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Math Help - cyclic quadrilateral

  1. #1
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    cyclic quadrilateral

    In the attached pic file,
    ABCD is a cyclic quadrilateral
    AB is parallel to CD
    DA and CB produced meet at P
    theta= angle PAB

    I need to prove the arc AD and arc BC are equal.

    I have no idea where to begin on this...my circular trig is pretty rusty, Thanks for any help or direction

    PS I've sent off an email to my instructor hoping to get a hint or two...
    Attached Thumbnails Attached Thumbnails cyclic quadrilateral-untitled.jpg  
    Last edited by solarscott; September 13th 2009 at 10:27 PM.
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  2. #2
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    Exterior angle of a cyclic quadrilateral

    Hello solarscott
    Quote Originally Posted by solarscott View Post
    In the attached pic file,
    ABCD is a cyclic quadrilateral
    AB is parallel to CD
    DA and CB produced meet at P
    theta= angle PAB

    I need to prove the arc AD and arc BC are equal.

    I have no idea where to begin on this...my circular trig is pretty rusty, Thanks for any help or direction

    PS I've sent off an email to my instructor hoping to get a hint or two...
    In addition to well-known things about isoceles triangles and parallel lines, there are two key facts you need to know:

    • Two equal arcs will subtend equal angles at the centre of the circle. And so will two chords of equal length. So if we can prove the chords AD and BC are equal in length we're there.


    • The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. So in this case \angle DCB = \theta.

    OK, so now all you need to prove is that:

    • \triangle PCD is isosceles


    • \triangle PBA is isosceles

    which you can do by finding some more angles equal to \theta. Then look at the equal length sides of these two triangles, and you should see why AD = BC.

    Can you put all this together now?

    Grandad
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  3. #3
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    Thanks heaps for this! I can put it together! You're a life saver hahaha. thanks.

    Im just finishing up my teaching degree, and this was one of the questions on the last assignment designed to test my mathematical knowledge. I havnt done maths like this in, well since I was in high school a good 10 years ago. Thanks for the help!

    Would I have to prove that equal arcs (and equal chords) subtended have equal angles at the centre? Or is that just an identity of isosceles trianges?
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  4. #4
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    Quote Originally Posted by solarscott View Post
    Thanks heaps for this! I can put it together! You're a life saver hahaha. thanks.

    Im just finishing up my teaching degree, and this was one of the questions on the last assignment designed to test my mathematical knowledge.
    [snip]
    Unfortunately it appears that this assignment question has failed in its task - all it's done is tested someone elses mathematical knowledge.

    Thread closed.
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