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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 A revised solution: | Comment 36 of 51 |
(In reply to I'm no mathemetician: by Benjamin J. Ladd)

Every chord can be represented by only it's midpoint (the radius always perpendicularly bisects a chord). If we take any chord whose length is equal to that of the radius and use its midpoint to draw an inner circle, we will have divided the areas of the circle into possible places for chords to fall. If a chord is selected at random, it's midpoint will either lie in the region whose chord is larger than, smaller to, or equal to the radius.

For a circle of radius r, the radius of the inner circle will be 1/2r√3. To compare the probability of all chords produced that are greater than the radius, we can divide the area of the inner circle by the area of the entire circle. The equation for this is:

pi*(1/2*r*√3)²/pir²

And the final result after the pi*r² is cancelled is 3/4. Therefore there is a 75% chance that a randomly drawn chord will be longer than the radius of the circle.
  Posted by Benjamin J. Ladd on 2003-11-16 13:10:01

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