# 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
${A}^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}$
.

This translation can be described in coordinate notation as $(x,y)\to (x-5,y+3)$ .

## Rotations

A second type of transformation is the
**
rotation
**
. The figure below shows triangle
$ABC$
rotated
$90\xb0$
clockwise about the origin.

This rotation can be described in coordinate notation as $(x,y)\to (y,-x)$ . (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
$ABC$
reflected across the line
$y=x+2$
.

This reflection can be described in coordinate notation as $(x,y)\to (y-2,x+2)$ . (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.

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

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