Reflection and Rotation

Date: 02/20/2003 at 10:57:36
From: Anita Bechtold
Subject: Reflection and rotation 

I am a math teacher. I have a strong math background but not for these 
topics. Can rotation of a figure and reflection of that same figure 
yield the same result at times? If a shape with one line of symmetry, 
such as a club (from playing cards), is reflected across a line, this 
same image could be obtained by rotating the shape, right? 

Date: 02/20/2003 at 12:30:45
From: Doctor Peterson
Subject: Re: Reflection and rotation 

Hi, Anita.

I think you are talking about something like this:

 

Here I drew a T and an L, and reflected both across a line. The image 
of the L clearly can be obtained only by reflection; but the T image 
can also be obtained by rotation about a point on the line, as shown.

The reason for this is that any rotation or translation is equivalent 
to two reflections; when the object has a reflection symmetry, a 
single reflection produces the same effect as that reflection 
followed by a reflection in its line of symmetry, and so is 
equivalent to a rotation; specifically, it will be a rotation about 
the point of intersection of the line in which it was reflected, and 
the line of symmetry (of both the original object and the image).

So if a problem asks for any transformation that will take the first 
T into the other, you can use either the reflection or the rotation. 
Of course, if the corresponding points on the original and the image 
were marked, that would force you to choose either the reflection or 
the rotation.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 02/20/2003 at 14:55:48
From: Anita Bechtold
Subject: Thank you (reflection and rotation )

Wow! That was a prompt response!  Thank you so 
much.  I especially like the illustration.  Yes, that's 
exactly what I meant.  You were very helpful.