Reflecting shapes

Learn how to find the image of a given reflection.
In this article we will find the images of different shapes under different reflections.

The line of reflection

A reflection is a transformation that acts like a mirror: It swaps all pairs of points that are on exactly opposite sides of the line of reflection.
The line of reflection can be defined by an equation or by two points it passes through.

Part 1: Reflecting points

Let's study an example of reflecting over a horizontal line

We are asked to find the image $A'$ of $A(-6,7)$ under a reflection over $y=4$.

Solution

Step 1: Extend a perpendicular line segment from $A$ to the reflection line and measure it.
Since the reflection line is perfectly horizontal, a line perpendicular to it would be perfectly vertical.
Step 2: Extend the line segment in the same direction and by the same measure.
Answer: $A'$ is at $(-6,1)$.

Let's study an example of reflecting over a diagonal line

We are asked to find the image $C'$ of $C(-2,9)$ under a reflection over $y=1-x$.

Solution

Step 1: Extend a perpendicular line segment from $C$ to the reflection line and measure it.
Since the reflection line passes exactly through the diagonals of the unit squares, a line perpendicular to it should pass through the other diagonal of the unit square. In other words, lines with slopes $\textit 1$ and $\textit{-1}$ are always perpendicular.
For convenience, let's measure the distance in "diagonals":
Step 2: Extend the line segment in the same direction and by the same measure.
Answer: $C'$ is at $(-8,3)$.

Part 2: Reflecting polygons

Let's study an example problem

Consider rectangle $EFGH$ drawn below. Let's draw its image $E'F'G'H'$ under a reflection over the line $y=x-5$.

Solution

When we reflect a polygon, all we need is to perform the reflection on all of the vertices (this is similar to how we translate or rotate polygons).
Here are the original vertices and their images. Notice that $E$, $F$, and $H$ were on an opposite side of the reflection line as $G$. The same is true about their images, but now they switched sides!
Now we simply connect the vertices.