Math101 · Independent Ontario learning library
Math101learn.math101.caTransformations
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.
Worked example
Representations and interpretation
Arrow diagrams show input, intermediate image, and final image. Coordinate rules provide exact mappings, while tracing paper displays congruence or scale visually.
Reasoning about variations
Two reflections across parallel lines compose to a translation; across intersecting lines they compose to a rotation. These results show that different sequences can produce a familiar single transformation.
Common mistakes
How to check your work
- Compare corresponding lengths and angles according to the transformation type.
- Track orientation before and after.
- Apply inverses in reverse order and recover the original.
Practice
- Translate $(2,-1)$ by $(-5,4)$.
- Which basic transformation reverses orientation?
- Does a dilation with $k=2$ preserve length?
Answers and brief solutions
Show answers
- $(-3,3)$ Add the vector componentwise.
- Reflection Translations and rotations preserve orientation.
- No It doubles lengths while preserving angles.
Synthesis and transfer
A sequence of moves on a logo can be tracked vertex by vertex; reversing the sequence in reverse order should recover the original figure when every step is invertible.
A sequence matters: reflecting a triangle across the $y$-axis and then translating right usually differs from translating first and reflecting afterward. Coordinate tracking makes the noncommutativity visible at a single test vertex. Rigid motions preserve all distances and angles, so their composition is another rigid motion; including a dilation preserves shape but changes size. To undo a composition, apply inverse transformations in reverse order, just as inverse operations reverse an algebraic process. Invariants such as orientation, parallelism, and scale factor help classify an unknown sequence without reconstructing every intermediate point.
Related topics
Teaching and accessibility note
Explore the idea
Vector and matrix transform
Change one quantity at a time and connect what moves to Transformations.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Translate $(2,-1)$ by $(-5,4)$.
- Add the vector componentwise.
End of lesson
Nice work making it this far.
Understanding grows through return visits. Save this lesson, try the practice, or continue when you are ready.
