# Two circles are tangent to each other, find the ratio of line that splits the area into $1:2$

There is one circle with radius $1$. There is another circle with radius $2$. They are tangent to each other and touch each other at point $c$. A line through $c$ splits the area formed by the two circles into two parts. The ratio of the two parts is $1:2$. In what ratio does the line split the area of the smaller circle (the circle with radius $1$)? This was one of the questions in last year's AMC. I have no idea how to solve it. I thought that you could try and work out the of the two parts of the smaller circle and figure out what the ratio was between them. I tried that but failed. Thanks for any help.

• If anyone has a picture for this problem, could you please edit my question and upload the picture? – Joao Sep 17 '14 at 6:39
• Are the two circles and dividing chords similar figures? – Henry Sep 17 '14 at 6:56
• @Henry Well, two circles are always similar right? I don't know about the chords. – Joao Sep 17 '14 at 7:00
• I would have thought the chords had the same angle with the tangent, and so the same angles with the radii. – Henry Sep 17 '14 at 7:03
• @Henry I don't know... sorry I don't know much about geometry. – Joao Sep 17 '14 at 7:09 Regions P and S are similar, as are regions Q and R. And the area of the right-hand circle is four times the area of the left-hand circle. So $P = 4S$ and $Q = 4R$.

And we are told that $Q+S = 2(P+R)$.

Substituting, $4R+S = 2(4S+R)$. You can take it from there.

Updated to add: Riffing on André's comment (now deleted), this doesn't just work for circles. For instance, it works for Australias too: • Yes I can! Thanks its all clear now. – Joao Sep 17 '14 at 7:17