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Strike a Chord (..Any Chord) (Posted on 2003-10-09) Difficulty: 4 of 5
What is the probability that a randomly drawn chord will be longer than the radius of the circle?

Prove it.

No Solution Yet Submitted by DJ    
Rating: 4.5263 (19 votes)

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Solution Another solution | Comment 4 of 51 |
Take an arbitrary point on the circle. Then the problem is equivalent to asking: if we pick another point on the circle randomly, what is the probability that the line from the original point to the randomly chosen point is longer than the radius? If we divide the circle into six equal arcs (each determined by a 60 degree angle situated at the centre of the circle) such that two of the arcs meet at the original point, then the line in question will be less than the length of the radius if and only if the randomly chosen point lies on one of the two arcs which have the original point as one of their endpoints. Thus, the probability in question reduces to asking what the probability is that a point chosen randomly from one of six different regions of equal measure will not fall into one of two particular such regions. The answer is 4/6, or 2/3.
Edited on October 9, 2003, 3:37 pm
  Posted by RoyCook on 2003-10-09 15:36:28
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