Math101 · Independent Ontario learning library
Math101Transformations
A geometric transformation maps each point of a figure to an image point.
Transformations unify congruence, similarity, symmetry, coordinate rules, animation, and design. Invariants reveal what a mapping changes and preserves.
Intuition and core definition
A geometric transformation maps each point of a figure to an image point. Translations, rotations, and reflections are rigid motions preserving distance and angle; dilations preserve angle and proportional shape but scale length. Composition applies transformations in a specified order.
Notation, language, and conditions
$T(P)=P\prime$ names an image. An isometry preserves distance. Orientation is preserved by translations and rotations, reversed by reflections, and usually preserved by positive dilations. Composition $T_2\circ T_1$ means apply $T_1$ first, then $T_2$.
Why this idea matters
Transformations describe geometric change through precise rules, separating rigid motions from size-changing dilations.
A dependable method
- Identify invariant features: lengths, angles, parallelism, orientation, and scale.
- Use a point-image pair to infer displacement, centre/angle, mirror line, or scale factor.
- Apply the rule to every vertex in order.
- For a composition, complete and record each intermediate image.
- Check invariants and apply an inverse transformation when possible.
