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# Dilation Worksheet, Formula, and Definition

Get the free Dilation Worksheet and other resources for teaching & understanding Dilations

What is a Dilation in Math? (Examples and Definition of Dilation in Math)

### Key Points

• Dilation is a fundamental concept in mathematics that involves scaling a figure to a new size while maintaining its shape and orientation.
• To dilate a figure, one needs to determine the scale factor and the center of dilation.
• Dilation has numerous applications in various fields, including architecture, engineering, and physics.

## What is Dilation in Math?

Dilation Definition in Math involves enlarging or shrinking figures on a coordinate grid. Dilation on a Coordinate Grid takes place when a figure is multiplied by a Scale Factor. The scale factor tells you if the figure is going to get larger or smaller If the scale factor is greater then one, the figure will get larger. If the scale factor is less then one, the figure will get smaller. You can Dilate on a Coordinate Grid by multiplying the coordinates of the figure by the scale factor. The last step for Dilation on a Coordinate Grid is to write the coordinates of the new location of the figure.

Dilation is a fundamental concept in mathematics that involves the transformation of a figure to a new size, either larger or smaller, while maintaining its shape and orientation. It is an essential concept in geometry, which involves the study of shapes, sizes, and positions of objects in space. Dilation is used in various fields, including architecture, engineering, and physics, to scale objects and structures.

To dilate a figure, one needs to determine the scale factor, which is the ratio of the size of the image to the size of the pre-image. The scale factor can be greater than 1, indicating that the image is larger than the pre-image, or less than 1, indicating that the image is smaller than the pre-image. The center of dilation is the point about which the figure is scaled, and it can be located anywhere in the plane.

Dilation is a powerful tool in mathematics that has numerous applications in various fields. Understanding the concept of dilation is essential for students in geometry and other related fields. In this article, we will explore the basics of dilation, including how to dilate a figure, the formula for dilation, dilation in the coordinate plane, and examples of dilation in real life.

Common Core Standard: 8.G.4
Related Topics: Congruent Shapes, Similar Figures, Translation on a Coordinate Grid, Rotation on a Coordinate Grid, Reflection on a Coordinate Grid

## How to Dilate a Figure

Dilation is a transformation in which a figure is enlarged or reduced by a certain scale factor. It involves multiplying the coordinates of each point in the figure by the scale factor. Here are the steps to perform a dilation:

1. Determine the center of dilation: The center of dilation is the fixed point around which the figure is enlarged or reduced. It can be any point in the plane.
2. Determine the scale factor: The scale factor is the ratio of the length of the corresponding sides of the original figure and the dilated figure. It can be greater than 1 (enlargement) or less than 1 (reduction).
3. Draw the image: Draw the original figure and the center of dilation on a coordinate plane. Then, mark the corresponding points of the dilated figure by multiplying the coordinates of each point by the scale factor.
4. Connect the points: Connect the corresponding points of the original figure and the dilated figure to form the dilated image.

It is important to note that the direction of the dilation depends on the sign of the scale factor. If the scale factor is positive, the image is enlarged. If the scale factor is negative, the image is reflected and then enlarged or reduced.

In addition, dilation is a type of similarity transformation, which means that the shape of the figure remains the same but the size changes. Therefore, the corresponding angles of the original figure and the dilated figure are congruent, and the corresponding sides are proportional.

Overall, dilating a figure is a simple process that involves determining the center of dilation, the scale factor, and connecting the corresponding points. By following these steps, one can easily enlarge or reduce any figure on a coordinate plane.

## Formula for Dilation

Dilation is a transformation that changes the size of a figure but not its shape. It is performed by multiplying the coordinates of each point by a scale factor. The scale factor can be greater than 1, less than 1, or equal to 1. The formula for dilation varies depending on the center of dilation.

### Formula for Dilation about the Origin

When dilating a point with coordinates (x, y) about the origin by a scale factor of k, the new coordinates are (kx, ky). This means that each coordinate is multiplied by the scale factor.

For example, if a point has coordinates (2, 3) and is dilated about the origin by a scale factor of 2, the new coordinates are (4, 6). The point is twice as far from the origin as it was before.

### Formula for Dilation about a Point

When dilating a point with coordinates (x, y) about a point with coordinates (a, b) by a scale factor of k, the new coordinates are (k(x-a)+a, k(y-b)+b). This means that each coordinate is first shifted by -a or -b, then multiplied by the scale factor, and finally shifted back by a or b.

For example, if a point has coordinates (2, 3) and is dilated about the point (1, 1) by a scale factor of 2, the new coordinates are (3, 5). The point is twice as far from (1, 1) as it was before.

