Describing Reflections

 When things are placed in front of a mirror, we see a reflection.  In geometry, a shape in the plane can be reflected across a line of reflection or a mirror in the plane to give an image.  In this activity you will learn properties of reflections.

Students should be placed into groups of three or four and the following materials should be provided for each group:

• 1 Mirror

• 1 Plane Shaped Polygon

Begin the exercise by asking students to place the plane shaped polygon in front of the mirror and then describe what they see. Students should express the reflection (the image) as appearing to be the plane-shaped-polygon (the pre-image) that was flipped over the mirror.  Some may go on to explain that the image appears to be at a distance behind the mirror equal to the pre-image’s distance from the mirror.  With this preliminary understanding of a reflection, the following questions should be posed and further guided explorations on The Geometers Sketchpad should be conducted to help answer these questions.

1. What information must be given to describe a reflection?

2. How can you describe the process of reflecting an object or shape?

The first step is to understand how a point is reflected.  Click here for a GSP activity

Upon completion of this exercise, have students write down the most important things that they learned while conducting this investigation.  They will build upon this exercise to establish a definition of the reflection of a plane shaped polygon within the plane.  Click here for a GSP activity

At this point students should have enough experience to enable them to answer the previously posed questions.

1. What information must be given to describe a reflection?

2. How can you describe the process of reflecting a plane shaped object or shape?

Once the students have come up with their conclusions, discuss the definitions amongst the class and finally present and compare with the following formal definition.

A transformation of a plane is a mapping from the plane to itself such that

a.  no two points have the same image

b.  each point in the plane is the image of a point in the plane

If g is a line in a plane, a reflection with axis g is a transformation of the plane where each point P has image P’ such that

a.      if  P is not on g, then g is the perpendicular bisector of PP

b.       If P is on g, then P is a fixed point and P=P’.

This information provides a base that is uses full in determining optimal angle settings for department store three-paneled mirrors.