2004 AMC 10B 第 16 题

Question

三个半径为

1

的圆两两外切，并且都与一个较大的圆内切。大圆的半径是多少？

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?

\dfrac{2 + \sqrt{6}}{3}

2

\dfrac{2 + 3\sqrt{2}}{3}

\dfrac{3 + 2\sqrt{3}}{3}

\dfrac{3 + \sqrt{3}}{2}

三个小圆的圆心形成边长为

2

的等边三角形。

The three small centers form an equilateral triangle of side

2

大圆半径等于

1

加上该等边三角形中心到顶点的距离。

The large radius is

1

plus the distance from that triangle’s center to a vertex

三个单位圆的圆心形成边长为

2

的等边三角形；它的中心就是大圆的圆心。

对于边长为

2

的等边三角形，中心到顶点的距离为

\dfrac{2}{\sqrt3} = \dfrac{2\sqrt3}{3}

。

再加上单位圆半径，大圆半径为

1 + \dfrac{2\sqrt3}{3} = \dfrac{3 + 2\sqrt3}{3}

。

所以正确答案是 D。

The centers of the three unit circles form an equilateral triangle with side

2.

Its center is the center of the large circle.

For an equilateral triangle of side length

2,

the distance from its center to a vertex is

\dfrac{2}{\sqrt3} = \dfrac{2\sqrt3}{3}.

Adding the unit radius, the large radius is

1 + \dfrac{2\sqrt3}{3} = \dfrac{3 + 2\sqrt3}{3}.

Thus, the correct answer is D.

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