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three points are randomly chosen on a circle. what the probability that

1.triangle formed is right angled triangle.

2.triangle formed is acute angled triangle.

3.triangle formed is obtuse angled triangle.

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    What are your thoughts? What have you tried so far?2012-03-03
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    My thinking is that the probability of acute and obtuse is the same = 89/180 and that of right angle triangle is 1/90.. I correlated the problem to the hands of a clock and the probability of clock being 90 degrees apart in 60 minutes of duration - and that is 4 times in 60 minutes.. The acute and obtuse have equal probability of 89/1802012-03-03
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    The right-angled question has an immediate answer: the probability is $0$.2012-03-03
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    @Sharat Chandra: Your number suggests that you have a discrete model in mind. Perhaps you have mentally marked $360$ points, equally spaced around the circle, and are choosing from these, and only from these. Is that what you have in mind? That is perhaps not the most reasonable geometric model, since $1$ degree has no special geometric significance. Division into $360$ equal parts is "local." That's probably not what a native of Mars would choose.2012-03-03
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    @AndréNicolas Then what is the probability of acute and obtuse angle triangle ? How can probability of right angle be 0, when that is not true ?2012-03-03
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    Do you know Thales' theorem?2012-03-03
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    I'm sure you agree that if the angle of the triangle is $89^\circ$ or $91^\circ$ then it is not a right angle. But you also have to remember that it is also not a right angle if it is $90.1^\circ$, or $90.01^\circ$, or $90.000001^\circ$, and so on. Only when it is $90.000000000\!\ldots^\circ$ is it a right angle. So maybe now you can appreciate why this can only occur with vanishingly small probability.2012-03-03
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    @Sharat Chandra: The reason that the probability is $0$ has been very well explained by Rahul Narain. Here is a simpler situation. A **real** number is picked "at random" from the interval $[0,1]$. What is the probability the number is $1/\pi$? The probability that our chosen number lies in a specific interval of width $\epsilon$ is $\epsilon/1$, that is, $\epsilon$. Now think of smaller and smaller intervals around $1/\pi$.2012-03-03

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