Math Help Forum: 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 (My◦Mx)◦Mx therefore after re associating I ge the identity and then My. Correct?

Help is greatly appreciated.

Hello RoboMyster5

Originally Posted by RoboMyster5

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 (My◦Mx)◦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