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Math Help - Identifying translations and rotations

  1. #1
    Senior Member Pinkk's Avatar
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    Identifying translations and rotations

    Given the following diagram (note that the intersection point of the segment that connects triangles XYB and ABC on the segment AB should be point I), where the triangles are all equilateral, and the points H, I, and J bisect the segments AC, AB, and XB, respectively:



    Identity the following translations and rotations:

    * R_{A, \frac{\pi}{2}}  \circ R_{B, \frac{\pi}{2}}
    * R_{C, \frac{\pi}{2}}  \circ R_{A, \frac{\pi}{2}}
    * R_{A, \frac{\pi}{2}}  \circ T_{CA}
    * R_{A, \frac{\pi}{3}}  \circ R_{B, \frac{\pi}{3}}
    * R_{A, \frac{\pi}{2}}  \circ R_{B, \frac{\pi}{6}}
    * R_{A, \frac{3\pi}{4}}  \circ R_{B, \frac{\pi}{4}}

    I am absolutely confused and lost in the modern geometry class and have no idea how to approach these.

    Edit: Is R_{A, \frac{\pi}{2}}  \circ R_{B, \frac{\pi}{2}} = R_{J, \frac{4\pi}{3}} ?
    Last edited by Pinkk; February 11th 2010 at 08:16 PM.
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  2. #2
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    Quote Originally Posted by Pinkk View Post
    Given the following diagram (note that the intersection point of the segment that connects triangles XYB and ABC on the segment AB should be point I), where the triangles are all equilateral, and the points H, I, and J bisect the segments AC, AB, and XB, respectively:



    Identity the following translations and rotations:

    * R_{A, \frac{\pi}{2}}  \circ R_{B, \frac{\pi}{2}}
    * R_{C, \frac{\pi}{2}}  \circ R_{A, \frac{\pi}{2}}
    * R_{A, \frac{\pi}{2}}  \circ T_{CA}
    * R_{A, \frac{\pi}{3}}  \circ R_{B, \frac{\pi}{3}}
    * R_{A, \frac{\pi}{2}}  \circ R_{B, \frac{\pi}{6}}
    * R_{A, \frac{3\pi}{4}}  \circ R_{B, \frac{\pi}{4}}

    I am absolutely confused and lost in the modern geometry class and have no idea how to approach these.

    Edit: Is R_{A, \frac{\pi}{2}}  \circ R_{B, \frac{\pi}{2}} = R_{J, \frac{4\pi}{3}} ?
    No, why in the world would you think so? You cannot possible get a rotation of 4\pi/3 by 2 rotations of \pi/2. If you wrote around point B \pi/2 radians the vertical line BH becomes a horizontal line, extending what is originally XB. Then rotating around the new position of A sends C into the original position of B. I'm not sure what is meant by "identifying" these but R_{A,\frac{\pi}{2}} \circ R_{B,\frac{\pi}{2}} is a rotation around B by \pi followed by a translation from B to C.
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  3. #3
    Senior Member Pinkk's Avatar
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    From the way the book seems to describe, all the rotations are around the original points A,B, and C, if even they have already been translated. And by identity, they seem to always be able to reduce it to only one translation or only one rotation.

    But don't worry, I'm fairly certain I'm dropping this class so it doesn't matter.
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  4. #4
    Senior Member Pinkk's Avatar
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    Proof of rotations

    Nevermind.
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  5. #5
    Senior Member Pinkk's Avatar
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    Proof regarding rotations/reflections

    Double posted by mistake.
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