2004 AMC 10B Problems/Problem 16

Problem

Three circles of radius $1$ are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle?

The situation in shown in the picture below. The radius we seek is $SD = AD + AS$ . Clearly $AD=1$ . The point $S$ is clearly the center of the equilateral triangle $ABC$ , thus $AS$ is $2/3$ of the altitude of this triangle. We get that $AS = \frac23 \cdot \sqrt 3$ . Therefore the radius we seek is $1 + \frac23 \cdot \sqrt 3 = \boxed{\frac{3+2\sqrt{3}}3}$ .

WARNING. Note that the answer does not correspond to any of the five options. Most probably there is a typo in option D.