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

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

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.

• The vertical diagonal of the rhombus is $84$. – TonyK Jan 28 at 13:36
• @TonyK Yes, I know that. But how am I supposed to find the other diagonal of that rhombus? – Shromi Jan 28 at 13:41
• If you know that, why didn't you say it? – TonyK Jan 28 at 13:51
• I wrote that in my question. – Shromi Jan 28 at 13:53
• I give up.${}{}$ – TonyK Jan 28 at 14:22

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 $$42=R-r$$

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.

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 $$r=4s$$.