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Math Help - Similarity problem, tangents

  1. #1
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    Exclamation Similarity problem, tangents

    Hi,
    So I have a problem that reads as follows:

    [PQ] is a chord of a circle. R lies on the major arc of the circle. Tangents are drawn through P and through Q. From R, perpendiculars [PA], [PB] and [PC] are drawn to the tangent at A, the tangent at B and [AB] respectively. Prove that RA.RB = RC^2.

    It's not that I don't get similarity, it's rather the phrasing of the problem...how can you have perpendiculars PA, PB and PC from R???
    Last edited by HelenaStage; August 31st 2009 at 08:33 AM.
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  2. #2
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    Quote Originally Posted by HelenaStage View Post
    Hi,
    So I have a problem that reads as follows:

    [PQ] is a chord of a circle. R lies on the major arc of the circle. Tangents are drawn through P and through Q. From R, perpendiculars [PA], [PB] and [PC] are drawn to the tangent at A, the tangent at B and [AB] respectively. Prove that RA.RB = RC^2.

    It's not that I don't get similarity, it's rather the phrasing of the problem...how can you have perpendiculars PA from R???

    You can't !

    At least not until you define where points A, B and C are.
    The problem is incomplete in its current form.
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  3. #3
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    Quote Originally Posted by bowline View Post
    You can't !

    At least not until you define where points A, B and C are.
    The problem is incomplete in its current form.
    That's all the info I get...
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  4. #4
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    Quote Originally Posted by HelenaStage View Post
    Hi,
    So I have a problem that reads as follows:

    [PQ] is a chord of a circle. R lies on the major arc of the circle. Tangents are drawn through P and through Q. From R, perpendiculars [PA], [PB] and [PC] are drawn to the tangent at A, the tangent at B and [AB] respectively. Prove that RA.RB = RC^2.

    It's not that I don't get similarity, it's rather the phrasing of the problem...how can you have perpendiculars PA from R???
    the phrasing is the problem
    A simple drawing would explain it.
    Unfortunately, I do not understand the phrasing well enough to construct a sketch.

    You are drawing tangents to A and B, therefore A & B must be on the circle. It does not so state but C can be anywhere.

    how can you have perpendiculars PA from R???
    If D is on the line PA, then line RD is perpendicular to line PA; it is a line perpendicular to PA that passes through point R.
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  5. #5
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    Quote Originally Posted by aidan View Post
    A simple drawing would explain it.
    Unfortunately, I do not understand the phrasing well enough to construct a sketch.

    You are drawing tangents to A and B, therefore A & B must be on the circle. It does not so state but C can be anywhere.


    If D is on the line PA, then line RD is perpendicular to line PA; it is a line perpendicular to PA that passes through point R.
    Yes, OK, but the problem says perpendiculars PA, PB and PC...?

    I believe point C will be the result of a line from R cutting the line joining A and B...
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  6. #6
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    Quote Originally Posted by HelenaStage View Post
    [PQ] is a chord of a circle. R lies on the major arc of the circle. Tangents are drawn through P and through Q. From R, perpendiculars [PA], [PB] and [PC] are drawn to the tangent at A, the tangent at B and [AB] respectively. Prove that RA.RB = RC^2.
    That's all the info I get...
    See the attached sketches for my initial understanding of the question.
    Attached Thumbnails Attached Thumbnails Similarity problem, tangents-mhf_0001.jpg   Similarity problem, tangents-mhf_0002.jpg   Similarity problem, tangents-mhf_0003.jpg   Similarity problem, tangents-mhf_0004.jpg   Similarity problem, tangents-mhf_0005.jpg  

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  7. #7
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    Since I'm limited to attaching 5 files
    this is a continuation of the above:

    Additional images:
    Attached Thumbnails Attached Thumbnails Similarity problem, tangents-mhf_0006.jpg  
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  8. #8
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    This is a continuation also.

    This is what I believe the question was intended to be:

    It's a simple question about the orthocenter of a triangle.
    Attached Thumbnails Attached Thumbnails Similarity problem, tangents-mhf_0007.jpg  
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