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So, I've got a triangle, ABC, inscribed in a circle--Thale's theorem states that it is therefore a right triangle. It is also given that $\overline{BA}$ is the diameter of the circle, and hence angle ACB is the right angle. That's all well and good.

My question is... What is the probabilty that the measure of angle CAB is less than/equal to 60 degrees?

My thinking: Since it's a right triangle, for CAB to be == 60 degrees, CBA would have to be == 30 degrees. This seems...perfectly plausible. But then what if CBA is 45 degrees...that's possible too... This is where I kind of ran into a dead end, because I couldn't figure out the probabilities of each scenario occurring. What am I missing?

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@amWhy I am given that AB is the diameter. – Roy Brewer Jun 15 '13 at 18:36
up vote 1 down vote accepted

Let's say we have our triangle $\triangle ABC$ inscribed in a circle, with $\overline{BA}$ its diameter.

It is not too hard to show that the angle labeled $\theta$ in the diagram is equal to $2\angle CAB$. This is sometimes known as the central angle theorem.

enter image description here

This means that, assuming a "random" triangle is chosen by randomly choosing the point $C$ on the top half of the circle, we are randomly choosing an angle $\theta$ between $0^\circ$ and $180^\circ$, and wanting it to come up less than or equal to $120^\circ$. The probability of this would be $\frac{2}{3}$.

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cute graphic! ;-) Good ole' drawing by hand is still in style! +1 – amWhy Jun 15 '13 at 18:48
Thanks! I am too rusty with TikZ to come up with it fast enough for math.SE answers anyway :) – Zev Chonoles Jun 15 '13 at 18:51
Awesome, thanks! – Roy Brewer Jun 15 '13 at 18:56

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