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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 Sez me ... | Comment 1 of 51

To draw a chord with a pencil, you put the pencil point somewhere on the circle, then draw a line to any other point on the circle. Orient a polar coordinate system such that the origin is at the center of the circle and the first point of the chord is at zero degrees. Then the second point of the chord can be anywhere from 0 to 360 degrees, exclusive.

A chord exactly as long as the circle'd radius will intersect the circle at +/- 60 degrees. Chords with their second point within this sector will be shorter than the radius, while chords with their second point outside this sector will be longer than the radius. since the area outside the sector from -60 to +60 degrees is 2/3 of the circle, the odds are 2/3.
  Posted by Bryan on 2003-10-09 14:49:53

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