2007 AMC 10B Problem 18

Question

A circle of radius

1

is surrounded by

4

circles of radius

r

as shown. What is

r?

\sqrt2

1+\sqrt2

\sqrt6

3

2+\sqrt2

Connect the centers of the four outer circles to form a square

The square has side

2r

and diagonal

2+2r

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.

Problem 18 in other years