The Concept of Plane Motion (Transformations)
Big Idea: A plane motion (also called a rigid transformation or isometry) is a movement of every point in a plane that preserves distances and angles. The shape and size of figures remain unchanged — only their position or orientation changes.
Formal Definition
A plane motion is a mapping $T: \mathbb{R}^2 \to \mathbb{R}^2$ such that for every pair of points $A$ and $B$:
$$d\big(T(A),\, T(B)\big) = d(A, B)$$
This means the distance between any two points is the same before and after the transformation. Because distances are preserved, so are:
- Segment lengths — sides of a figure keep their measure
- Angle measures — corners stay the same
- Parallelism — parallel lines remain parallel
- Area — the region enclosed doesn't shrink or grow