Question

在一个半径为 $r$ 的圆周上，独立随机选取两个点。从每个点沿顺时针方向画一条长度为 $r$ 的弦。两条弦相交的概率是多少？

Two points on the circumference of a circle of radius $r$ are selected independently and at random. From each point a chord of length $r$ is drawn in a clockwise direction. What is the probability that the two chords intersect?

$\dfrac{1}{6}$

$\dfrac{1}{5}$

$\dfrac{1}{4}$

$\dfrac{1}{3}$

$\dfrac{1}{2}$

答案：D

知识点：几何概率弦弧

难度评级：1840

解答：

半径为 $r$ 的圆中，长度为 $r$ 的弦所对的圆心角为 $60^\circ$ 。固定第一条弦，使其端点的角度为 $0^\circ$ 和 $60^\circ.$ 若第二条弦从角度 $\theta,$ 开始，则另一端点在其顺时针方向 $60^\circ$ 处。

两条弦的端点恰好交替出现，当且仅当 $\theta$ 位于紧邻固定弦两个端点的两段 $60^\circ$ 圆弧之一。因此有利的起点共占圆周的 $120^\circ$ 。

所求概率为 $\dfrac{2}{6} = \dfrac{1}{3}.$

所以正确答案是 D。

A chord of length $r$ in a circle of radius $r$ subtends a $60^\circ$ arc. Fix the first chord, with endpoints at angles $0^\circ$ and $60^\circ.$ If the second chord starts at angle $\theta,$ its other endpoint is $60^\circ$ clockwise from there.

The endpoints of the two chords alternate exactly when $\theta$ lies in either of the two $60^\circ$ arcs immediately adjacent to the fixed chord's endpoints. Thus the favorable starting positions occupy $120^\circ$ of the circle.

The desired probability is then $\dfrac{2}{6} = \dfrac{1}{3}.$

Thus, D is the correct answer.

2011 AMC 10A 第 20 题

答案：D

难度评级：1840

解答：

其他年份的第 20 题