What do these figures have in common?
- the isosceles triangle
- the equilateral triangle
- the isosceles trapezoid
- the rectangle
- the kite
- the rhombus
- the square
- the circle.
Answer: they all have reflection symmetry! Being able to predict the effect of a reflection, to recognize figures that have reflection symmetry, and to make inferences about the attributes of line-symmetric figures are all key skills for students of geometry. But in order to reason about reflections, we first need to formulate a precise definition of this term. The CCSSM call for students to be able to do this in Standard G-CO.4.
Let line be given, and let be the reflection of the plane over line . Intuitively, line acts like a mirror, and the reflection over sends each point to its “mirror image” . Another way to think about reflections is to represent the plane with a sheet of paper, then crease the paper along line . Folding the paper in half should superpose onto its image .
Click here to interact with this figure in GeoGebra.
The definition below captures our intuitions about reflections in a mathematically precise way.
- If point is on line , then fixes . That is, .
- Let be a point that is not on line . In this case, takes to a point for which is the perpendicular bisector of the segment .
What kinds of inferences are available from this definition?
- If line is the perpendicular bisector of segment , then .
- A reflection interchanges pairs of points: if , then .
- If the reflection over line takes point to point , then must be the perpendicular bisector of segment .
The task of defining a reflection is described further at Illustrative Mathematics.
Homework: which of these theorems can be proved by performing a single reflection?
- Theorems about lines and angles: G-CO.9
- Theorems about triangles: G-CO.10
- Theorems about parallelograms: G-CO.11
By my count, 4 of the 15 theorems listed above have a very simple and clear link to reflections. (Although technically, they can all be expressed in terms of reflections!! But that’s a story for another day.)
Next up: let’s use reflections and line-symmetry to prove some theorems.