How to Translate, Rotate & Reflect Polygons

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we will study transformations of figures and show how this concept could be used to decode a message. Using examples, we will learn about translations, rotations and reflections of polygons.

Deciphering a Coded Message

It's been a fun week, and now the gang plans to meet for coffee and pastries on Friday morning. Fred doesn't know the time. Fortunately, he receives instructions in the form of a coded transformation message.

Along with this picture of colored polygons is this message:

  • purple square: translate 2 units to the left
  • green polygon: translate 4 units to the left
  • orange polygon: reflect across the y-axis
  • blue rectangle: rotate 90o clockwise about the lower left-hand corner

In this lesson, a transformation is a change to a polygon which does not alter its shape. We can help Fred with three types of transformations: translation, reflection and rotation.


The idea of a translation is moving all the points of a figure in the same direction by the same amount. It's like a shift.

The first of Fred's instructions (translate 2 units to the left) means move the purple figure over to the left by 2 units.

Same idea for the green polygon:

This time, the translation is four units to the left. The translation does not alter the shape of the figure because all of the points in the figure are moved by the same amount and in the same direction.

Right now, Fred thinks he sees a 5 or maybe an 8 in the image. The image is the resulting figure after the transformation. Wow! 5am or even 8am would be way too early for Fred. Let's continue with the transformations.


A reflection is the mirror image of a figure. We transform the figure in this way by folding it across some line. This line is called a line of symmetry. In the third instruction, Fred is asked to reflect the orange figure across the y-axis.

We imagine folding the orange polygon across the vertical y-axis to get:

Do you see the similarity between these transformations? Translation and reflection are similar because they both preserve the shape of the figure.

Do you see how these transformations are different? A translation needs the direction and the distance for the shift. A reflection needs the line of symmetry.

Can you think of a shape which would look the same whether we translated or reflected it?

How about a square? The purple square could have been reflected about a vertical line instead of translated, and the resulting image would be the same. Do you agree? What would be the vertical line? How about the line passing through the x-axis at 0.5?

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