2021 AMC 10B Spring 第 14 题

Question

三条等距平行线与一个圆相交，形成三条长度分别为

38

、

38

和

34

的弦。相邻两条平行线之间的距离是多少？

Three equally spaced parallel lines intersect a circle, creating three chords of lengths

38,

38,

and

34.

What is the distance between two adjacent parallel lines?

5\frac12

6

6\frac12

7

7\frac12

答案：B

知识点：弦圆勾股定理

难度评级：1540

小提示：

长度相等的弦到圆心的距离相等。

Equal chord lengths are equally far from the circle center

大提示：

若两条相等的弦在相邻直线上，则第三条直线到圆心的距离是半间距的三倍。

If the equal chords are adjacent lines, the third line is three half-spacings from the center

视频讲解：

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文字解答：

两条长度为

38

的弦到圆心的距离相等。它们不能位于两条最外侧的直线上：否则圆心会在中间直线上，该线所截的弦会是直径，长度大于

38

，而不是给定的

34

。因此两条相等弦位于相邻直线上，圆心在两线中间。设这个半间距为

d

。则每条长

38

的弦到圆心的距离是

d

，长

34

的弦到圆心的距离是

3d

。

若圆的半径为

r

，则

r^2=19^2+d^2=17^2+(3d)^2\text{。}

因此

19^2-17^2=8d^2

，所以

72=8d^2

，得到

d=3

。相邻平行线的距离是

2d=6

。

所以答案是 B。

The two chords of length

38

are equally far from the center of the circle. They cannot lie on the two outer lines: then the center would lie on the middle line, whose chord would be a diameter longer than

38,

not the given

34.

Therefore, the equal chords lie on adjacent lines, with the center halfway between them. Let that half-distance be

d.

Then each

38

-chord is distance

d

from the center, and the

34

-chord is distance

3d

from the center.

If the circle has radius

r,

then

r^2=19^2+d^2=17^2+(3d)^2.

Thus

19^2-17^2=8d^2,

so

72=8d^2,

and

d=3.

The distance between adjacent parallel lines is

2d=6.

Thus, the answer is B .