2007 AMC 10B Problem 18

Connect the centers of the four outer circles to form a square. Adjacent outer circles are tangent, so each side has length $2r.$ The diagonal of the square passes through the center of the central circle, giving length $1+r+r+1=2+2r.$ Since a square with side $2r$ has diagonal $2r\sqrt2,$ we get $2(2r)^2=(2+2r)^2.$ Expanding gives $1+2r+r^2=2r^2,$ so $r^2-2r-1=0.$ The positive root is $r=1+\sqrt2.$ Thus, the correct answer is B.