Transformations

Compositions of Transformations

Describing a sequence of rigid motions that maps one figure onto another.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Translations, Reflections & Rotations

Chaining Transformations Together

A figure is translated, and then the result is reflected. Before reading on, think about this: since both individual steps preserve every distance and angle, must the combined two-step result also preserve them — or could two isometries chained together somehow produce something that isn't an isometry?

Definition — Composition of Transformations

Applying two or more transformations in sequence, where the output of the first becomes the input of the next. A composition of rigid motions is itself always a rigid motion.

This must be true, and the reasoning is straightforward: the first isometry preserves every distance in the figure exactly, producing an intermediate image that's already fully congruent to the original. The second isometry then acts on that intermediate image, and since it's also an isometry, it preserves every distance in that image too — including the distances that were already preserved from the very first step. Distance preservation carries through every stage of the chain, no matter how many transformations are composed.

Worked Example — Composing a Translation and a Reflection

Apply a translation by (2, 0) to the point (1, 3), then reflect the result over the x-axis. First, translate: (1 + 2, 3) = (3, 3). Then reflect over the x-axis (negate y): (3, −3). The final image is (3, −3), reached through two isometries applied in sequence.

Worked Example — Describing a Sequence That Maps One Figure onto Another

Triangle A has vertices (0, 0), (2, 0), (0, 3). Triangle B has vertices (5, 0), (5, 2), (2, 0). Describe a sequence of rigid motions mapping A onto B. A 90° clockwise rotation about the origin sends (x, y) to (y, −x), mapping A's vertices to (0, 0), (0, −2), (3, 0) — not quite matching B yet. Following that with a translation by (5, 2) maps those points to (5, 2), (5, 0), (8, 2) — testing and adjusting the sequence this way (rotation, then translation) is the general strategy: find a rigid motion aligning one key feature, then compose additional rigid motions until every vertex matches.

Geometry Canvas

Point mode (click to add a point)LabelsMeasurementsSnap to grid

Construct

Tool

Objects

1.
Color
P1
P1: a free point, draggable on the plane
2.
Color
P2
P2: a free point, draggable on the plane
3.
Color
P3
P3: a free point, draggable on the plane
4.
Color
poly1
poly1: the polygon through P1, P2, P3

Measurements

poly1area = 15perimeter = 17.66

Tip

When searching for a sequence of transformations mapping one figure onto another, start by matching a single distinctive point or angle first, then adjust with additional transformations until the entire figure aligns — rather than trying to guess the whole sequence at once.

Common Mistakes

Applying the transformations in the wrong order, such as reflecting first and translating second when the problem specifies the opposite order.
A composition's order matters — apply each transformation in the exact sequence given, since translating then reflecting can produce a different result than reflecting then translating.
Assuming any single rigid motion, on its own, must map one figure onto any other congruent figure.
Two congruent figures are always related by some composition of rigid motions, but that composition often requires more than one transformation — don't stop searching after the first transformation fails to complete the mapping.

Key Takeaways

A composition of rigid motions is itself always a rigid motion, since distance preservation carries through each stage of the sequence.
The order in which transformations are composed generally matters and changes the final result.
Describing a sequence of transformations that maps one figure onto another is a strategy for proving two figures are congruent.

Summary

Compositions of rigid motions stay rigid motions, a fact used constantly to establish congruence between figures. The next unit formalizes exactly what it means for two triangles to be congruent, and how to prove it.

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