Lesson 4.2.5

Identifying Sequences of Rigid Motions

Given a pre-image and an image, can you reverse-engineer the sequence of rigid motions that connects them? This capstone lesson combines all transformation skills into a single analysis framework.

Introduction

This is the detective work of transformations. You see the “before” and “after” — now figure out what happened. By analyzing orientation, position, and coordinate changes, you can identify the exact rigid motion(s) that were applied.

Past Knowledge

All transformations (4.1). Compositions (4.2.2). Glide reflections (4.2.3). Congruence (4.2.4).

Today's Goal

Given pre-image and image, identify the type and parameters of the rigid motion(s) applied.

Future Success

Triangle congruence proofs (Unit 5) and coordinate geometry proofs.

Key Concepts

Decision Flowchart

Check orientation. Is the vertex order (clockwise vs counterclockwise) preserved?
- Preserved → Translation or Rotation
- Reversed → Reflection or Glide Reflection
If orientation preserved:
- All points shifted by the same vector → Translation
- Points moved different directions → Rotation (find the center)
If orientation reversed:
- Midpoints of corresponding segments are collinear on one line → Reflection
- Midpoints are not collinear on a single perpendicular bisector → Glide Reflection

Finding the Parameters

Translation vector: for any corresponding pair.
Reflection line: Perpendicular bisector of .
Rotation: The center is equidistant from each and pair; find it via perpendicular bisector intersection.

Worked Examples

Basic

Identifying a Translation

, , . Identify the transformation.

Step 1 — Check orientation: Original order A→B→C is counterclockwise (or clockwise). Same order in image? A'→B'→C' maintains the same relative ordering. Orientation preserved.

Step 2 — Check displacement:

All three have the same displacement vector.

Translation by .

Vector

⟨,⟩

Adjust the vector to explore

A(2,5)→A′(-1,2)

B(4,3)→B′(1,0)

C(6,5)→C′(3,2)

Intermediate

Identifying a Rotation

, . Identify the transformation.

Step 1 — Check displacement: , . Different vectors → not a translation.

Step 2 — Check rotation rule: is 90° CCW about origin.

✓, ✓

90° counterclockwise rotation about the origin.

Rotateabout(,)

P(3,1)→P′(-1,3)

Q(5,2)→Q′(-2,5)

Advanced

Identifying a Glide Reflection Sequence

with , , maps to with , , . Identify the sequence.

Step 1 — Orientation: The y-coordinates all flip sign → orientation is reversed.

Step 2 — Is it a simple reflection? Across x-axis would give for , but we need . There's a horizontal shift of +4.

Step 3 — Decompose: Translation (horizontal, parallel to x-axis) + reflection across x-axis.

Verify: ✓

Glide reflection: translate by , then reflect across the x-axis.

T⟨4,0⟩

→Reflect x-axis

A(1,4)→A′′(5,-4)

B(3,4)→B′′(7,-4)

C(2,7)→C′′(6,-7)

Common Pitfalls

Skipping the Orientation Check

Always check orientation first — it immediately eliminates half the possibilities. Preserved = translation or rotation. Reversed = reflection or glide reflection.

Jumping to a Rotation Without Verifying the Center

If displacement vectors differ, it might be a rotation — but verify by checking if a specific center and angle work for ALL points, not just one.

Assuming a Single Transformation Suffices

Sometimes a glide reflection or a composition of two steps is required. If no single transformation works, try decomposing into two.

Real-Life Applications

Computer Vision & Image Recognition

AI systems that recognize objects in photos use transformation analysis: is this rotated? Reflected? Translated from the training image? Understanding the sequence of rigid motions helps machines identify objects regardless of their position or orientation in a photo.

Forensic Handwriting Analysis

Analysts compare letter shapes by identifying what rigid motions would map one sample onto another. If translation and rotation suffice, it's likely the same hand. If scaling (a non-rigid motion) is needed, it may indicate forgery.