Properties of Dilations

HSG.SRT.A.1

In this lesson:

• Define dilations using center and scale factor
• Verify three fundamental properties experimentally
• Distinguish dilations from rigid motions

Learning Objectives

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

1. Define a dilation in terms of a center point and a scale factor
2. Verify experimentally that a dilation takes a line not passing through the center to a parallel line
3. Verify experimentally that a dilation leaves a line passing through the center unchanged
4. Verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor
5. Perform dilations on geometric figures using graph paper, tracing paper, or geometry software
6. Distinguish dilations from rigid motions by recognizing that dilations change size but preserve shape

Real-World Scaling

Dilations appear everywhere in daily life:

• Photocopying: Enlarge to 200% or reduce to 50%
• Digital Zoom: Zoom in on phone photos to see details
• Architectural Blueprints: Scale buildings down to paper
• Map Scales: Represent cities on portable maps

In all cases: same shape, different size

What Is a Dilation?

A dilation is a transformation that produces an image with:

• The same shape as the original
• A different size from the original

Key distinction: Unlike rigid motions (translations, rotations, reflections), dilations change size while preserving shape.

Formal Definition: Dilation

A dilation with center C and scale factor k (where ) is defined by:

For any point , the image satisfies:

1. lies on the ray from through
2. Distance from to equals times distance from to

In symbols: on ray and

The Dilation Rule

To find the image of point P:

1. Draw the ray starting at center and passing through
2. Measure the distance
3. Multiply:
4. Mark on ray at distance from

Result: is the dilated image of

Scale Factor

The scale factor determines the type of dilation:

Enlargement ():

• Image is larger than original
• Points move farther from center
• Example: doubles all distances

Reduction ():

• Image is smaller than original
• Points move closer to center
• Example: halves all distances

Watch out: means reduction, NOT enlargement!

Visualizing Dilation ()

Process:

• Center is fixed anchor point
• Draw rays from through each vertex
• Mark new vertices at twice the distance from

Result: Enlarged Triangle

The dilated triangle:

• Is twice as large (all sides doubled)
• Has the same shape (all angles preserved)
• Looks like an "expansion" radiating from center

Key observation: Size changed, shape unchanged

Performing a Dilation

Five-step process:

1. Draw rays from center through each vertex
2. Measure distance from to each vertex
3. Multiply each distance by scale factor
4. Mark image points on rays at new distances
5. Connect image points to form dilated figure

Remember: Every point transforms, not just vertices!

Worked Example: Reduction

Given: Triangle , center outside, scale factor

Process:

• Ray from through vertex : distance cm
• New distance: cm → Mark at 4 cm
• Repeat for vertices and
• Connect

Result: Triangle half the size, closer to center

Note: Center is fixed—distance from to is 0!

Check-In

Question: If and a point is 4 cm from the center, how far from the center is the image point?

Think: Image distance = original distance

Investigation: Lines Not Through Center

Question: What happens to a line that does not pass through the center of dilation?

Predictions:

• Will the image be a line?
• Will it be parallel to the original?
• Will it be perpendicular?
• Will it intersect the original?

Let's experiment and observe...

Experiment: Dilate a Line

Setup:

• Line does not pass through center
• Pick multiple points on
• Dilate each point

Observation: Image points lie on line parallel to

Verification Method 1: Slopes

On a coordinate plane:

• Measure the slope of line
• Measure the slope of line
• Compare: Are the slopes equal?

Result: If slopes are equal → lines are parallel ✓

This can be tested by measuring rise over run for each line

Verification Method 2: Corresponding Angles

Draw a transversal crossing both and :

• Measure corresponding angles where transversal crosses
• Measure corresponding angles where transversal crosses
• Compare: Are corresponding angles congruent?

Result: If congruent → lines are parallel ✓

Misconception alert: Dilations preserve direction—slopes don't change!
No rotation happens. Lines don't become perpendicular.

