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Math Help - Quadrilateral inscribed in a circle

  1. #1
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    Quadrilateral inscribed in a circle

    Let a quadrilateral ABCD be inscribed in a circle such that AB=5, BC=3, CD=2 and Angle B=60 degrees.
    Find length AC
    Find length DA

    Okay, i found AC but i can't seem to find DA. I know im suppose to relate the 60 degrees to some other triangle but I just can't seem to find any relation. Am I missing a postulate about internal angles of a quadrilateral inscribed in a circle? XD
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  2. #2
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    Quote Originally Posted by gundanium View Post
    Let a quadrilateral ABCD be inscribed in a circle such that AB=5, BC=3, CD=2 and Angle B=60 degrees.
    Find length AC
    Find length DA

    Okay, i found AC but i can't seem to find DA. I know im suppose to relate the 60 degrees to some other triangle but I just can't seem to find any relation. Am I missing a postulate about internal angles of a quadrilateral inscribed in a circle? XD
    I have a result, but I hope for you that there is a more simple and elegant way to do the question. Here is my method:

    1. Use triangle ABC to calculate the length of the radius:
    r = \dfrac{\frac12 \overline{AC}}{\sin(60^\circ)}

    2. You'll get 4 isosceles triangles with the radius as their equal legs. Calculate the interior angles of the 3 known isosceles triangles at the center of the circle. the 4th angle add up to 360. I've got \delta \approx 73.174^\circ

    3. Calculate the length of
    \overline{AD} = 2 \cdot r \cdot \sin\left(\frac12 \delta\right)

    I've got AD = 3.
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  3. #3
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    This problem can be solved by coordinate geometry not as simple as using trig but certainly more elegant.It appears that the quadrilateral is a regular trapezoid and AB a chord close to the center of the circumscribed circle
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