1966 AMC 12 第 8 题

Question

两个相交圆的公共弦长为

16

英尺。若两个半径分别为

10

英尺和

17

英尺，则两圆心之间的距离可能为（单位：英尺）：

The length of the common chord of two intersecting circles is

16

feet. If the radii are

10

feet and

17

feet, a possible value for the distance between the centers of the circles, expressed in feet, is:

27

21

\sqrt{389}

15

无法确定

undetermined

两圆心的连线垂直平分公共弦

The line of centers perpendicularly bisects the common chord

利用半弦长

8

求每个圆心到弦的距离

Find each center’s distance from the chord using half-chord

8

半弦长为

8

。两个圆心到弦所在直线的垂直距离分别为

\begin{gathered} \sqrt{10^2-8^2}=6,\\ \sqrt{17^2-8^2}=15 \end{gathered}\text{。}

若两圆心位于弦的两侧，则它们的距离为

6+15=21

，这是一个可能值。

因此，正确答案是 B。

The half-chord has length

8.

The two perpendicular distances from the centers to its line are

\begin{gathered} \sqrt{10^2-8^2}=6,\\ \sqrt{17^2-8^2}=15. \end{gathered}

If the centers lie on opposite sides of the chord, their distance is

6+15=21,

which is a possible value.

Therefore, the correct answer is B.

其他年份的第 8 题