## klimenkov one year ago 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?

1. 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?

2. nipunmalhotra93

no wait

3. klimenkov

|dw:1401618750847:dw|

4. 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

5. Miracrown

|dw:1401618885857:dw|

6. nipunmalhotra93

7. nipunmalhotra93

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

8. 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

9. 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.

10. Miracrown

|dw:1401619212117:dw|

11. nipunmalhotra93

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

12. Miracrown

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

13. Miracrown

|dw:1401619275225:dw|

14. nipunmalhotra93

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

15. 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|

16. myko

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

17. nipunmalhotra93

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

18. myko

it is included

19. myko

bigger circle includes the small one

20. nipunmalhotra93

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

21. myko

no

22. Miracrown

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

23. myko

yes

24. nipunmalhotra93

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

25. myko

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|

26. myko

△ABC means the angle at vertex B @nipunmalhotra93

27. Miracrown

C "is" obtuse for that diagram, yes

28. nipunmalhotra93

@myko dude... that means Triangle ABC.

29. myko

oh, never mind. You right

30. nipunmalhotra93

|dw:1401619854190:dw|

31. myko

32. nipunmalhotra93

np... that happens... :)

33. 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|

34. myko

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

35. myko

<4 correction

36. Miracrown

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

37. Miracrown

Very interesting problem, that was.

Find more explanations on OpenStudy