Mathematics: Geometry - Transformations

Rotation
A rotation is a transformation that turns a figure around a point called the center of rotation, which can lie anywhere in the plane. If a line is drawn from a point on a figure to the center of rotation, and another line is drawn from the center to the rotated image of that point, the angle between the two lines is the angle of rotation.

The vertex of the angle of rotation is the center of rotation.


Translation and Dilation
A translation is a transformation which slides a figure from one position in the plane to another position in the plane. The original figure and the translated figure have the same size, shape, and orientation. A dilation is a transformation which proportionally stretches or shrinks a figure by a scale factor. The dilated image is the same shape and orientation as the original image but a different size. A polygon and its dilated image are similar.
 

Dilation



Reflection
A reflection of a figure over a line (a 'flip') creates a congruent image that is the same distance from the line as the original figure but on the opposite side. The line of reflection is the perpendicular bisector of any line segment drawn from a point on the original figure to its reflected image (unless the point and its reflected image happen to be the same point, which happens when a figure is reflected over one of its own sides).

A reflection of a figure over a point (an inversion) in two dimensions is the same as the rotation of the figure 180° about that point. The image of the figure is congruent to the original figure.
The point of reflection is the midpoint of a line segment which connects a point in the figure to its image (unless the point and its reflected image happen to be the same point, which happens when a figure is reflected in one of its own points).

Reflection of a figure over a line
Reflection of a figure over a point