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# 10.4: Dilations of Geometric Shapes

Difficulty Level: At Grade Created by: CK-12

Objectives

The lesson objectives for dilations of Geometric Shapes are:

• What is a dilation?
• Graphing an image that has undergone a dilation.
• Writing a rule for a dilation
• The properties of a dilation

## What Is a Dilation?

Concept Content

As you learned in the previous lessons, there are four types of transformations. In the previous lesson, you learned about one of these transformations, rotations. The other three transformations are dilations, reflections, and translations. In this lesson, you will learn about dilations. The transformations known as dilations involve changing the size of a preimage to create the dilation image. When performing a dilation you use a scale factor. The scale factor will determine how much bigger or smaller the dilation image will be. The scale factor is sometimes called the scalar factor and has the symbol r\begin{align*}r\end{align*}. The figure below shows that the image \begin{align*}A^\prime\end{align*} is a dilation by a scale factor of 2. The key to remember is that the shape does not change; just the size does.

In this concept you will learn to describe dilations and, in the process, learn what they are. You will learn how to recognize dilations and to determine the scale factor. When an object is dilation, each of the points in the preimage is multiplied by the same value to create the dilation image. Dilations need a scale factor but they also need a center point. The center point is the center of the dilation. You use the center point to measure the distances to the preimage and the dilation image. It is these distances that determine the scale factor.

In later concepts, you will also learn to graph dilations, write rules to describe them, and finally understand some of the properties of dilations.

Guidance

Which one of the following figures represents a dilation. Explain.

You know that a dilation is a transformation that produces an image of the same shape but larger or smaller. Both of the figures above represent objects that involve dilations. In the figure with the triangles, the scale factor is 3.

The second figure with the squares also represents a dilation. In this figure, the center point \begin{align*}B\end{align*} is used to dilate the square \begin{align*}CDEF\end{align*} by a factor of \begin{align*}\frac{1}{2}\end{align*}.

Examples

Example A

Describe the dilation about the center point \begin{align*}B\end{align*} in the diagram below.

Looking at the diagram below, in particular, you can see the measures of the dotted lines. In the figure, the center point \begin{align*}B\end{align*} is used to dilate the polygon \begin{align*}CDEF\end{align*} by a factor of \begin{align*}\frac{1}{3}\end{align*}.

Example B

Using the measurement below and the scale factor, determine the measure of the dilation image.

\begin{align*}m \overline{A B} &= 15 \ cm \\ r &=\frac{1}{3}\end{align*}

You need to multiply the scale factor by the measurement of \begin{align*}AB\end{align*} in order to find the measurement of the dilation image \begin{align*}A^\prime B^\prime\end{align*}.

\begin{align*}m \overline{A^\prime B^\prime} = |r| m \overline{A B}\end{align*}

Since \begin{align*}|r|=\frac{1}{3}\end{align*}, and \begin{align*}m \overline{A B} = 15\end{align*},

\begin{align*}& m \overline{A^\prime B^\prime}=\frac{1}{3}(15) \\ & m \overline{A^\prime B^\prime}= 5 \ cm \end{align*}

Example C

Using the measurement below and the scale factor, determine the measure of the preimage.

\begin{align*}m \overline{H^\prime I^\prime} &= 24 \ cm \\ r &=2\end{align*}

Here, you need to divide the scale factor by the measurement of \begin{align*}H^\prime I^\prime\end{align*} in order to find the measurement of the preimage \begin{align*}HI\end{align*}.

\begin{align*}m \overline{H^\prime I^\prime} = |r| m \overline{H I}\end{align*}

Since \begin{align*}|r|=2\end{align*}, and \begin{align*}m \overline{H^\prime I^\prime} = 24\end{align*},

\begin{align*}&24 = 2m\overline{H I} \\ & m \overline{H I}=\frac{24}{2} \\ & m \overline{H I}= 12 \ cm \end{align*}

Vocabulary

Center Point
The center point is the center of the dilation. You use the center point to measure the distances to the preimage and the dilation image. It is these distances that determine the scale factor.
Dilations
Dilations in transformations involve changing the size of a preimage to create the dilation image.
Scale Factor
The scale factor will determine how much bigger or smaller the dilation image will be. The scale factor is sometimes called the scalar factor and has the symbol \begin{align*}r\end{align*}.

