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Thread: Ptolemy's Theorem and Cyclic Quadrilaterals

  1. #1
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    Ptolemy's Theorem and Cyclic Quadrilaterals

    In need of some help for 3 problems that I can't seem to understand. I would greatly appreciate some help.

    Point P on side AB of a right Triangle, ABC, is placed so that BP = PA = 2. Point Q is on hypotenuse AC such that PQ is perpendicular to AC. If CB = 3 what is the measure of BQ?

    What is the area of quadrilateral CBPQ?

    As P is translated from B to A along BA, find the range of values of BQ, where PQ remains perpendicular to CA.


    I don't understand the part where you translate P along BA at all. I can't seem to even draw the diagram correctly for the quadrilateral. I am also having a hard time drawing the diagram to even solve the first part, I just know the pyhgreoen theorem might help out. Thank you so much for the help.
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  2. #2
    Senior Member BAdhi's Avatar
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    i think the diagram comes likes this
    Attached Thumbnails Attached Thumbnails Ptolemy's Theorem and Cyclic Quadrilaterals-triangle.jpg  
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  3. #3
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    Oh thank you so much for the diagram ! This helps a lot but I'm still wondering how I can find the measure of BQ when I connect Point B to Q. When I do that is the angle PBQ a right angle ?
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  4. #4
    Senior Member BAdhi's Avatar
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    i doubt it.

    i think you were able to find the length of AQ.
    If so, since you know AB and $\displaystyle \angle PAQ$ you can use cosine equation to find BQ
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  5. #5
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    Quote Originally Posted by zzz123 View Post
    Oh thank you so much for the diagram ! This helps a lot but I'm still wondering how I can find the measure of BQ when I connect Point B to Q. When I do that is the angle PBQ a right angle ?
    1. You are dealing with similar right triangles:

    $\displaystyle \Delta(AQP) \sim \Delta(ABC)$

    and

    $\displaystyle \Delta(QFC) \sim \Delta(ABC)$

    2. The distance BQ is the hypotenuse of the right triangle $\displaystyle \Delta(BFQ)$
    Attached Thumbnails Attached Thumbnails Ptolemy's Theorem and Cyclic Quadrilaterals-simrw3eck.png  
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  6. #6
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    I'm not sure I follow. How do I find length of AQ ? Im given AB but unsure of what Angle PAQ.

    Oh seeing that they are similar helped a lot ! But now im unsure on how to find the length of QF and FC.

    Thanks again for both your help.
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