The distance between the centers of the large two circles in the figure

circlescontest-mathgeometry

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In figure, if the distance between two small circles is 84,
what will be the distance between two large circles?

Source: Bangladesh Math Olympiad 2016 Junior Category

I can't solve this this problem. But I found that the connected lines between the centers of the 4 circles create a rhombus. The diagonal of the rhombus is 84.

Best Answer

Hint:

If the large circles each have radius $R$ and the small circles $r$ then:

  • the distance between the centres of the small circles is $84=2R-2r$ and half of this is $42=R-r$
  • the distance between the centre of a small circle and the centre of a large circle is $R+r$
  • the distance between the centres of the large circles is $2R$ and half of this is $R$
  • You have right-angled triangles so can get a second equality to solve simultaneously with $42=R-r$
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