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 m be given, and let r be the reflection of the plane over line m. Intuitively, line m acts like a mirror, and the reflection over m sends each point A to its “mirror image” A'. Another way to think about reflections is to represent the plane with a sheet of paper, then crease the paper along line m. Folding the paper in half should superpose A onto its image A'.


Click here to interact with this figure in GeoGebra.

The definition below captures our intuitions about reflections in a mathematically precise way.

  1. If point P is on line m, then r fixes P. That is, r(P) = P.
  2. Let A be a point that is not on line m. In this case, r takes A to a point A'=r(A) for which m is the perpendicular bisector of the segment \overline{AA'}.

What kinds of inferences are available from this definition?

  • If line m is the perpendicular bisector of segment \overline{PQ}, then r(P) = Q.
  • A reflection interchanges pairs of points: if r(P) = Q, then r(Q) = P.
  • If the reflection over line j takes point X to point Y, then j must be the perpendicular bisector of segment \overline{XY}.

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?

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.