Non-Rigid Transformations — Dilation and Classification

Lesson 2 of 2: What Is — and Isn't — Preserved

In this lesson:

• Dilation: scales from a center, preserves angles but not distances
• Horizontal stretch: preserves neither distances nor angles
• How to classify any transformation by what it preserves

Learning Objectives for This Lesson

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

3. Identify dilation as a basic transformation and write its coordinate rule
4. Distinguish rigid motions (distance- and angle-preserving) from non-rigid transformations
5. Predict the effect of a given transformation by reasoning about what is preserved

Does Photo Enlargement Preserve Distance?

You've enlarged a photo before:

• The shape of the image stays the same — proportions preserved
• The size changes — a 4×6 photo becomes 8×12

Does enlarging a photo preserve distances? Does it preserve angles?

Dilation — Precise Definition and Coordinate Rule

• Dilation centered at the origin, scale factor
• Every point moves toward or away from the center by factor
• : enlargement; : reduction; : identity

Worked Example — Triangle Dilation by Factor Two

Dilate : , , by scale factor :

Verify: , — distances doubled. — angles preserved!

Dilation Preserves Shape but Not Size

What dilation preserves:

• Angle measures (all angles unchanged)
• Shape (the figure looks the same — proportions maintained)

What dilation does NOT preserve:

• Distances (all distances multiplied by )
• Therefore: dilation is not a rigid motion

Quick Check — Predicting a Dilation's Effect

After dilating a triangle by :

• The original was 40°. What is ?
• An original side had length 5. What is the image side length?

Apply what you know about dilation before advancing.

Answer: Angles Preserved, Distances Scaled

• — dilation preserves all angle measures
• Image side — distances are multiplied by

Dilation scales distances uniformly — same factor in every direction.

Horizontal Stretch — Non-Uniform Scaling

• Horizontal stretch : doubles -coordinates; unchanged
• Distances change — horizontal lengths grow but vertical lengths stay the same
• Angles also change — the figure's shape is distorted, unlike dilation

Dilation vs. Stretch — Side-by-Side Comparison

• Dilation (): both and scaled equally — shape preserved
• Stretch (): scaled, unchanged — shape distorted
• Key test: are all distances scaled by the same factor?

Quick Check — Square vs. Rectangle After Transformation

A square with side length 4 undergoes two transformations:

1. Dilated by
2. Horizontally stretched by factor 2:

Is the image still a square after each? Or does it become a rectangle?

Answer: Dilation Preserves Shape; Stretch Does Not

1. Dilation : side in every direction — still a square (shape preserved)
2. Stretch : horizontal sides , vertical sides — becomes an rectangle (shape distorted)

Watch out: "Bigger" does not mean "dilated" — dilation scales uniformly; stretch does not.

Classification Table — What Each Transformation Preserves

The Key Insight — Necessary vs. Sufficient

Rigid motion = preserves distances AND angles (both required)

• Angle preservation alone is not enough to conclude rigid motion
• Dilation preserves angles — but is NOT a rigid motion
• You must check distances too

"Preserving angles is necessary for a rigid motion — but not sufficient."

Quick Check — Identifying from What Is Preserved

A transformation is applied to a triangle. Afterward:

• All angle measures are unchanged
• The sides are all 3 times longer

What type of transformation was it? Is it a rigid motion?

Name the type and justify before advancing.

Answer: Dilation with Scale Factor 3

The transformation is a dilation with :

• Angles preserved ✓ (consistent with dilation)
• Distances multiplied by 3 ✓ (consistent with scale factor )
• Distance preservation fails → not a rigid motion

Use the classification table: angle-yes, distance-no → dilation.

Synthesis — Classifying from a Pre-image and Image

Given a pre-image triangle with , and an image with , :

Step 1: Check distances — preserved () ✓

Step 2: Check angles — preserved () ✓

Conclusion: Both preserved → rigid motion (translation, reflection, or rotation)

Preview — Rigid Motions and Similarity Compose

If you compose a rigid motion with a dilation, what is preserved?

• The rigid motion preserves distances and angles
• The dilation then scales all distances by — distances no longer preserved
• But angles: the rigid motion preserves them, and the dilation preserves them too
• Result: angles preserved, distances not → this composition defines similarity

Key Takeaways and Misconception Warnings

✓ Dilation : uniform scaling — angles preserved, distances scaled by

✓ Horizontal stretch: non-uniform — neither distances nor angles preserved

✓ Rigid motion requires both distance AND angle preservation

Angle preservation alone does not imply rigid motion — dilation preserves angles but is not rigid

Dilation ≠ stretch: uniform scaling vs. one-axis scaling — shapes distort in a stretch

What Comes Next in This Course

• HSG.CO.A.3 — Symmetries use the reflections and rotations from Deck 1
• HSG.CO.B.6 — Congruence defined as existence of a rigid motion between figures
• HSG.SRT.A.1 — Dilation: center, scale factor, and invariant properties
• HSG.SRT.A.2 — Similarity via dilation composed with rigid motions