Guided Practice

1. Using the measurement below and the scale factor, determine the measure of the preimage.

\begin{align*}m \overline{T^\prime U^\prime}&=12 \ cm \\ r&= 4 \ cm \end{align*}

If the center point is \begin{align*}S\end{align*} draw the image and dilation image.

2. Describe the dilation in the diagram below.

3. Quadrilateral \begin{align*}STUV\end{align*} has vertices \begin{align*}S(-1, 3), T(2, 0), U(-2, -1),\end{align*} and \begin{align*}V(-3, 1)\end{align*}. The quadrilateral undergoes a dilation about the origin with a scale factor of \begin{align*}\frac{4}{5}\end{align*}. Sketch the preimage and the dilation image.

Here, you need to divide the scale factor by the measurement of \begin{align*}H^\prime I^\prime\end{align*} in order to find the measurement of the preimage \begin{align*}HI\end{align*}.

1. \begin{align*}m \overline{T^\prime U^\prime} = |r| m \overline{T U}\end{align*}

Since \begin{align*}|r|=4\end{align*}, and \begin{align*}m \overline{T^\prime U^\prime} = 12\end{align*},

\begin{align*}&12 = 4m\overline{T U} \\ & m \overline{T U}=\frac{12}{4} \\ & m \overline{T U}= 3 \ cm \end{align*}

2. Look at the diagram below:

In the figure, the center point \begin{align*}D\end{align*} is used to dilate the \begin{align*}A\end{align*} by a factor of \begin{align*}\frac{1}{2}\end{align*}.

3. Look at the diagram below:

In the figure, the center point \begin{align*}O\end{align*} is used to dilate the \begin{align*}A\end{align*} by a factor of \begin{align*}\frac{4}{5}\end{align*}.

Summary

In this concept you began your study of dilations, the forth type of transformations. Remember dilations involve increasing or decreasing the size of a figure by a scale factor. The scale factor uses the symbol \begin{align*}r\end{align*}. Also, the preimage undergoes a dilation about the center point to produce the dilation image. Remember the size of the figure changes but not the shape. In later concepts, you will also learn to graph dilations, write rules to describe them, and finally understand some of the properties of dilations.

Problem Set

Find the measure of the dilation image given the following information:

1. .
\begin{align*}m \overline{A B} &= 12 \ cm \\ r&=2\end{align*}
1. .
\begin{align*}m \overline{C D} &= 25 \ cm \\ r&=\frac{1}{5}\end{align*}
1. .
\begin{align*}m \overline{E F} &= 18 \ cm \\ r&=\frac{2}{3}\end{align*}
1. .
\begin{align*}m \overline{G H} &= 18 \ cm \\ r&=3\end{align*}
1. .
\begin{align*}m \overline{I J} &= 100 \ cm \\ r&=\frac{1}{10}\end{align*}

Find the measure of the preimage given the following information:

1. .
\begin{align*}m \overline{K^\prime L^\prime} &= 48 \ cm \\ r&=4\end{align*}
1. .
\begin{align*}m \overline{M^\prime N^\prime} &= 32 \ cm \\ r&=4\end{align*}
1. .
\begin{align*}m \overline{O^\prime P^\prime} &= 36 \ cm \\ r&=6\end{align*}
1. .
\begin{align*}m \overline{Q^\prime R^\prime} &= 20 \ cm \\ r&=\frac{1}{4}\end{align*}
1. .
\begin{align*}m \overline{S^\prime T^\prime} &= 40 \ cm \\ r&=\frac{4}{5}\end{align*}

