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 AA' of A(6,7)A(-6,7) under a reflection over y=4y=4.

Solution

Step 1: Extend a perpendicular line segment from AA 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: AA' is at (6,1)(-6,1).

Your turn!

Practice problem

Challenge problem

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

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

Solution

Step 1: Extend a perpendicular line segment from CC 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 1\textit 1 and -1\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: CC' is at (8,3)(-8,3).

Your turn!

Practice problem

Challenge problem

Part 2: Reflecting polygons

Let's study an example problem

Consider rectangle EFGHEFGH drawn below. Let's draw its image EFGHE'F'G'H' under a reflection over the line y=x5y=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 EE, FF, and HH were on an opposite side of the reflection line as GG. The same is true about their images, but now they switched sides!
Now we simply connect the vertices.

Your turn!

Problem 1

Problem 2