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Math Help - Circle Geometry: Arc Lengths

  1. #1
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    Circle Geometry: Arc Lengths

    If two chords of a circle AB and CD meet at right angles, show that the length of the arc AC plus the arc BD is equal to half the circumference of the circle.

    Here i tried to prove that angle AOC and angle BOD = 180, where O is centre and hence arc lengths would be half the circumference. However i cannot do that.

    I am pretty sure the approach of this question is using arc lengths and angles at centre.

    However if any other approach is found that would be brilliant.
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  2. #2
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    Hello, Lukybear!

    If two chords of a circle AB and CD meet at right angles,
    show that arc AC + arc BD is equal to half the circumference of the circle.

    Let the two chords intersect at P.
    . . Then: . \angle APC = 90^o.


    By the "intersecting chords theorem", we have:

    . . \angle APC \;=\;\dfrac{\text{arc}(AC) + \text{arc}(BD)}{2} \:=\:90^o


    Therefore: . \text{arc}(AC) + \text{arc}(BD) \:=\:180^o
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