Describe the following dilations:

## Graphing an Image that Has Undergone a Dilation

Concept Content

In this second concept of lesson Dilations of Geometric Shapes, you will learn how to graph an image that has undergone a dilation. Remember that a dilation occurs when a preimage is increased or decreased by a scale factor that is based on the center point. In the last lesson you learned to recognize what happens to the points and images when undergoing a dilation. In this lesson you will create the images and preimages by first graphing them on a Cartesian plane. When graphing the dilation image, it is often helpful to remember that what happens to the points in the preimage depends on the scale factor.

Guidance

Quadrilateral \begin{align*}WXYZ\end{align*} has coordinates \begin{align*}W(-5, -5), X(-2, 0), Y(-1, 3)\end{align*} and \begin{align*}Z(2, 3)\end{align*}. Draw the quadrilateral on the Cartesian plane. The quadrilateral has a dilation centered at the origin of scale factor \begin{align*}\frac{1}{3}\end{align*}. Show the resulting image.

Test to see if the dilation is correct by determining the scale factor.

\begin{align*}& scale \ factor = \frac{dilation \ image \ length}{preimage \ length} \\ & scale \ factor = \frac{10.63}{3.54} \\ & scale \ factor = 3\end{align*}

Examples

Example A

Line \begin{align*}\overline{A B}\end{align*} drawn from (–4, 2) to (3, 2) has undergone a dilation about the origin to produce \begin{align*}A^\prime(-6, 3)\end{align*} and \begin{align*}B^\prime(4.5, 3)\end{align*}. Draw the preimage and dilation image and determine the scale factor.

\begin{align*}& scale \ factor = \frac{dilation \ image \ length}{preimage \ length} \\ & scale \ factor = \frac{10.5}{7.0} \\ & scale \ factor = \frac{3}{2}\end{align*}

Example B

The diamond \begin{align*}ABCD\end{align*} undergoes a dilation about the origin to form the image \begin{align*}A^\prime B^\prime C^\prime D^\prime\end{align*}. Find the coordinates of the dilation image. Using the diagram, determine the scale factor.

\begin{align*}& scale \ factor = \frac{dilation \ image \ length}{preimage \ length}\\ & scale factor = \frac{7.21}{3.61} \\ & scale factor = 2\end{align*}

Example C

The diagram below undergoes a dilation about the origin to form the dilation image. Find the coordinates of \begin{align*}A\end{align*} and \begin{align*}B\end{align*} and \begin{align*}A^\prime\end{align*} and \begin{align*}B^\prime\end{align*} of the dilation image. Using the diagram, determine the scale factor.

\begin{align*}& scale factor = \frac{dilation \ image \ length}{preimage \ length}\\ & scale \ factor = \frac{2.00}{10.00} \\ & scale \ factor = \frac{1}{5}\end{align*}

Vocabulary

Dilations
Dilations in transformations involve changing the size of a preimage to create the dilation image.
Scale Factor
The scale factor will determine how much bigger or smaller the dilation image will be. The scale factor is sometimes called the scalar factor.

Guided Practice

1. Line \begin{align*}\overline{S T}\end{align*} drawn from (–3, 4) to (–3, 8) has undergone a dilation of scale factor 3 about the point \begin{align*}A (1, 6)\end{align*}. Draw the preimage and image and properly label each.

2. The polygon below has undergone a dilation about the origin with a scale factor of \begin{align*}\frac{5}{3}\end{align*}. Draw the dilation image and properly label each.

3. The triangle with vertices \begin{align*}J(-5, -2), K(-1, 4)\end{align*} and \begin{align*}L(1, -3)\end{align*} has undergone a dilation of scale factor \begin{align*}\frac{1}{2}\end{align*}. about the center point \begin{align*}L\end{align*}. Draw and label the dilation image and the preimage then check the scale factor.

