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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 A B C 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 ( x , y ) ( x 5 , y + 3 ) .

Rotations

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

Math diagram

This rotation can be described in coordinate notation as ( x , y ) ( y , x ) . (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 A B C reflected across the line y = x + 2 .

Math diagram

This reflection can be described in coordinate notation as ( x , y ) ( y 2 , x + 2 ) . (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 ( x , y ) ( 2 x , 2 y ) . (Again, you can check this by plugging in the coordinates of each vertex.)

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