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Symmetry of a circle proof

  1. Dec 22, 2007 #1
    1. The problem statement, all variables and given/known data
    [​IMG]

    I'm trying to prove that the circle is symmetrical by showing that x² + y² = a² holds when the circle rotates.

    I know that this is proved given the following two formulae:
    x = x'cosθ - y'sinθ
    y = x'sinθ + y'cosθ

    but I don't know where those two equations have come from based on my diagram. Help!
     
  2. jcsd
  3. Dec 22, 2007 #2

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    Drop a perpendicular from where the x' axis cuts the circle to the x-axis and another perp from where the y' axis cuts the circle to the y-axis. Use some properties of similar triangles and right angled triangles.
     
  4. Dec 22, 2007 #3
    I've got the x'cosθ part of the expression for x, but I just cannot see how the -y'sinθ is found...

    [​IMG]
     
  5. Dec 22, 2007 #4

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    My mistake for giving a hasty answer. Sorry.

    Take a point P:(x,y) in the x-y system. Now draw x' and y' axes, rotated by some theta. If you drop perps from P on the x-axis and the x' axis, the first perp cuts the x-axis at a dist x from O and the 2nd perp cuts the x'-axis at a dist x' from O. Now, find x in terms of x' and y', using elementary geometry.
     
  6. Dec 22, 2007 #5

    HallsofIvy

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    Just out of curiosity, why was this posted under "Introductory Physics"?
     
  7. Dec 22, 2007 #6
    OK, i think I'm almost there,

    I have the x'cosθ term, and I know I need to minus the purple section, which I trust is y'sinθ -- but I can't seem to show that it is, lol, it's the last stumbling block

    [​IMG]
     
  8. Dec 22, 2007 #7

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    (HallsofIvy has asked you a question. I am also curious.)

    Have you drawn the diagram as I said in my 2nd post? You can show us, if possible.
     
  9. Dec 22, 2007 #8

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    EDIT: ignore
     
  10. Dec 23, 2007 #9

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    Hi raintrek,

    I'm not able to see the pictures you posted initially. Have you removed them, or is something wrong with my browser settings? Please answer asap.
     
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