Question

在一条直线的同一侧画三个圆：一个半径为 $4$ 英寸的圆与该直线相切，另两个圆全等，且每个圆都与该直线及另外两个圆相切。两个全等圆的半径为：

On the same side of a straight line three circles are drawn as follows: a circle with a radius of $4$ inches is tangent to the line, the other two circles are equal, and each is tangent to the line and to the other two circles. The radius of the equal circles is:

$24$

$20$

$18$

$16$

$12$

答案：D

知识点：相切圆距离公式

难度评级：1590

小提示：

小圆对称地位于两个全等圆之间

The small circle lies symmetrically between the two equal circles

大提示：

若全等圆半径为 $R$ ，比较圆心距 $\sqrt{R^2+(R-4)^2}$ 与 $R+4$

If an equal circle has radius $R,$ compare the center distance $\sqrt{R^2+(R-4)^2}$ with $R+4$

解答：

设两个全等圆的半径为 $R$ 。它们的圆心相距 $2R$ ，所以半径为 $4$ 的圆的圆心位于两者正中间。小圆圆心与任一大圆圆心的水平距离和竖直距离分别为 $R$ 和 $R-4$ 。由相切条件得 $R^2+(R-4)^2=(R+4)^2\text{。}$ 化简得 $R^2-16R=0$ ，由正性知 $R=16$ 。

因此，正确答案是 D。

Let the equal circles have radius $R.$ Their centers are $2R$ apart, so the center of the radius- $4$ circle lies midway between them. The horizontal and vertical separations between its center and either large center are $R$ and $R-4.$ Tangency gives $R^2+(R-4)^2=(R+4)^2.$ Simplifying yields $R^2-16R=0,$ and positivity gives $R=16.$

Thus, the correct answer is D.

1959 AMC 12 第 41 题

答案：D

难度评级：1590

小提示：

大提示：

解答：

其他年份的第 41 题