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Introduction

QUICK LINKS


Introduction

INTRO
A dilation is a transformation that changes the size of an image without changing its shape or proportions.
It's like how when Ant-Man gets bigger or smaller, each body part changes size by the same amount, so he looks the same - just bigger or smaller than before. Dilating shapes is similar 👯‍♀️.
When we dilate shapes on a grid, we need to know the center of dilation and the scale factor.
To help us understand these terms, let’s imagine we have a movable screen and a fixed projector that can’t be moved:

Closer

✨ Drag to move the projection ✨

Farther

The center of dilation is like the fixed projector: it’s a fixed point from which the image is drawn. The scale factor determines how much smaller or larger the dilated image will be.
If the scale factor is , we can imagine keeping the distance between the projector and the screen the same, which will make the new image the same as the original.
Check out our
Calculator
or explore our
Lesson
and
Practice
sections to learn more about dilations on a graph and test your understanding.

You can also use the Quick Links menu on the left to jump to a section of your choice.

You can also use the Quick Links dropdown above to jump to a section of your choice.

CALCULATOR

Dilation on a Graph Calculator

Step 1. Identify the center of dilation.

Imagine the center of dilation as the fixed location of a projector.

Step 2. Identify the original points of the polygon.

How many points does the shape have?
Imagine this as an image projected onto a screen by the projector.

Step 3. Identify the scale factor .

What is the scale factor?

KEY STEPS

How to Perform Dilations

Step 1. Identify the center of dilation.

Imagine this as the fixed location of the projector.

Step 2. Identify the original points of the polygon.

Imagine this as the original image before the screen is moved.

Step 3. Identify the scale factor .

If the scale factor is , we can imagine keeping the distance between the projector and the screen the same, which will make the new image the same as the original.

Step 4. Multiply each original point of the polygon by the scale factor to get the new points.

Original PointScale FactorNew Point

Step 5. Plot the new points and connect the dots to get your dilated shape.

Step 4. Find the difference between the and values of each original point and the center of dilation .

Imagine this as finding the distance from the projector to the screen’s original position.
Original PointCenter of DilationDifference

Step 5. Multiply each difference by the scale factor.

Imagine this as finding the new distance from the projector to where the screen’s new position will be.
DifferenceScale FactorNew Difference

Step 6. Add the new difference to the center of dilation to get the new points.

Imagine this as finding the position of the new image.
New DifferenceCenter of DilationNew Point

Step 7. Plot the new points and connect the dots to get your dilated shape.

LESSON
Dilations Centered at the Origin

Dilations Centered at the Origin

When a dilation is centered at the origin, it means that the dilation is projected from the point .

In order to find the new points of a dilation centered at the origin, all we need to do is multiply the and coordinates of the original points by the scale factor.
Once we have the new points, we can plot them on the graph and connect the dots to reveal the new image. 🤩
Let’s walk through a couple examples!

EXAMPLE 1

Let's start by dilating this image ⬇️ by a scale factor of around the origin.

Step 1. Identify the center of dilation.

Imagine the center of dilation as the fixed location of a projector.

Step 2. Identify the original points of the polygon.

Imagine this as an image projected onto a screen by the projector. The original points of our image are .

Step 3. Identify the scale factor .

Our scale factor is .
Since the scale factor is greater than 1, we can imagine increasing the distance between the projector and the screen, which will make the new image larger than the original.

Step 4. Multiply each original point of the polygon by the scale factor to get the new points.

This is how we find the new points of our dilated image.
Original PointScale FactorNew Point

Step 5. Plot the new points and connect the dots to get your dilated shape.

EXAMPLE 2

Amazing! Let's try another problem.
Dilate this image ⬇️ by a scale factor of around the origin.

Step 1. Identify the center of dilation.

Imagine the center of dilation as the fixed location of a projector.

Step 2. Identify the original points of the polygon.

Imagine this as an image projected onto a screen by the projector. The original points of our image are .

Step 3. Identify the scale factor .

Our scale factor is .
Since the scale factor is less than 1, we can imagine decreasing the distance between the projector and the screen, which will make the new image smaller than the original.

Step 4. Multiply each original point of the polygon by the scale factor to get the new points.

This is how we find the new points of our dilated image.
Original PointScale FactorNew Point

Step 5. Plot the new points and connect the dots to get your dilated shape.

