Transformations

There are three kinds of isometric transformations of 2-dimensional shapes: translations, rotations, and reflections. (Isometric means that the transformation doesn't change the size or shape of the figure.) A fourth type of transformation, a dilation, is not isometric: it preserves the shape of the figure but not its size.

Translations

A translation is a sliding of a figure. For example, in the figure below, triangle ABC is translated 5 units to the left and 3 units up to get the image triangle A'B'C'.

translation of a triangle 5 units left and 3 units up

This translation can be described in coordinate notation as .

Rotations

A second type of transformation is the rotation. The figure below shows triangle ABC rotated 90° clockwise about the origin.

This rotation can be described in coordinate notation as . (You can check that this works by plugging in the coordinates (x, y) of each vertex.)

Reflections

A third type of transformation is the reflection. The figure below shows triangle ABC reflected across the line y = x + 2.

This reflection can be described in coordinate notation as . (Again, you can check this by plugging in the coordinates of each vertex.)

Dilations

A dilation is a transformation which preserves the shape and orientation of the figure, but changes its size. The scale factor of a dilation is the factor by which each linear measure of the figure (for example, a side length) is multiplied.

The figure below shows a dilation with scale factor 2, centered at the origin.

dilation with scale factor 2

This dilation can be described in coordinate notation as . (Again, you can check this by plugging in the coordinates of each vertex.)