1960 AMC 12 第 7 题

Question

圆 I 经过圆 II 的圆心，并与圆 II 相切。圆 I 的面积为

4

平方英寸。那么圆 II 的面积（以平方英寸计）是：

Circle I passes through the center of, and is tangent to, circle II. The area of circle I is

4

square inches. Then the area of circle II, in square inches, is:

8

8\sqrt2

8\sqrt\pi

16

16\sqrt2

画出经过两个圆心和切点的直线

Draw the line through the two centers and the tangency point

圆 II 的半径是圆 I 半径的两倍

The radius of circle II is twice the radius of circle I

因为圆 I 经过圆 II 的圆心，所以两圆心之间的距离等于圆 I 的半径

r

。内切又使该距离等于

R-r

，其中

R

是圆 II 的半径。因此

R=2r

，面积便乘以

2^2=4

。所以圆 II 的面积为

4\cdot4=16

。

因此，正确答案是 D。

Because circle I passes through the center of circle II, the distance between the centers equals the radius

r

of circle I. Internal tangency also makes that distance

R-r,

where

R

is the radius of circle II. Thus

R=2r,

and the area is multiplied by

2^2=4.

It is therefore

4\cdot4=16.

Thus, the correct answer is D.

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