What is the distance between the centers of the large two circles in the figure?

Question

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.

Answer 1

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 $42=R-r$

Answer 2

Let the radii of the large and small circles be $R$ and $r$. Then Pythagoras gives $$(R+r)^2=R^2+(R-r)^2$$ which simplifies to $$4r=R$$ We are given that $2R=84+2r$. You should now be able to work out what $2R$ must be.

Answer 3

Hint:

If $r$ is the radius of large circle and $s$ of small circle then:$$r^2+(r-s)^2=(r+s)^2$$leading to $r=4s$.