A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

klimenkov

  • 2 years 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?

  • This Question is Closed
  1. nipunmalhotra93
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    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
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    no wait

  3. klimenkov
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1401618750847:dw|

  4. Miracrown
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    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
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    |dw:1401618885857:dw|

  6. nipunmalhotra93
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    is the answer|dw:1401618840569:dw|

  7. nipunmalhotra93
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  8. Miracrown
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    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
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    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
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    |dw:1401619212117:dw|

  11. nipunmalhotra93
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  12. Miracrown
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

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

  13. Miracrown
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    |dw:1401619275225:dw|

  14. nipunmalhotra93
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  15. nipunmalhotra93
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    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. anonymous
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  17. nipunmalhotra93
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  18. anonymous
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    it is included

  19. anonymous
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    bigger circle includes the small one

  20. nipunmalhotra93
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  21. anonymous
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    no

  22. Miracrown
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

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

  23. anonymous
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes

  24. nipunmalhotra93
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  25. anonymous
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    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. anonymous
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    △ABC means the angle at vertex B @nipunmalhotra93

  27. Miracrown
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    C "is" obtuse for that diagram, yes

  28. nipunmalhotra93
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    @myko dude... that means Triangle ABC.

  29. anonymous
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh, never mind. You right

  30. nipunmalhotra93
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1401619854190:dw|

  31. anonymous
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    read it bad

  32. nipunmalhotra93
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    np... that happens... :)

  33. Miracrown
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    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. anonymous
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  35. anonymous
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    <4 correction

  36. Miracrown
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

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

  37. Miracrown
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    Very interesting problem, that was.

  38. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.