the centre. That is why a 18° rotation generates the same figure.
According to the properties of regular polygon,
[tex]OA=OB\ \ \Rightarrow m\angle OAB=m\angle OBA[/tex]
And
[tex]m\angle OAC=m\angle OAB[/tex]
(as the radius of a regular polygon bisects the interior angle)
As in the triangle OAB,
[tex]\Rightarrow m\angle AOB+m\angle OAB+m\angle OBA=180^{\circ}[/tex]
[tex]\Rightarrow 18^{\circ}+2m\angle OAB=180^{\circ}[/tex]
[tex]\Rightarrow m\angle OAB=81^{\circ}[/tex]
So,
[tex]\Rightarrow m\angle CAB=2m\angle OAB=162^{\circ}[/tex]
So, interior angle of polygon is 162°
We know that, the interior angle of regular polygon with n side is,
[tex]=\dfrac{(n-2)180}{n}[/tex]
Hence,
[tex]\Rightarrow \dfrac{(n-2)180}{n}=162[/tex]
[tex]\Rightarrow (n-2)180=162n[/tex]
[tex]\Rightarrow 180n-360=162n[/tex]
[tex]\Rightarrow 180n-162n=360[/tex]
[tex]\Rightarrow 18n=360[/tex]
[tex]\Rightarrow n=20[/tex]
Therefore, the polygon has 20 sides (Icosagon)
