Question

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?

$\dfrac{1}{6}$

$\dfrac{1}{5}$

$\dfrac{1}{4}$

$\dfrac{1}{3}$

$\dfrac{1}{2}$

Answer: D

Concepts:geometric probability chord arc

Difficulty rating: 1840

Solution:

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 Problem 20

Answer: D

Difficulty rating: 1840

Solution:

Problem 20 in Other Years