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 step-by-step dilation calculator to practice dilating polygons on a grid or keep scrolling to learn how we find the new points of a dilated image.

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.

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.

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 Difference

$+$

Center of Dilation

$=$

New Point

$((x_{1}−x_{c})s,(y_{1}−y_{c})s)$

$+$

$(x_{c},y_{c})$

$=$

$((x_{1}−x_{c})s+x_{c},(y_{1}−y_{c})s+y_{c})$

$((x_{2}−x_{c})s,(y_{2}−y_{c})s)$

$+$

$(x_{c},y_{c})$

$=$

$((x_{2}−x_{c})s+x_{c},(y_{2}−y_{c})s+y_{c})$

$...$

$+$

$(x_{c},y_{c})$

$=$

$...$

$((x_{n}−x_{c})s,(y_{n}−y_{c})s)$

$+$

$(x_{c},y_{c})$

$=$

$((x_{n}−x_{c})s+x_{c},(y_{n}−y_{c})s+y_{c})$

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

Try out some practice problems - starting with some where the center of dilation is the origin!

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Dilations Centered at the Origin

When a dilation is centered at the origin, it means that the dilation is projected from the point $(0,0)$.

In order to find the new points of a dilation centered at the origin, all we need to do is multiply the $x$ and $y$ 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!

Let's start by dilating this image ⬇️ by a scale factor of $2$ 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 $(2,1),(3,4),(4,1)$.

Step 3. Identify the scale factor .

Our scale factor is $2$.

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.

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

Amazing! Let's try another problem.

Dilate this image ⬇️ by a scale factor of $21 $ 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 $(4,4),(4,8),(6,8),(6,4)$.

Step 3. Identify the scale factor .

Our scale factor is $0.5$.

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.

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

Practice with some additional problems or keep scrolling to learn more about how to complete dilations not centered at the origin.

Practice: Dilations Centered at the Origin

Question 1 of 3: Dilate this figure by a scale factor of $2$ with a center of dilation at the origin.

Step 1. Identify the center of dilation.

Imagine this as the fixed location of the projector.

$center of dilation=$

$($$,$$)$

Step 2. Identify the original points of the polygon.

Enter the original points of your polygon.

Original Points

$(x_{original},y_{original})$

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.

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 Difference

$+$

Center of Dilation

$=$

New Point

$(−6,6)$

$+$

$(0,0)$

$=$

$($$,$$)$

$(8,6)$

$+$

$(0,0)$

$=$

$($$,$$)$

$(8,10)$

$+$

$(0,0)$

$=$

$($$,$$)$

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

Awesome job! Our dilated shape is the blue shape:

Nice work! Now let's try some problems where the center of dilation is not 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!

Let's start by dilating this image ⬇️ by a scale factor of $2$ around the point $(1,1)$.

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 $(2,1),(3,4),(4,1)$.

Step 3. Identify the scale factor .

Our scale factor is $2$.

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 $x$ 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 points−center of dilation(x_{original},y_{original})−(x_{center},y_{center}) =difference=(x_{original}−x_{center},y_{original}−y_{center}) $

Original Point

$−$

Center of Dilation

$=$

Difference

$(2,1)$

$−$

$(0,2)$

$=$

$(2,−1)$

$(3,4)$

$−$

$(0,2)$

$=$

$(3,2)$

$(4,1)$

$−$

$(0,2)$

$=$

$(4,−1)$

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 Difference

$+$

Center of Dilation

$=$

New Point

$(4,−2)$

$+$

$(0,2)$

$=$

$(4,0)$

$(6,4)$

$+$

$(0,2)$

$=$

$(6,6)$

$(8,−2)$

$+$

$(0,2)$

$=$

$(8,0)$

Nice work! Now, let's dilate this image ⬇️ by a scale factor of $21 $ around the point $(−1,−1)$.

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 $(4,4),(4,8),(6,8),(6,4)$.

Step 3. Identify the scale factor .

Our scale factor is $0.5$.

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 points−center of dilation(x_{original},y_{original})−(x_{center},y_{center}) =difference=(x_{original}−x_{center},y_{original}−y_{center}) $

Original Point

$−$

Center of Dilation

$=$

Difference

$(4,4)$

$−$

$(−1,−1)$

$=$

$(5,5)$

$(4,8)$

$−$

$(−1,−1)$

$=$

$(5,9)$

$(6,8)$

$−$

$(−1,−1)$

$=$

$(7,9)$

$(6,4)$

$−$

$(−1,−1)$

$=$

$(7,5)$

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 Difference

$+$

Center of Dilation

$=$

New Point

$(2.5,2.5)$

$+$

$(−1,−1)$

$=$

$(1.5,1.5)$

$(2.5,4.5)$

$+$

$(−1,−1)$

$=$

$(1.5,3.5)$

$(3.5,4.5)$

$+$

$(−1,−1)$

$=$

$(2.5,3.5)$

$(3.5,2.5)$

$+$

$(−1,−1)$

$=$

$(2.5,1.5)$

If you’d like additional practice with dilations not centered at the origin, take a moment to complete the quick practice below before closing out this lesson! ⚡

Practice: Dilations NOT Centered at the Origin

Question 1 of 3: Dilate this figure by a scale factor of $0.5$ with a center of dilation at $(1,3)$.

Step 1. Identify the center of dilation.

Imagine this as the fixed location of the projector.

$center of dilation=$

$($$,$$)$

Step 2. Identify the original points of the polygon.

Enter the original points of your polygon.

Original Points

$(x_{original},y_{original})$

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 points−center of dilation(x_{original},y_{original})−(x_{center},y_{center}) =difference=(x_{original}−x_{center},y_{original}−y_{center}) $

Original Point

$−$

Center of Dilation

$=$

Difference

$(1,1)$

$−$

$(1,3)$

$=$

$($$,$$)$

$(3,5)$

$−$

$(1,3)$

$=$

$($$,$$)$

$(5,3)$

$−$

$(1,3)$

$=$

$($$,$$)$

$(−1,−3)$

$−$

$(1,3)$

$=$

$($$,$$)$

$(−3,3)$

$−$

$(1,3)$

$=$

$($$,$$)$

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.