Define Transformations Formally

Lesson 4 of 5: Congruence Cluster A

In this lesson:

• Define translation, reflection, and rotation geometrically
• Connect each definition to CO.A.1 vocabulary
• Use formal definitions to verify transformation claims

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. State the formal definition of translation using parallel segments
2. State the formal definition of reflection using perpendicular bisectors
3. State the formal definition of rotation using circles and angles
4. Identify which CO.A.1 terms appear in each definition
5. Use formal definitions to verify whether a mapping is a specific transformation

Is a Coordinate Rule a Definition?

Can a coordinate rule define a transformation — or just describe one instance?

Coordinate Rules Depend on Coordinate Systems

• The rule describes one specific translation
• Rotate the axes 45° — the same translation gets a different rule
• A geometric definition must work without any coordinate system

Goal: use only CO.A.1 terms — parallel lines, perpendicular lines, circles, angles, line segments.

How CO.A.4 Connects to Earlier Work

• CO.A.1: Replaced informal notions with precise definitions — angle, circle, perpendicular, parallel
• CO.A.2: Explored transformations using coordinates and technology
• CO.A.4 (today): Define transformations using only those CO.A.1 geometric terms

Just as "point" needs no simpler definition, "translation" shouldn't depend on a coordinate system.

Challenge: Try Defining a Translation

Try this (really try — before the next slide):

Define a translation without using coordinates, without the word "slide," and without pointing to an example.

Think for a moment before advancing...

Informal Attempts Lack Geometric Precision

Common attempts and their problems:

• "Same direction" — what does "direction" mean geometrically?
• "Fixed amount" — "amount" requires coordinates, not geometry
• "Parallel line" — closer, but which line? How far?

Fix: use — a directed segment captures direction and distance.

Formal Definition of Translation (CO.A.4)

Translation along directed segment maps every point to such that:

1. is parallel to
2. (same length)
3. points in the same direction as

Equivalently: is congruent and parallel to , same direction.

Translation Definition Applied to Multiple Points

• Given , pick any point
• From : draw a segment parallel to , equal length, same direction → endpoint is
• All segments are parallel and congruent to each other

CO.A.1 Terms Used in Translation Definition

• Not used: angles, circles, perpendicular lines
• Translation: the "parallel segments" transformation

Check-In: Verify a Translation Claim

Given pointing right with .

A student claims is the image of .

Verify all three conditions:

1. Is parallel to ?
2. Is ?
3. Same direction?

Check before advancing...

Translation Verification Answer: Valid Translation

, , rightward,

1. ✓ is horizontal — parallel to
2. ✓
3. ✓ Same direction

Conclusion: valid translation.

Formal Definition of Reflection (CO.A.4)

Reflection across line maps every point to such that:

1. on : (fixed points)
2. off : is the perpendicular bisector of

That is: and passes through midpoint of .

Reflection: Constructing the Image Point

• Draw and point not on
• Drop a perpendicular from to — meets at midpoint
• Extend to so that

Both Reflection Conditions Serve a Purpose

: without this, and are not directly across — the image is skewed.

is the midpoint: without this, and are at unequal distances from .

The fold model works because the fold is along the perpendicular bisector.

CO.A.1 Terms in the Reflection Definition

• Reflection is the "perpendicular bisector" transformation

Check-In: Verify a Reflection Claim

A diagram claims is the reflection of across .

Given: meets at , and the foot of the perpendicular from is not the midpoint of .

Is this a valid reflection? Which condition(s) are violated?

Think through both conditions before advancing...

Reflection Verification Answer: Not Valid

Condition 1 (): violated — meets at 70°

Condition 2 ( through midpoint): violated — foot ≠ midpoint

Conclusion: Not a valid reflection. Both conditions fail.

One violated condition is enough to disprove the claim.

Formal Definition of Rotation (CO.A.4)

Rotation by about center maps every point to such that:

1. : (center is fixed)
2. :
   • ( on the circle at through )
   • in the specified direction

Rotation: The Circle Carries the Point

• Draw center and the circle through
• lies on this circle at angular displacement from
• traces a circular arc to reach — not a straight line

CO.A.1 Terms in the Rotation Definition

• Rotation: "circle + angle" transformation

Rotation: Verify P=(4,0) Rotated 90° About O

Given: Rotate by CCW about

1. → circle of radius 4
2. CCW from →
3. ✓ and ✓

Check-In: Does Center Location Matter?

A rotation maps triangle to triangle .

True or False: The center of rotation must be inside the original triangle.

Think of a counterexample before advancing...

Rotation Center Is Not Restricted to Figure

False — the center can be anywhere: inside, on, or outside the figure.

Example: Center 10 units to the right of triangle → the triangle moves to a different region.

Definition requires only: and .

No restriction on where is.

All Five CO.A.1 Terms Accounted For

Together: all five CO.A.1 terms used — angle, circle, perpendicular, parallel, segment.

Formal Definitions Make Proof Arguments Possible

Each definition implies distance preservation:

• Translation: → is a parallelogram →
• Reflection: ⊥ bisector → by symmetry
• Rotation: by SAS →

These informal arguments preview the proofs in CO.B.

Key Takeaways and Misconception Warnings

✓ Translation: ,

✓ Reflection: ⊥ bisects

✓ Rotation: on circle at ,

Coordinate rules ≠ definitions

Reflection: 2D only — no lifting

Rotation paths: arcs, not segments

Center: anywhere in the plane

These Definitions Unlock the Rest of Geometry

CO.A.5 (next): use definitions to draw images with compass and straightedge

CO.B: use definitions to prove rigid motions produce congruent figures

CO.D.12: ⊥ bisector, parallel segments, circle-arc constructions become formal techniques

Goal in CO.B: prove via rigid motion — only possible with today's precise definitions.