Dilation and Comparing Transformations

In this lesson:

• Apply the dilation rule to scale a figure
• See why dilation preserves shape but changes size
• Sort transformations as rigid or non-rigid

Learning Objectives for This Lesson

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

1. Apply the dilation rule for a scale factor
2. Explain why dilation preserves shape but multiplies side lengths
3. Compare rigid motions and dilation as congruent versus similar

Same Shape, But a Different Size

No rule from last lesson can do this. So what move changes size?

What Does Multiplying a Coordinate Do?

Take the point and multiply both coordinates by .

• You get — twice as far from the origin in both directions
• Last lesson we only added; now we multiply

Predict: does the point move closer to the origin, or farther?

The Dilation Rule and the Effect of k

A dilation centered at the origin with scale factor :

• enlarges; shrinks; leaves it unchanged
• The origin stays fixed:

Why Shape Survives but Size Does Not

Multiplying both coordinates by multiplies every distance by .

• Every side length scales by the same factor
• Every angle stays exactly the same
• Same proportions, same shape — just resized

Enlargement With a Scale Factor of Three

Dilate with . Multiply every coordinate by .

• , ,
• Side becomes — tripled, as expected

Reduction With a Scale Factor of One-Half

Dilate the same triangle with . Multiply every coordinate by .

• , ,
• Side becomes — halved, and it moved inward

Does Scale Factor Two Mean Add Two?

A tempting wrong move: dilating by by adding to each coordinate.

• "Add ": — same size, just shifted (a translation!)
• "Times ": — actually larger

Quick Check: Find the Scale Factor

A triangle's side goes from length to length after a dilation.

• The image side is the original times
• What is the scale factor ? Commit before advancing.

Four Rules: Three Keep Size, One Changes It

You now know four coordinate rules.

• Translation, reflection, rotation — every length stays the same
• Dilation — every length multiplies by

Let's line all four up against one triangle and compare.

Read Rigid or Non-Rigid From the Rule

The coordinate rule itself tells you whether size is preserved.

• Add or rearrange coordinates distances unchanged rigid
• Multiply by distances scaled non-rigid

The operation in the rule reveals the type of transformation.

One Triangle Under Four Transformations

Three images are the same size as the original. One is not — find it.

Comparing Congruent Versus Similar Images

• Rigid motions same size congruent
• Dilation different size similar

Explain It in Your Own Words

A dilation with scale factor produces an image that is similar but not congruent.

• Why does the size change but the shape stay the same?
• Use the words "multiply," "distance," and "angle" in your answer
• Write two sentences before advancing

On Your Own: Dilate and Classify

Dilate the quadrilateral , , , with .

• Apply the rule to all four vertices and plot the image
• Is the image congruent to the original, or similar? Why?

Congruent, Similar, and the Coordinate Clue

✓ Dilation: multiplies every length by

✓ Rigid motions keep size congruent; dilation changes size similar

✓ Read the rule: adding keeps size, multiplying changes it

Scale factor means multiply, not add; dilation keeps shape — it doesn't distort it

Coming Up: Defining True Similarity

You can now make a similar figure with one dilation.

Next, we combine dilation with rigid motions to define what similar truly means.

A figure is similar to another if a sequence of rigid motions and a dilation maps one onto it.