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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 Solution | Comment 15 of 51 |
Without loss of generality, choose any point on the circumference of the circle as the starting poinrt for the chord (Q), Draw a chord equal to the length of the radius from this point in each direction - these join the circumference at points P & R. Any chord from Q that joins the circumference on the larger arc PQ of the circle is longer than the radius; any that joins the circumference on the smaller arc PQ is equal to or shorter than the radius. We know that we can draw exactly 6 consecutive arcs equal to the length of the radius around the circumference of the circle; the smaller arc, PQ, comprises 2 of them. Hence the probability of choosing a point on that arc is 1/3. Thw probability of choosing another point on the circumference, giving a chord longer than the radius, is thus 2/3
  Posted by DrBob on 2003-10-10 03:31:11
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