2007 AMC 10B Problems/Problem 18
Problem
A circle of radius 
is surrounded by 
circles of radius 
as shown. What is ?
![[asy] unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(7pt)); dotfactor=4; real r1=1, r2=1+sqrt(2); pair A=(0,0), B=(1+sqrt(2),1+sqrt(2)), C=(-1-sqrt(2),1+sqrt(2)), D=(-1-sqrt(2),-1-sqrt(2)), E=(1+sqrt(2),-1-sqrt(2)); pair A1=(1,0), B1=(2+2sqrt(2),1+sqrt(2)), C1=(0,1+sqrt(2)), D1=(0,-1-sqrt(2)), E1=(2+2sqrt(2),-1-sqrt(2)); path circleA=Circle(A,r1); path circleB=Circle(B,r2); path circleC=Circle(C,r2); path circleD=Circle(D,r2); path circleE=Circle(E,r2); draw(circleA); draw(circleB); draw(circleC); draw(circleD); draw(circleE); draw(A--A1); draw(B--B1); draw(C--C1); draw(D--D1); draw(E--E1); label("$1$",midpoint(A--A1),N); label("$r$",midpoint(B--B1),N); label("$r$",midpoint(C--C1),N); label("$r$",midpoint(D--D1),N); label("$r$",midpoint(E--E1),N); [/asy]](http://latex.artofproblemsolving.com/6/b/8/6b83570902439de3661af1cf5b186d453fed927f.png)
Solution
Solution 1
You can express the line connecting the centers of an outer circle and the inner circle in two different ways. You can add the radius of both circles to get 
You can also add the radius of two outer circles and use a 
triangle to get 
Since both representations are for the same thing, you can set them equal to each other.
![\[r = \frac{1}{\sqrt{2}-1} = \frac{\sqrt{2}+1}{2-1} = \boxed{\mathrm{(B) \ } 1 + \sqrt{2}}\]](http://latex.artofproblemsolving.com/d/6/4/d64f905ef4ff1b6f1d16f74c629cc22ffae06d8d.png)
Solution 2
You can solve this problem by setting up a simple equation with the Pythagorean Theorem. The hypotenuse would be a segment that includes the radius of two circles on opposite corners and the diameter of the middle circle. This results in 
. The two legs are each the length between two large, adjacent circles, thus 
.
Using the Pythagorean Theorem:
\begin{align*}
(2r+2)^2 = 2(2r)^2\\
4r^2+8r+4=8r^2\\
r^2+2r+1=2r^2\\
r^2-2r-1=0\\
r=\frac{2+\sqrt{4-(-4)}}{2}=\frac{2+\sqrt{2}}{2}=\boxed{\mathrm{(B) \ } 1 + \sqrt{2}} (Error compiling LaTeX. Unknown error_msg)
See Also
| 2007 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
