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 $A'$.

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.