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Math Help - Geometry of a Circle

  1. #1
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    Geometry of a Circle

    Hi, I have 3 problems relating to the geometry of a circle. Also, can someone please explain the theorem relating to 'the tangent-chord angle being equal to an angle in the alternate segment'. I don't really understand that (the alternate segment part). Thanks!

    1) In the figure, PQ is a tangent to the circle at A and OB is parallel to PQ. Find angle BAQ.


    2) In the figure, the inscribed circle of triangle ABC meets the sides BC, CA and AB at P, Q and R respectively. If angle CAB is 50 degrees and angle ABC is 84 degrees, find all the angles of triangle PQR.


    3) In the figure, O is the centre of the circle. A and B are two points of the circle such that OAB is an equilaterla triangle. OA is produced to C such that OA = AC. Find angle ABC and is CB tangent to the circle at B, giving a reason.


    Thanks for your help.
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  2. #2
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    Quote Originally Posted by BG5965 View Post
    Hi, I have 3 problems relating to the geometry of a circle. Also, can someone please explain the theorem relating to 'the tangent-chord angle being equal to an angle in the alternate segment'. I don't really understand that (the alternate segment part). Thanks!

    1) In the figure, PQ is a tangent to the circle at A and OB is parallel to PQ. Find angle BAQ.


    ...
    I've modified your sketch. Since ACBO is a square the angle \angle(QAB) = 45^\circ

    With my figure you easily can prove that the tangent-chord angle is have as large as the center angle \angle(AOB).
    Attached Thumbnails Attached Thumbnails Geometry of a Circle-sehnen_tangwinkel.png  
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