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klimenkov

  • 6 months 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?

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  1. nipunmalhotra93
    • 6 months ago
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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?

  2. nipunmalhotra93
    • 6 months ago
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    no wait

  3. klimenkov
    • 6 months ago
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    |dw:1401618750847:dw|

  4. Miracrown
    • 6 months ago
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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

  5. Miracrown
    • 6 months ago
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    |dw:1401618885857:dw|

  6. nipunmalhotra93
    • 6 months ago
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    is the answer|dw:1401618840569:dw|

  7. nipunmalhotra93
    • 6 months ago
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    the answer is ratio of the area of the shaded region to total area.

  8. Miracrown
    • 6 months ago
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    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
    • 6 months ago
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    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
    • 6 months ago
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    |dw:1401619212117:dw|

  11. nipunmalhotra93
    • 6 months ago
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    @bswan this is obtuse |dw:1401619197813:dw|

  12. Miracrown
    • 6 months ago
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    There certainly "appears" to be an obtuse angle at C .

  13. Miracrown
    • 6 months ago
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    |dw:1401619275225:dw|

  14. nipunmalhotra93
    • 6 months ago
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    @bswan |dw:1401619255446:dw| and this is obtuse. And hence the answer I previously proposed.

  15. nipunmalhotra93
    • 6 months ago
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    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
    • 6 months ago
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    |dw:1401619217208:dw| so I would say the probability is the area of double shaded area vs total area of bigger circle

  17. nipunmalhotra93
    • 6 months ago
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    @myko we also need to include the area of the smaller circle.

  18. myko
    • 6 months ago
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    it is included

  19. myko
    • 6 months ago
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    bigger circle includes the small one

  20. nipunmalhotra93
    • 6 months ago
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    I meant that the double shaded area must cover the smaller circle too.

  21. myko
    • 6 months ago
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    no

  22. Miracrown
    • 6 months ago
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    so, it looks like it need to be qualified with the condition that R > 1 .

  23. myko
    • 6 months ago
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    yes

  24. nipunmalhotra93
    • 6 months ago
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    @myko and @Miracrown |dw:1401619632107:dw| is angle C not obtuse here?

  25. myko
    • 6 months ago
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    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
    • 6 months ago
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    △ABC means the angle at vertex B @nipunmalhotra93

  27. Miracrown
    • 6 months ago
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    C "is" obtuse for that diagram, yes

  28. nipunmalhotra93
    • 6 months ago
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    @myko dude... that means Triangle ABC.

  29. myko
    • 6 months ago
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    oh, never mind. You right

  30. nipunmalhotra93
    • 6 months ago
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    |dw:1401619854190:dw|

  31. myko
    • 6 months ago
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    read it bad

  32. nipunmalhotra93
    • 6 months ago
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    np... that happens... :)

  33. Miracrown
    • 6 months ago
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    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
    • 6 months ago
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    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
    • 6 months ago
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    <4 correction

  36. Miracrown
    • 6 months ago
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    I think you can just say that R > 1. If R <= 1 , the prob (obtuse angle) = 1 .

  37. Miracrown
    • 6 months ago
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    Very interesting problem, that was.

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