Two points on the plane \(A\) and \(B\) are given. \(|AB| = 2\). \(C\) is a randomly picked point in the circle of the radius \(R\) with the center in the midpoint of \(AB\). What is the probability that the \(\triangle ABC\) has an obtuse angle?

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Two points on the plane \(A\) and \(B\) are given. \(|AB| = 2\). \(C\) is a randomly picked point in the circle of the radius \(R\) with the center in the midpoint of \(AB\). What is the probability that the \(\triangle ABC\) has an obtuse angle?

Mathematics
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The angle will be obtuse if the point lies inside the smaller circle. So the ratio of the areas should be the answer. Sounds right?
no wait
|dw:1401618750847:dw|

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points A and B are fixed and they are 2 units apart |dw:1401618811902:dw| the center of our circle is the midpoint of segment AB
|dw:1401618885857:dw|
is the answer|dw:1401618840569:dw|
the answer is ratio of the area of the shaded region to total area.
we don't know the actual numerical radius of this circle , so we use R it could be R < 1 , in which case A , B would be outside the circle it could be R > 1 , in which case A , B are inside the circle or with R = 1 , A and Bb are on the circle
not exactly sure yet Let's try the case with R = 1 , so that A and B are on the circle . |dw:1401619069277:dw| C must be a point "in" the circle, so I interpret that to be in the interior of the circle.
|dw:1401619212117:dw|
@bswan this is obtuse |dw:1401619197813:dw|
There certainly "appears" to be an obtuse angle at C .
|dw:1401619275225:dw|
@bswan |dw:1401619255446:dw| and this is obtuse. And hence the answer I previously proposed.
It should be noted that the question is asking for an obtuse angle in the triangle. So the angles at A and B could be obtuse too. Which is possible only if C lies in the following shaded region: |dw:1401619378172:dw|
|dw:1401619217208:dw| so I would say the probability is the area of double shaded area vs total area of bigger circle
@myko we also need to include the area of the smaller circle.
it is included
bigger circle includes the small one
I meant that the double shaded area must cover the smaller circle too.
no
so, it looks like it need to be qualified with the condition that R > 1 .
yes
@myko and @Miracrown |dw:1401619632107:dw| is angle C not obtuse here?
nowyou need to calculate what is the angle in black and later integrate from negative to positive of this angle to find the area|dw:1401619688850:dw|
△ABC means the angle at vertex B @nipunmalhotra93
C "is" obtuse for that diagram, yes
@myko dude... that means Triangle ABC.
oh, never mind. You right
|dw:1401619854190:dw|
read it bad
np... that happens... :)
If you choose C to be in the black-shaded region , is where it appears that we will "not" get an obtuse angle for C . |dw:1401619882929:dw|
so then R must include the condition\(AC^2+CB^2>4\) from one side and what I said befor from the other.
<4 correction
I think you can just say that R > 1. If R <= 1 , the prob (obtuse angle) = 1 .
Very interesting problem, that was.

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