### Formula for Dilation in the Coordinate Plane

When dilating a figure in the coordinate plane, the formula for dilation can be applied to each point in the figure. The center of dilation can be any point in the plane, including the origin.

It’s important to note that dilation changes the size of a figure but not its shape. If the scale factor is greater than 1, the figure is enlarged. If the scale factor is less than 1, the figure is reduced. If the scale factor is equal to 1, the figure is unchanged.

## Dilation in the Coordinate Plane

Dilation is a transformation of a figure in which all points are moved away from or towards a fixed point called the center of dilation. This transformation changes the size of the figure, but not its shape. Dilation can be performed in a coordinate plane by multiplying the coordinates of each point by a scale factor.

To dilate a point on a coordinate plane, the coordinates of the point are multiplied by a scale factor. If the scale factor is greater than 1, the point moves away from the origin. If the scale factor is less than 1, the point moves closer to the origin. The center of dilation is the fixed point about which the dilation is performed.

Dilations in the coordinate plane have several properties that are important to know. These properties include:

• A dilation is a similarity transformation, which means that the shape of the figure is preserved, but its size is changed.
• All dilations have a center of dilation, which is the fixed point about which the dilation is performed.
• The scale factor of a dilation determines how much the figure is enlarged or reduced. If the scale factor is greater than 1, the figure is enlarged. If the scale factor is less than 1, the figure is reduced.
• Dilations can be performed on any figure, including polygons, circles, and other shapes.

Dilation can be used in many different applications, including in art, architecture, and engineering. For example, architects may use dilation to scale a building design up or down to fit a particular site or to meet specific design requirements. Engineers may use dilation to scale a model of a product to test its performance under different conditions.

In summary, dilation is a transformation of a figure in which all points are moved away from or towards a fixed point called the center of dilation. Dilations in the coordinate plane have several properties, including the fact that they are similarity transformations and have a fixed center of dilation and scale factor. Dilations can be used in many different applications, including in art, architecture, and engineering.

## Center of Dilation

In mathematics, dilation is a transformation that rescales the size of an object. It involves multiplying the coordinates of each point of the object by a constant factor called the scale factor. The center of dilation is the fixed point around which the dilation occurs.

The center of dilation can be inside or outside the object being transformed. If the center of dilation is inside the object, the object is enlarged or reduced uniformly. If the center of dilation is outside the object, the object is reflected and enlarged or reduced.

One way to find the center of dilation is to locate the corresponding points of the pre-image and the image. The center of dilation is the intersection of the lines connecting the corresponding points.

Another way to find the center of dilation is to use the scale factor. The center of dilation is the point that does not move during the dilation. It is the fixed point that remains in the same position before and after the transformation.

The center of dilation is an important concept in geometry. It helps to determine the properties of the dilation, such as the direction and magnitude of the transformation. It also helps to identify the relationship between the pre-image and the image.

To summarize, the center of dilation is the fixed point around which the dilation occurs. It can be located by finding the intersection of the lines connecting the corresponding points or by identifying the point that does not move during the transformation. Understanding the center of dilation is crucial in analyzing the properties and relationships of dilated objects.

## 3 Simple Math Dilation Examples

1. When completing a Dilation Worksheet, you must shrink or enlarge a figure.
2. You multiply each coordinate by the scale factor.
3. If the scale factor is above one, the figure gets larger.
4. If the scale factor is under one, the figure gets smaller.

Dilation is a transformation in geometry that resizes an object without changing its shape. It is one of the five major transformations in geometry. This section will discuss some math dilation examples in triangles, circles, and polygons.

### Dilation Triangle

Dilation in triangles involves resizing the triangle by a scale factor. The scale factor is a ratio of the length of the corresponding sides of the image and pre-image. For example, if the scale factor is 2, the image will be twice the size of the pre-image. To perform a dilation, the center of dilation and the scale factor must be given.

Consider a triangle ABC with vertices at (2, 3), (4, 3), and (3, 5). If the center of dilation is (1, 2) and the scale factor is 2, the image of the triangle will be:

VertexImage
A(-1, 1)
B(1, 1)
C(0, 3)

### Dilation in Circles

Dilation in circles involves resizing the circle by a scale factor. The scale factor is a ratio of the radius of the image and pre-image. For example, if the scale factor is 2, the image will be twice the size of the pre-image. To perform a dilation, the center of dilation and the scale factor must be given.

Consider a circle with center at (2, 3) and radius 4. If the center of dilation is (1, 2) and the scale factor is 2, the image of the circle will have center at (-1, -1) and radius 8.

### Dilation in Polygons

Dilation in polygons involves resizing the polygon by a scale factor. The scale factor is a ratio of the length of the corresponding sides of the image and pre-image. To perform a dilation, the center of dilation and the scale factor must be given.

