Similarity Through Sequences of Transformations

Lesson 1 of 2: Defining and Describing Similarity

In this lesson:

• Define similarity using a sequence of transformations
• Describe a sequence that maps one figure onto another

What You Will Be Able to Do

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

1. Define similarity using a sequence of rigid motions and dilations
2. Tell congruence from similarity by the transformations involved
3. Describe a sequence that maps one similar figure onto another
4. Use the similarity symbol with vertices in matching order

Same Shape but a Different Size

Can sliding, flipping, or turning ever make the small one land on the big one?

A Move That Changes Size

You already know this rule from dilations:

Multiply every coordinate by — the scale factor.

Check the Coordinates One by One

Triangle : , ,
Triangle : , ,

Every coordinate of is exactly twice the matching one in .

What It Means to Be Similar

Two figures are similar if a sequence of rotations, reflections, translations, and dilations maps one onto the other.

We write .

Congruence Is Just a Special Case

• Congruent figures: rigid motions only
• Similar figures: rigid motions plus dilation
• If the dilation has scale factor , nothing resizes

So every congruent pair is also similar — with .

The Scale Factor Sets the Size

The scale factor is the common ratio of corresponding sides:

• , , so ratio
• , , so ratio
• , , so ratio

Are These Two Squares Similar?

Two squares have side lengths and .

• Are they similar?
• What is the scale factor?

Decide on your own before the next slide.

When the Dilation Isn't Enough

The dilated copy is the right size — but the wrong place.

A Reliable Strategy in Four Steps

1. Find the scale factor from corresponding sides
2. Dilate the pre-image to match size
3. Compare the dilated figure to the target
4. Align it with a rigid motion

Dilate First, Then Slide It Over

Scale factor , dilate, then translate right and up.

Dilate First, Then Turn or Flip

Scale factor , dilate, then rotate about the origin.

Order Matters; Many Sequences Work

• Dilate first, then apply rigid motions — reliable
• Translating before dilating can shift the result
• More than one valid sequence usually exists

Your Turn to Finish the Move

Triangle dilated by now sits at , , .
Target: , , .

What single rigid motion finishes the job?

Spot the Mistake in This Work

A student writes: ", and the scale factor is ."

What went wrong?

Describe the Whole Sequence Yourself

Triangle at , , is similar to one at , , .

Find the scale factor and describe a full sequence.

What Similarity Really Means for Figures

✓ Similar = a resized, repositioned copy
✓ Dilation sets the size; rigid motions set the place
✓ Congruence is similarity with

Watch out: compare corresponding sides; flips and turns are allowed

What Is Coming Up Next

So far we've been told two figures are similar and built the sequence.

Next lesson: how to test two figures and decide whether any similarity sequence could exist — using only measurements.