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Math Help - Positions of points in intersecting circles (DIFFICULT)

  1. #1
    Newbie NatalieSv94's Avatar
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    Lightbulb Positions of points in intersecting circles (DIFFICULT)

    Hi! here is the task I was talking about in my introductory thread I just posted a few minutes ago. I got this task yesterday and I have no idea how to even start with it. Here is an image of the task. I read it 100 times but I still don't know what to do!
    Positions of points in intersecting circles (DIFFICULT)-screen-shot-2013-02-02-11.53.17-am.png
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    Senior Member Paze's Avatar
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    Re: Positions of points in intersecting circles (DIFFICULT)

    You should post this here: Trigonometry

    To get you started, try reading this: Intersecting Circles
    Last edited by Paze; February 2nd 2013 at 05:22 AM.
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    Super Member ILikeSerena's Avatar
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    Re: Positions of points in intersecting circles (DIFFICULT)

    Hi NatalieSv94!

    The triangle OPA is an isosceles triangle.
    Furthermore the triangle OP'A is also an isosceles triangle.
    Moreover this second triangle has a common angle with the first, meaning that these two triangles are similar.

    Can you say anything about what that means for OP'?
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  4. #4
    Newbie NatalieSv94's Avatar
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    Re: Positions of points in intersecting circles (DIFFICULT)

    I dunno . OP'/AO=AP'/PO=AO/AP ? Is that right? And then, what am I supposed to do?
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  5. #5
    Super Member ILikeSerena's Avatar
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    Re: Positions of points in intersecting circles (DIFFICULT)

    That is good!

    So suppose r=1 and OP=2, as your problem states.
    Then AO, which is the radius, is also 1.
    What is OP' then?
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  6. #6
    Newbie NatalieSv94's Avatar
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    Re: Positions of points in intersecting circles (DIFFICULT)

    It must be 1/2! Right?
    Thank you so much
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  7. #7
    Super Member ILikeSerena's Avatar
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    Re: Positions of points in intersecting circles (DIFFICULT)

    Yep! It is 1/2!

    Can you find the general statement?
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  8. #8
    Newbie NatalieSv94's Avatar
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    Re: Positions of points in intersecting circles (DIFFICULT)

    OP'=1/OP Right?
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  9. #9
    Newbie NatalieSv94's Avatar
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    Re: Positions of points in intersecting circles (DIFFICULT)

    OP'=1/OP Right?
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  10. #10
    Super Member ILikeSerena's Avatar
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    Re: Positions of points in intersecting circles (DIFFICULT)

    Right!
    That is for r=1.

    So the next challenge is, what if r=2?
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  11. #11
    Newbie NatalieSv94's Avatar
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    Re: Positions of points in intersecting circles (DIFFICULT)

    I'll find it! OMG! I'll find it! it's OP'=r2/OP !!!! Is that right?
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  12. #12
    Super Member ILikeSerena's Avatar
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    Re: Positions of points in intersecting circles (DIFFICULT)

    Yep, that is right!
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  13. #13
    Newbie NatalieSv94's Avatar
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    Re: Positions of points in intersecting circles (DIFFICULT)

    OMG! Thank you sooooooooooooooooooooo much! You're a great teacher you know!
    In the last part when it says Discuss the scope and/or limitations of the general statement. I know that OP' approaches infinity as r approaches infinity and OP' approaches 0 as OP approaches infinity. Is that all? Or is there any situation in which OP' is not defined or the general statement is not working? (I know that it's not defined if OP is 0, but ...)
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  14. #14
    Super Member ILikeSerena's Avatar
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    Re: Positions of points in intersecting circles (DIFFICULT)

    Thanks!

    The only real limitation is that OP may not be zero, so the point P may not be chosen to coincide with the point O.

    EDIT: Hold on! See below.
    Last edited by ILikeSerena; February 3rd 2013 at 03:16 AM.
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  15. #15
    Newbie NatalieSv94's Avatar
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    Re: Positions of points in intersecting circles (DIFFICULT)

    Thanks again.
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