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Math Help - IGCSE Maths Geometry

  1. #1
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    IGCSE Maths Geometry

    pls help me on this;

    two circles touch externally at T. A chord of the first circle XY is produced and touches the other at Z. The chord ZT of the second circle. when produced, cuts the first circle at W. prove that angle XTW IS equal to angle YTZ
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  2. #2
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    Angles in circles

    Hello josemalawi
    Quote Originally Posted by josemalawi View Post
    pls help me on this;

    two circles touch externally at T. A chord of the first circle XY is produced and touches the other at Z. The chord ZT of the second circle. when produced, cuts the first circle at W. prove that angle XTW IS equal to angle YTZ
    Draw the common tangent at T, meeting YZ at A.

    Let angle ATZ = \theta, and angle ATY = \phi

    \therefore angle YTX = \theta + \phi

    Now AT = AZ (tangents from A to the circle)

    \therefore angle AZT = \theta (angles in isosceles triangle)

    \therefore angle XYT = 2\theta + \phi (exterior angle of triangle YTZ)

    Angle YXT = angle YTA = \phi (angle in alternate segment)

    \therefore angle XTY = 180 - (2\theta + 2\phi) (angle sum of triangle XYT)

    \therefore angle XTW = 180 - angle XTY - angle YTZ (angles on a straight line)

    = 180 - (180 -(2\theta + 2\phi)) - (\theta + \phi)

    = \theta + \phi

    = angle YTZ

    QED

    Grandad
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  3. #3
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    Pls, can somebody send me a sketch of this example (English is not my native language and I am an amateur).
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  4. #4
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    Hello x-mather
    Quote Originally Posted by x-mather View Post
    Pls, can somebody send me a sketch of this example (English is not my native language and I am an amateur).
    Please see the attachment.

    Grandad
    Attached Thumbnails Attached Thumbnails IGCSE Maths Geometry-untitled.jpg  
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