Whether you're just getting started with geometric transformations, preparing for a test, or simply looking to brush up on your geometry, you’re in the right place.
Read on to find easy-to-follow definitions, real-life examples, and a simple guide to dilations in math.
What Is Dilation in Math?
Dilation is a geometric transformation in which we change the size of a figure without changing its shape.
When we dilate a figure, a triangle or circle for example, we can enlarge or shrink it, but its proportions and angles must remain the same.
Dilation is one of the four geometric transformations which include:
- Translation: Moving a figure from one location to another while keeping its shape and direction the same
- Rotation: Turning a figure around a fixed point
- Reflection: Flipping a figure over a line to create a mirror image
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How We Use Dilation in Real-Life
The concept of dilation is often used in everyday life.
For example, it helps us scale photos.
Photo editing apps use the principle of dilation to help us crop or resize our pictures without changing their proportions.
When we crop or resize images, dilation helps maintain their proportions.
In cartography and mapmaking, dilation helps us to accurately show geographical features on maps of various sizes while keeping their positions and proportions correct.
Dilation in mapmaking keeps features accurate at any size.
In architecture and engineering, dilation is used to enhance drawings and blueprints. It helps find edges, clean up messy images, and pick out features like roads or boundaries.
The list doesn’t end there. Dilation is also used in:
- road construction
- medical imaging (e.g. X-rays),
- artand design
- and many other professions and practices
Now let’s explore this geometric transformation, including relevant terms, dilation types, and principles.
Important Terms in Dilation
Here are some of the most important dilations terms we’ll be using throughout this guide:
1. Pre-Image & Image
Once dilation is performed, the figure we get is called image.
The original figure, the one we had before dilation, is called pre-image.
The image keeps the same shape as the pre-image but may be larger or smaller.
This is how it would look like in practice.
2. Center of Dilation
The center of dilation is the point from which a figure expands or contracts evenly in all directions.
During dilation, this point serves as the anchor and remains fixed while the rest of the figure transforms around it.
When students first learn dilation, they're dealing with the origin, which is the point (0,0) on a coordinate plane, as the center of dilation.
3. Scale Factor
The scale factor is a number that represents how much larger or smaller a figure becomes after dilation. It shows the ratio between the lengths of the pre-image and the image so we can see how much the figure changed in size.
The scale factor is denoted by "r" or "k." For consistency, we will use "k" throughout this blog.
- When the scale factor is greater than 1 (k > 1), the image is expanded.
- When the scale factor is between 0 and 1 (0 < k < 1), the image is contracted.
- When the scale factor is 1 (k = 1), the image stays the same.
- The scale factor can never be zero.
- Scale factors can be negative (k < 0), which means we perform both a dilation and a reflection across a specified line.
4. Scale Factor Formula
To precisely calculate the expansion or the contraction of a figure, we follow the scale factor formula:
For example, consider a rectangle with a length of 6 and a width of 4.
After dilation, the length becomes 12, and the width becomes 8.
Using the scale factor formula:
So, the scale factor of the rectangle is 2.
Types of Dilation
We’ve already said that dilation is the process of changing the size of a figure without changing its proportions or angles.
As such, dilation has two types:
- Expansion or Enlargement: when dilation creates a larger figure
- Contraction: when dilation creates a smaller figure
Now, let’s explore how we can expand and contract figures using the principles of dilation.
How to Dilate Figures on the Coordinate Plane
Now that we’ve defined the key terms and types, it’s time to put that knowledge to use.
This illustration shows an expanded image on the coordinate plane.
Here’s a step-by-step guide to dilating figures using the coordinate plane:
- Find the scale factor and the center of dilation. In our case, the scale factor will be 4 (k = 4), and the center of dilation is the origin (0,0).
- Find the vertex of the figure (a square in our case). Draw a line from the center of dilation through one vertex.
- Determine the distance between the center of dilation and the vertices vertically and horizontally. In our case, those will be A (1,2), B (2,2) C (2,1), and D (1,1). Notice that these horizontal and vertical distances are just the coordinates of A, B, C, and D.
This step is important as it helps us see how dilation changes the figure. When we dilate, we're essentially changing the distance between the center of dilation and the
points that make up the pre-image. In our case, with a scale factor of 4, this distance expands by 4 times.
- Multiply the original distances by the scale factor to get the horizontal and vertical distances from the center of dilation to a new vertex.
A x k = A' or 1 x 4 = 4 and 2 x 4 = 8, so we get A' (4,8)
B x k = B' or 2 x 4 = 8 and 2 x 4 = 8, so we get B' (8,8)
C x k = C' or 2 x 4 = 8 and 1 x 4 = 4, so we get C'(8,4)
D x k = D' or 1 x 4 = 4 and 1 x 4 = 4, so we get D' (4,4)
- Connect the new vertices to produce your new image.
How to Calculate the Scale Factor on the Coordinate Plane
To round out our understanding of dilation, let's see how we calculate the scale factor on a coordinate plane.
On this coordinate plane, we see the pre-image (ABC) of a triangle and its dilated image (A’B’C’):
The pre-image has been expanded and the coordinates of its vertices have changed too.
The coordinates are:
A (1,1) → A’ (3,3)
B (2,3) → B’ (6,9)
C (3,1) → C’ (9, 3)
If we look at each vertex individually, we’ll see that:
- Vertex A changed its x-coordinate from 1 to 3, and its y-coordinate from 1 to 3.
- Vertex B changed its x-coordinate from 2 to 6, and its y-coordinate from 3 to 9.
- Vertex C changed its x-coordinate from 3 to 9, and its y-coordinate from 1 to 3.
To calculate the scale factor, we can:
- Choose any of the corresponding vertices. In this case, we’ll take C and C’.
- Next, divide the x-coordinate of C’ by the x-coordinate of C:
We can conclude that our scale factor is 3.
Note that you could have chosen any other vertices and gotten the same result.
Frequently Asked Questions About Dilation
Here are the questions we usually get from our students as they learn about dilation.
1. What kind of shapes can we dilate?
We can dilate various geometric shapes, including polygons, circles, lines, and irregular shapes.
2. Can we dilate figures without using a coordinate plane?
Yes, we can dilate figures without a coordinate plane. We just use the scale factor to expand or contract the image by changing the lengths of the sides
Here’s an example:
3. Can we dilate figures if the scale factor is negative?
Yes, dilation can be performed even if the scale factor is negative. This results in a reflection
.
4. Can we perform dilation if the scale factor is a fraction?
Yes, if the scale factor is a fraction, the object will shrink proportionally. For example, if the scale factor is 1/2
, the object will be halved in size.
If the scale factor is a fraction greater than 1 (e.g. 3/2
), the object will enlarge proportionally.
5. Can we dilate three-dimensional figures?
Yes, dilation applies to 3D objects as well. We can make a 3D object larger or smaller by scaling all its dimensions uniformly.
Just like with 2D shapes, where you might make a square larger or smaller by multiplying its side length, in 3D, you can stretch or shrink objects along all three axes.
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