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'Running Race' printed from http://nrich.maths.org/
In a leisure park there are three running tracks, all with the same Start and Finish, and all made from either one or two semicircles with centres on the same line.
Three runners $P$, $Q$ and $R$ start together at the Start and run at the same constant speed along the tracks shown. In what order do they finish?
Imagine a circle of diameter 1 unit, and another with the same diameter, and another.
What is the total perimeter of these three circles (you can leave $\pi$ in your answer)?
What is the total perimeter of a circle of diameter $3$?
What do you notice about these two answers?
Now imagine two circles, one of diameter $2$ units and one of diameter $3$ units.
What is the total perimeter of these two circles?
What is the perimeter of a circle of diameter $5$ units?
What do you notce this time?
Is this generally the case with combinations of circles whose total diameters are equal?
Can you explain why?
This problem is taken from the UKMT Mathematical Challenges.