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?

- klimenkov

- katieb

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- nipunmalhotra93

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?

- nipunmalhotra93

no wait

- klimenkov

|dw:1401618750847:dw|

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## More answers

- Miracrown

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

- Miracrown

|dw:1401618885857:dw|

- nipunmalhotra93

is the answer|dw:1401618840569:dw|

- nipunmalhotra93

the answer is ratio of the area of the shaded region to total area.

- Miracrown

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

- Miracrown

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.

- Miracrown

|dw:1401619212117:dw|

- nipunmalhotra93

@bswan this is obtuse
|dw:1401619197813:dw|

- Miracrown

There certainly "appears" to be an obtuse angle at C .

- Miracrown

|dw:1401619275225:dw|

- nipunmalhotra93

@bswan |dw:1401619255446:dw| and this is obtuse. And hence the answer I previously proposed.

- nipunmalhotra93

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|

- anonymous

|dw:1401619217208:dw|
so I would say the probability is the area of double shaded area vs total area of bigger circle

- nipunmalhotra93

@myko we also need to include the area of the smaller circle.

- anonymous

it is included

- anonymous

bigger circle includes the small one

- nipunmalhotra93

I meant that the double shaded area must cover the smaller circle too.

- anonymous

no

- Miracrown

so, it looks like it need to be qualified with the condition that R > 1 .

- anonymous

yes

- nipunmalhotra93

@myko and @Miracrown |dw:1401619632107:dw| is angle C not obtuse here?

- anonymous

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|

- anonymous

â–³ABC means the angle at vertex B @nipunmalhotra93

- Miracrown

C "is" obtuse for that diagram, yes

- nipunmalhotra93

@myko dude... that means Triangle ABC.

- anonymous

oh, never mind. You right

- nipunmalhotra93

|dw:1401619854190:dw|

- anonymous

read it bad

- nipunmalhotra93

np... that happens... :)

- Miracrown

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|

- anonymous

so then R must include the condition\(AC^2+CB^2>4\) from one side and what I said befor from the other.

- anonymous

<4 correction

- Miracrown

I think you can just say that R > 1. If R <= 1 , the prob (obtuse angle) = 1 .

- Miracrown

Very interesting problem, that was.

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