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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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re(3): Different Approach (Continued) | Comment 32 of 51 |
(In reply to re(2): Different Approach (Continued) by DJ)

when we are dealing with infinite something (chords, points etc.), normal rules just don't hold true any more. Here is a good example : given two fragments AB and CD (say length of AB larger than CD) , comparing the number of points on AB and CD . Now we know the total points number on AB and CD are both infinte, there seems no way to compare two infinte numbers. But an intuitive answer will be N ab is more than N cd because AB is longer than CD. But is that right? The answer is no. Now I will prove the number of points on AB is exact same as that on CD.

a ---------------------------b
A \ | C
\ |
\ |
B \ | D
c--------------------------d

To prove this, we might think about a reasoning first: If we can prove for any points on AB, we can found a counter part on CD, vice verser.then we can say the number of points on AB is exactly the same as that on CD. Now put these two fragments between two pararel lines,ab and cd ( see the picture above) , pick a random points on A, we call it E, make a parallel line start from E and cross CD at F, then F is the 1to1 correspondent point for E. Using the same way we will find that for all the points on AB (infinite) , we can find a counter part on CD, vice verser. So we can conclude the number of points on AB is same as to that on CD, although AB is longer than CD, and the numbers are infinite.

Finally, using this theory , we can prove that the chance of chord longer than
radius is 50%. If you are using other methods that can only apply to finite numbers, you will generate different answers depend on which method you use. But I think the truth is unique! hehe. good question, good luck guys
  Posted by Jonathan shao on 2003-10-22 15:43:33

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