Consider a regular pentagon with vertices at (0, 0), (1, 1), (3, 1), (4, 0), and (2, -2). If the center of dilation is (1, 1) and the scale factor is 2, the image of the pentagon will have vertices at (-1, -1), (1, 3), (5, 3), (7, -1), and (3, -5).

## 5 Quick Dilation Practice Problems

/5

Dilation Quiz

Click Start to begin the practice quiz!

1 / 5

Find the coordinate after a dilation with a scale factor of 1/4.

(-8,-4)

2 / 5

Find the coordinate after a dilation with a scale factor of 4.

(1,8)

3 / 5

Find the coordinate after a dilation with a scale factor of 2.

(3,4)

4 / 5

Find the coordinate after a dilation with a scale factor of 1/2.

(6,4)

5 / 5

Find the coordinate after a dilation with a scale factor of 3.

(5,7)

0%

## Scale Factor of Dilation

Dilation is a transformation that changes the size of a figure. It can make the figure larger or smaller, but it does not change its shape. The scale factor is a crucial component of dilation. It is a ratio of the size of the image to the size of the pre-image.

The scale factor is always greater than zero. If the scale factor is greater than one, the image will be larger than the pre-image. Conversely, if the scale factor is between zero and one, the image will be smaller than the pre-image. If the scale factor is equal to one, the image will be the same size as the pre-image.

The formula for the scale factor is:

``````scale factor = size of image ÷ size of pre-image
``````

The size of the image and the pre-image can be measured using any unit of measurement, such as inches, centimeters, or pixels. The scale factor does not depend on the unit of measurement used.

It is important to note that the scale factor is the same for all corresponding sides of the image and the pre-image. In other words, if one side of the pre-image is multiplied by the scale factor to get the corresponding side of the image, then all other sides of the pre-image must also be multiplied by the same scale factor to get their corresponding sides of the image.

Understanding the scale factor is critical when performing dilations in math. It helps to determine the size of the image and the pre-image and ensures that the corresponding sides are proportional.

## Scale Factor Formula

A scale factor is a ratio that describes how much larger or smaller an image is compared to its original size. It is used in dilation transformations to scale an object in the plane. The formula for calculating the scale factor is:

``````Scale Factor = Size of Image ÷ Size of Pre-image
``````

The size of the image refers to the dimensions of the transformed object, while the size of the pre-image refers to the dimensions of the original object. The scale factor can be expressed as a decimal, fraction, or percentage.

For example, if an object is dilated with a scale factor of 2, the image will be twice the size of the original object. If the scale factor is less than 1, the image will be smaller than the original object.

To calculate the scale factor, it is necessary to measure the dimensions of both the pre-image and the image. The size of the pre-image can be measured using any standard unit of measurement, such as inches or centimeters. The size of the image can then be measured using the same unit of measurement.

It is important to note that the scale factor is always positive. If the image is larger than the pre-image, the scale factor will be greater than 1. If the image is smaller than the pre-image, the scale factor will be less than 1.

## Dilation Math Definition

Dilation is a fundamental transformation in mathematics that involves the resizing of a geometric figure. It is a non-rigid transformation that changes the size of a shape while preserving its shape and orientation. In essence, dilation involves multiplying the coordinates of each point in a figure by a scale factor to create a new figure that is either larger or smaller than the original.

The scale factor is a positive number that determines the degree of enlargement or reduction of the figure. If the scale factor is greater than 1, the figure is enlarged, and if it is less than 1, the figure is reduced. If the scale factor is equal to 1, the figure remains the same size.

Dilation can be performed with respect to a point, a line, or a plane, called the center of dilation. When the center of dilation is the origin, the dilation is called a similarity transformation, which preserves the shape of the figure. When the center of dilation is not the origin, the dilation is called a non-similarity transformation, which changes the shape of the figure.

Dilation is a useful mathematical tool with many practical applications. It is used in art, architecture, engineering, and science to create scaled models, blueprints, and designs. It is also used in computer graphics and animation to create 3D models and special effects.

## Dilation Real Life Examples

Dilation is a transformation that can be observed in many real-life scenarios. Here are a few examples of dilation in real life:

• Pupil dilation: The pupil of the eye dilates when the amount of light entering the eye increases. This is a natural example of dilation that can be observed in humans and animals.
• Maps and blueprints: Maps and blueprints are often scaled-down versions of the actual object or location. For example, a map of a city may be scaled down to fit on a piece of paper. This is an example of dilation, where the original object is dilated to create a smaller version.
• Photography: In photography, zoom lenses are used to change the size of the image being captured. Zooming in on a subject results in a dilation of the image, making it appear larger.
• Medical imaging: Medical imaging techniques such as X-rays and CT scans use dilation to create images of the body. These techniques use a scale factor to dilate the image of the body, making it easier to see and analyze.
• Forensics: Forensic scientists use dilation techniques to analyze fingerprints and footprints. By dilating the image of the print, they can see more details and make more accurate comparisons.

