The number of times the image of the octagon will coincide with the preimage during rotation is determined by:
N = R/C
where
N is the number of times the preimage coincided with the rotated image during rotation
R is the angle of rotation
C is the central angle of the regular polygon
For an octagon, the central angle is
C = 360/8 = 45
So,
N = 360 / 45 = 8
Therefore, the rotated image of the octagon will coincide with the preimage 8 times during rotation.
Lets say that Williams current age is x, since it is currently unknown
From here we know that in 15 years (x+15) he will be 3 times his age 5 years prior (3x-5). Thus we can set these equations up as equal to one another
1) x + 15 = 3(x - 5)
2) x + 15 = 3x-15
3) 30= 2x
4) 15 = x
5) Now this age is his current age of 15 years, but they also want his age 15 years from now
The equation is x + 15 = 3(x - 5)
This should be correct now, sorry for the misconception. Good luck
