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Math Help - Isometries

  1. #1
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    Question Isometries

    Hey wall, I'm in collge geometry and my professor gave us practice problems before the final and I have some lingering questions regarding isometries. I'm in abstract algebra so I know that isometries are a groupd under composition.

    1. Could a glide reflection followed by a rotation be a reflection?
    2. Let Mx denote reflection in the x axis and let My dwnote reflection in the y axis. Show that Ro(180)Mx=My

    1. My instinct is to say no because a glide reflection translates a point and then reflects, so if you rotate that point is still translated and cannot be a reflection.

    2. By factoring Ro(180) into two reflections I get (MyMx)◦Mx therefore after re associating I ge the identity and then My. Correct?

    Help is greatly appreciated.

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  2. #2
    MHF Contributor
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    Hello RoboMyster5
    Quote Originally Posted by RoboMyster5 View Post
    Hey wall, I'm in collge geometry and my professor gave us practice problems before the final and I have some lingering questions regarding isometries. I'm in abstract algebra so I know that isometries are a groupd under composition.

    1. Could a glide reflection followed by a rotation be a reflection?
    2. Let Mx denote reflection in the x axis and let My dwnote reflection in the y axis. Show that Ro(180)Mx=My

    1. My instinct is to say no because a glide reflection translates a point and then reflects, so if you rotate that point is still translated and cannot be a reflection.

    2. By factoring Ro(180) into two reflections I get (MyMx)◦Mx therefore after re associating I ge the identity and then My. Correct?

    Help is greatly appreciated.

    Your proof for #2 is fine.

    But the answer to #1 is that, yes, it could be a reflection. Look at the attached diagram. I've given triangle #1 a glide reflection, to produce #2. Then a rotation to produce #3. The result is a reflection in the mirror-line shown. I'll leave you to verify that every point on the mirror-line is mapped to itself under these two isometries.

    Grandad
    Attached Thumbnails Attached Thumbnails Isometries-untitled.jpg  
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