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Math Help - Astroid

  1. #1
    MHF Contributor alexmahone's Avatar
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    Astroid

    Refer to the attached figure.

    The ratio of the radii of the circles is 4:1. Angles x and t are shown. Prove that x=3t.
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  2. #2
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    Re: Astroid

    Could you clarify how angle x is defined? What determines the orientation of the lines that form this angle?
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  3. #3
    MHF Contributor alexmahone's Avatar
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    Re: Astroid

    Quote Originally Posted by corsica View Post
    Could you clarify how angle x is defined? What determines the orientation of the lines that form this angle?
    x is the angle between the "horizontal through the centre of the small circle" and the "line joining the centre of the small circle and the red dot". See Astroid.
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  4. #4
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    Re: Astroid

    first, note the extension of the segment forming angle t in the diagram.

    for the small circle, consider the arclength from the point of tangency of the two circles to the red dot ...

    s = 1(x+t)

    for the large circle, the arclength from the same point of tangency to (4,0) is ...

    s = 4t

    the two arclengths are the same since the small circle "rolls" inside the larger.

    4t = x+t

    x = 3t
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