Dilation is a fundamental concept in geometry, but it has many practical applications in the real world. By understanding dilation, we can better understand and analyze the world around us.

## Math Dilation FAQ

### What is a dilation in math?

A dilation in math is a transformation that changes the size of an object, but not its shape. It is performed by multiplying the coordinates of each point by a scale factor. The scale factor determines whether the dilation will make the object bigger or smaller. Dilations are commonly used in geometry to create similar figures.

### How do you calculate dilations?

To calculate a dilation, you need to know the scale factor and the center of dilation. The scale factor is a ratio that determines how much the object will be stretched or shrunk. The center of dilation is the point about which the object is dilated. To dilate an object, you multiply the coordinates of each point by the scale factor and the distance between the point and the center of dilation.

### How do you dilate by a scale factor of 2?

To dilate an object by a scale factor of 2, you multiply the coordinates of each point by 2. This will make the object twice as big as it was before. To dilate an object by a scale factor of 1/2, you multiply the coordinates of each point by 1/2. This will make the object half as big as it was before.

### Is dilation in math getting bigger or smaller?

Dilation in math can make an object either bigger or smaller. The scale factor determines whether the object will be stretched or shrunk. If the scale factor is greater than 1, the object will be stretched and get bigger. If the scale factor is less than 1, the object will be shrunk and get smaller. If the scale factor is equal to 1, the object will not change size.

### What is Dilation Transformation?

Dilation transformation is a geometric transformation that changes the size of an object, but not its shape. It is performed by multiplying the coordinates of each point by a scale factor. The scale factor determines whether the dilation will make the object bigger or smaller. Dilations are commonly used in geometry to create similar figures.

### What is dilation of a triangle?

Dilation of a triangle is a transformation that changes the size of the triangle, but not its shape. It is performed by multiplying the coordinates of each vertex by a scale factor. The scale factor determines whether the triangle will be stretched or shrunk. Dilations of triangles are commonly used in geometry to create similar triangles.

## Dilation Worksheet Video Explanation

Watch our free video on how to solve Dilations. This video shows how to solve problems that are on our free Dilation worksheets that you can get by submitting your email above.

Watch the free Dilation video on YouTube here: Dilation Worksheet Video

Video Transcript:
This video is about our dilations worksheet. You can get this dilation worksheet for free by clicking on the link in the description below.

Dilation in math refers to either making a figure larger or making a figure smaller which is the dilation math definition. It’s a way to transform a figure on a coordinate grid. The way to dilate on the coordinate grid is to multiply all of your coordinates by a scale factor. If the scale factor is greater than 1, if it’s larger than one, your figure will get larger. If the scale factor is less than 1 your figure will get smaller. Most of the questions will ask which graph shows a dilation and you will have to decide on the right answer.

Looking at this square here, if we had a scale factor of let’s say 2 we would multiply all the coordinates by 2 and our figure would get larger. The square after a dilation with the scale factor of 2 would look like this. If we were to dilate our square by 0.5 or ½, the square would get smaller and it would look like this square in green.

The way to perform a dilation on the coordinate grid is to take your X and your Y coordinate and multiply it times whatever the scale factor is. So for example if the scale factor was 3 we would take 3 and we would multiply at times X so it would be like 3 X and times y so it would be like 3 Y.

The second problem on our dilation worksheet 8th grade says to graph the figure ABCD after a dilation with a scale factor of 4. We already know that the scale factor is going to be what we use to multiply times all of our coordinates below. In order to do this, we’re going to take each coordinate and multiply it times the scale factor which is 4. For example, in order to get a prime what we have to do is we have to take negative 1 multiply times 4 for our new x-coordinate and then 2 times 4 for a new y-coordinate. So negative 1 times 4 is negative 4 and then two times four is eight. This will be our new a prime.

We will then do that for the remainder of the coordinates. One times four is four. Two times four is eight. One times four is four again. Negative two times 4 is negative eight and then negative two times 4 is negative eight and then negative 1 times 4 is negative 4. In order to graph our new figure we just have to plot these vertices.

I went ahead and copy the new vertices down by the graph. We’re gonna go ahead and plot these. The next step is to draw our lines and create our new figure. You will notice that our new figure is larger and that’s because the scale factor was 4, which is of course greater than 1. This new figure is a solution for the second problem on our dilations on the coordinate plane worksheet.