Why It Works

Explanation:

• Each point on moves radially from center
• Direction is determined by ray from to each point
• Points that start on a line with slope end on a line with slope
• Direction (slope) is preserved → Image line is parallel

Property 1: A dilation takes a line not through the center to a parallel line.

Check-In

Question: Will a horizontal line dilated with a center above the line produce a horizontal, vertical, or diagonal image?

Think about slope preservation...

Contrast Question

We just saw: lines not through the center → parallel lines.

But what if the line does pass through the center?

Does it still move to a parallel position?
Or does something different happen?

Experiment: Line Through Center

Setup:

• Line passes through center
• Pick points on (both sides of )
• Dilate each point

Observation: Image points remain on line → line is unchanged

Clarification: "Unchanged"

What "unchanged" means:

• The line as a set of points is the same
• Individual points on the line do move to new locations (except )
• The line's position in the plane doesn't change

Think of it this way:

• Points slide along the line (like beads on a wire)
• The wire (line) itself stays put

Property 2: A dilation leaves a line through the center unchanged.

Worked Example: Line Through Origin

Setup: Center at origin , line (passes through origin)

Test a point: is on the line

Dilate by :

Check: Is on line ?

• Yes! Because ✓

Conclusion: Every point on maps to another point on → line unchanged

Check-In: Compare Properties

Compare:

• Line not through center → Maps to parallel line (new position)
• Line through center → Unchanged (same position)

What's different?
The key is whether the line contains the center point

Investigation: Segment Lengths

Question: How do dilations affect the length of a line segment?

Setup:

• Start with segment of known length
• Dilate with scale factor
• Measure the image segment
• Compare lengths

What pattern will we observe?

Experiment: Dilate a Segment

Example:

• Original segment : length cm
• Dilate with
• Image segment : length cm

Ratio: ✓

Data Table: Multiple Cases

Test multiple segments and scale factors:

Pattern: In every case, the ratio equals exactly!

Implication: Similar Figures

Example: Triangle with sides 3, 4, 5 (a right triangle)

Dilate by :

• Image triangle has sides , ,

Result:

• Same shape (right triangle, same angles)
• Proportional sides (all doubled)
• Different size (twice as large)

This is what "similar" means: same shape, proportional sides

Algebraic Proof (Advanced)

Given: Center at origin, scale factor

Points: and

Distance:

After dilation: and

Calculate:

Property 3: Image length original length ✓

Guided Practice

Your turn: Apply Property 3

Given:

• Segment length: 7 cm
• Scale factor:

Question: What is the length of the image segment?

Formula: Image length original length

Calculate:

Check-In: Why Properties Matter

Reflect: Why do these three properties matter for understanding similarity?

Think about:

• Property 1: Parallel lines → angles preserved
• Property 3: Segments scaled by → sides proportional
• Together: Same angles + proportional sides = similar figures

Dilations create similarity

Misconception Alert: Dilations vs Rigid Motions

Key distinction: Rigid motions preserve size; dilations change size

Misconception Alert: The Center is Fixed

Common mistake: Thinking the center point moves

Truth: The center is fixed — it does NOT move

Why:

• Distance from to is 0
• (always!)
• The center maps to itself

Role: The center is the anchor for all rays — everything else moves, but stays put

Summary

Definition: Dilation = scaling by factor from center

Three Properties Verified:

1. Lines not through → Parallel lines (direction preserved)
2. Lines through → Unchanged (line maps to itself)
3. Segments scaled by ratio (image length original)

Distinction: Dilations rigid motions

• Rigid motions → preserve size → congruent figures
• Dilations → change size → similar figures

Next Steps

Next lesson: Similarity Transformations

We'll combine dilations with rigid motions to create similarity transformations:

• Rigid motions (translate, rotate, reflect) + Dilation
• Use to prove figures are similar

Looking ahead: AA Criterion for Triangles

• Use dilation properties to prove: two angle pairs → similar triangles
• Foundation for all triangle similarity theorems

Dilations are the key to similarity!