Question

A circle of radius $1$ is surrounded by $4$ circles of radius $r$ as shown. What is $r?$

$\sqrt2$

$1+\sqrt2$

$\sqrt6$

$3$

$2+\sqrt2$

Answer: B

Concepts:tangent circles square (geometry)quadratic

Difficulty rating: 1680

Solution:

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 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.

2007 AMC 10B Problem 18

Answer: B

Difficulty rating: 1680

Solution:

Problem 18 in Other Years