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Math Help - Trigonometry, Circles, Chord length, Area of Segment problem

  1. #1
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    Trigonometry, Circles, Chord length, Area of Segment problem



    Ok basically i can figure out this equation from this info which is:

    8 x (sin a/2) = 6 x (sin b/2)

    I got the above equations by using the chord length formula as both circles share a similar chord length.

    It is pretty much two simultaneous equations subbed into each other.
    not sure how to go about solving this though
    All help is appreciated
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  2. #2
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    Quote Originally Posted by jibba View Post


    Ok basically i can figure out this equation from this info which is:

    8 x (sin a/2) = 6 x (sin b/2)

    I got the above equations by using the chord length formula as both circles share a similar chord length.

    It is pretty much two simultaneous equations subbed into each other.
    not sure how to go about solving this though
    All help is appreciated
    Hi jibba,

    you still need the 2 angles "a" and "b", but then you only have an equality.
    That equation doesn't give you either angle.

    There is an easy way to solve this, since the triangle from
    one point of intersection of the circles to both centres has sides 3, 4 and 5.
    This is right-angled since

    3^2+4^2=5^2

    Then, the angle "a" you mentioned is sin^{-1}\frac{3}{5}=cos^{-1}\frac{4}{5}=tan^{-1}\frac{3}{4}

    Also "b" is sin^{-1}\frac{4}{5} etc.

    Now draw the chord and work with the back to back right-angled triangles.

    Now that you have the angles, you can calculate the overlapping area
    by subtracting the triangle within the circle from the sector,
    doing that for both circles and adding the two results.
    Then double that to include the part underneath the line joining the circle centres.
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