Answer:
If the octagon is clockwise, then after 5 times 45 increments A' will coincide with F.
If the octagon is counterclockwise, then after 3 times 45 increments A' will coincide with F.
Step-by-step explanation:
Case 1: Let the octagon is clockwise
It is given that a regular octagon ABCDEFGH rotates 360 degree clockwise about it's center.
There are 8 vertices and the central angle between two consecutive vertices is 45 degrees.
Central angle between AB is 45 degree.
Central angle between AC is
[tex]\text{Angle between AB}+\text{Angle between BC}=45^{\circ}+45^{\circ}=2\times 45^{\circ}[/tex]
Central angle between AD is
[tex]\text{Angle between AB}+\text{Angle between BC}+\text{Angle between CD}=45^{\circ}+45^{\circ}+45^{\circ}=3\times 45^{\circ}[/tex]
Similarly,
Central angle between AE is
[tex]4\times 45^{\circ}[/tex]
Central angle between FA is
[tex]5\times 45^{\circ}[/tex]
Therefore after 5 times 45 increments A' will coincide with F.
Case 2: Let the octagon is counterclockwise
[tex]FA=AH+HG+GF[/tex]
[tex]FA=45^{\circ}+45^{\circ}+45^{\circ}[/tex]
[tex]FA=135^{\circ}[/tex]
[tex]FA=3\times 45^{\circ}[/tex]
Therefore after 3 times 45 increments A' will coincide with F.