Awesome! Expand the
Practice
below to try some more problems or expand the next
Lesson
to explore what we do when we're dealing with dilations not centered at the origin.
PRACTICE
Dilations Centered at the Origin

Practice: Dilations Centered at the Origin

Question 1 of 3: Dilate this figure by a scale factor of with a center of dilation at the origin.

Step 1. Identify the center of dilation.

Imagine this as the fixed location of the projector.

Step 2. Identify the original points of the polygon.

Enter the original points of your polygon.
Original Points
Point 1
Point 2
Point 3

Step 3. Identify the scale factor.

What is the scale factor?
Since the scale factor is greater than 1, we can imagine increasing the distance between the projector and the screen, which will make the new image larger than the original.

Step 4. Multiply each original point of the polygon by the scale factor to get the new points.

This is how we find the new points of our dilated image.
Original PointScale FactorNew Point

Step 5. Plot the new points and connect the dots to get your dilated shape.

Step 6. Add the new difference to the center of dilation to get the new points.

Imagine this as finding the position of the new image.
New DifferenceCenter of DilationNew Point

Step 7. Plot the new points and connect the dots to get your dilated shape.

Awesome job! Our dilated shape is the blue shape:

LESSON
Dilations NOT Centered at the Origin

Dilations NOT Centered at the Origin

Dilations that are not centered at the origin are a little trickier, but still 100% doable if we take it one step at a time.

Let’s walk through a couple examples using our projector-screen idea!

EXAMPLE 1

Let's start by dilating this image ⬇️ by a scale factor of around the point .

Step 1. Identify the center of dilation.

Imagine the center of dilation as the fixed location of a projector.

Step 2. Identify the original points of the polygon.

Imagine this as an image projected onto a screen by the projector. The original points of our image are .

Step 3. Identify the scale factor .

Our scale factor is .
Since the scale factor is greater than 1, we can imagine increasing the distance between the projector and the screen, which will make the new image larger than the original.

Step 4. Find the difference between the and values of each original point and the center of dilation .

Imagine this as finding the distance from the projector to the screen’s original position.
Original PointCenter of DilationDifference

Step 5. Multiply each difference by the scale factor.

Step 6. Add the new difference to the center of dilation to get the new points.

Imagine this as finding the position of the new image.
New DifferenceCenter of DilationNew Point

EXAMPLE 2

Nice work! Now, let's dilate this image ⬇️ by a scale factor of around the point .

Step 1. Identify the center of dilation.

Imagine the center of dilation as the fixed location of a projector.

Step 2. Identify the original points of the polygon.

Imagine this as an image projected onto a screen by the projector. The original points of our image are .

Step 3. Identify the scale factor .

Our scale factor is .
Since the scale factor is less than 1, we can imagine decreasing the distance between the projector and the screen, which will make the new image smaller than the original.

Step 4. Find the difference between the and values of each original point and the center of dilation .

Imagine this as finding the distance from the projector to the screen’s original position.
Original PointCenter of DilationDifference

Step 5. Multiply each difference by the scale factor.

Step 6. Add the new difference to the center of dilation to get the new points.

Imagine this as finding the position of the new image.
New DifferenceCenter of DilationNew Point
Awesome! If you’d like to practice additional problems with dilations not centered at the origin, expand the
Practice
below before closing out this lesson! ⚡
PRACTICE
Dilations NOT Centered at the Origin

Practice: Dilations NOT Centered at the Origin

Question 1 of 3: Dilate this figure by a scale factor of with a center of dilation at .

Step 1. Identify the center of dilation.

Imagine this as the fixed location of the projector.

Step 2. Identify the original points of the polygon.

Enter the original points of your polygon.
Original Points
Point 1
Point 2
Point 3
Point 4
Point 5

Step 3. Identify the scale factor.

What is the scale factor?
Since the scale factor is less than 1, we can imagine decreasing the distance between the projector and the screen, which will make the new image smaller than the original.

Step 4. Find the difference between the x and y values of each original point and the center of dilation.

Imagine this as finding the distance from the projector to the screen’s original position.
Original PointCenter of DilationDifference

Step 5. Multiply each difference by the scale factor.

Imagine this as finding the new distance from the projector to where the screen’s new position will be.
DifferenceScale FactorNew Difference

Step 6. Add the new difference to the center of dilation to get the new points.

Imagine this as finding the position of the new image.
New DifferenceCenter of DilationNew Point

Step 7. Plot the new points and connect the dots to get your dilated shape.

Awesome job! Our dilated shape is the blue shape:

CONCLUSION
Incredible job, look at you go! Thanks for checking out this lesson ☺️🙏. Where to next?
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