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Answer the question on the basis on the basis of the information given below.
Consider three circular parks of equal size with centres at $A_1,\:\: A_2,\:\:$ and $A_3$ respectively. The parks touch each other at the edge as shown in the figure (not drawn to scale). There are three paths formed by the triangles $A_1, \: A_2, \: A_3,\: B_1, \: B_2, \: B_3$, and $C_1, \: C_2, \: C_3,$ as shown. Three sprinters A, B, and C begin running from points $A_1,\: B_1$ and $C_1$ respectively. Each sprinter traverses her respective triangular path clockwise and returns to her starting point.

Let the radius of each circular park be $r,$ and the distances to be traversed by the sprinters A, B and C be $a, b$ and $c,$ respectively. Which of the following is true? 

  1. $b-a = c-b = 3 \sqrt{3} r$
  2. $b-a = c-b = \sqrt{3} r$
  3. $b= \frac{a+c}{2}=2(1+\sqrt{3})r$
  4. $c=2b-a=(2+\sqrt{3}r$
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