Transformations
A figure is slid, flipped or turned. Exactly where does each point go, and what about the figure is still the same afterwards?
Where you are 11 inside
What this area is about
A transformation sends every point P of the plane to exactly one image point P'. Slides, flips and turns are the rigid motions: they move a figure without changing any length or angle. That turns “the same shape and the same size” into something you can check, and it explains symmetry.
How these fit together
The three motions. Translations, reflections and rotations say where each point goes. Rotating about a point other than the origin is the procedure for a turn whose center is not (0, 0).
Combining them. Composing transformations does one motion after another, and order can matter. Two reflections in a row give a translation when the mirrors are parallel and a rotation when they cross.
What they keep. What rigid motions preserve lists lengths, angle measures, betweenness and parallel lines. Rigid motions defined precisely proves those facts. Rigid motions and congruence defines congruent figures by them.
Symmetry. Line symmetry and rotational and point symmetry are a figure mapped onto itself by a reflection, or by a turn strictly between 0^\circ and 360^\circ.
A path through it
Translations, then reflections, then rotations. After rotations come rotating about another point, composing, and what rigid motions preserve. Composing comes before congruence and two reflections; what rigid motions preserve comes before the precise definitions. The two symmetry lessons have no fixed place.
Next: similarity transformations and transformational proof.