1.

\begin{align*}& scale \ factor = \frac{dilation \ image \ length}{preimage \ length} \\ & scale \ factor = \frac{12.00}{4.00} \\ & scale \ factor = 3\end{align*}

2.

\begin{align*}& scale factor = \frac{dilation \ image \ length}{preimage \ length} \\ & scale \ factor = \frac{5.00}{3.00} \\ & scale \ factor = \frac{5}{3}\end{align*}

3.

\begin{align*}& scale \ factor = \frac{dilation \ image \ length}{preimage \ length}\\ & scale \ factor = \frac{7.21}{3.61} \\ & scale \ factor = \frac{1}{2}\end{align*}

Summary

In this concept of Chapter Is it a Slide, a Flip, or a Turn?, lesson Dilations of Geometric Shapes, you were introduced to graphing dilations knowing the scale factor and the center point. When graphing the dilation image, it is often helpful to remember what happens to the points in the preimage depends on the scalar factor. The preimage will be increased or decreased in size because of this scale factor. The position of the dilation image depends on the center point. Remember that the shape does not change; just the size does.

Problem Set

1. Draw a dilation image for the figure \begin{align*}MATH\end{align*} on the grid below about the origin with a scale factor of \begin{align*}\frac{1}{2}\end{align*}.
2. Draw a dilation image for Image A on the grid below about the origin with a scale factor of 3.
3. Draw a dilation image for the figure below on the grid below about the point \begin{align*}D\end{align*} with a scale factor of \begin{align*}\frac{3}{2}\end{align*}.
4. Draw a dilation image for the Image A on the grid below about the origin with a scale factor of 2.
5. Draw a dilation image for the figure on the grid below about the origin with a scale factor of \begin{align*}\frac{2}{3}\end{align*}.

For each of the following diagrams the images undergo a dilation about the origin by a scale factor of 2. On each diagram, draw and label the dilation image. Test the images to see if the scale factor is correct.

For each of the following diagrams the images undergo a dilation about the origin by a scale factor of \begin{align*}\frac{1}{2}\end{align*}. On each diagram, draw and label the dilation image. Test the images to see if the scale factor is correct.

## Writing a Rule for a Dilation

Concept Content

In this third concept with respect to dilations, you will learn how to describe a dilation using a rule. In the first lesson you used words to describe a dilation. By examining the coordinates of the dilation image, you could describe if the preimage underwent a dilation about the origin or about a point with a specific scale factor. You might have noticed that the coordinate points for positive scale factors are also multiplied by the scale factor to produce the dilation image. Look at the diagram below:

The Image A has undergone a dilation about the origin with a scale factor of 2. Notice that the points in the dilation image are all double the coordinate points in the preimage. A dilation with a scale factor \begin{align*}k\end{align*} about the origin can be described using the following notation:

\begin{align*}D_k(x, y)=(kx, ky)\end{align*}

If \begin{align*}k\end{align*} is greater than one, the dilation image will be larger than the preimage. If \begin{align*}k\end{align*} is between 0 and 1, the dilation image will be smaller than the preimage. If \begin{align*}k\end{align*} is equal to 1, you will have a dilation image that is congruent to the preimage. The mapping rule corresponding to a dilation notation would be:

\begin{align*}(x, y) \rightarrow (kx, ky)\end{align*}

In this concept you will write rules for dilation about the origin as well as draw graphs from such rules.

Guidance

The figure below shows a dilation of two trapezoids. Write the mapping rule for the dilation of Image A to Image B.

Look at the points in each image:

\begin{align*}& \text{Image} \ A \qquad \ B(-9, 6) \quad \qquad C(-5, 6) \quad \qquad D(-5, -1) \qquad \quad \ \ E(-10, -3) \\ & \text{Image} \ B \qquad B^\prime(-4.5, 3) \qquad C^\prime(-2.5, 3) \qquad D^\prime(-2.5, -0.5) \qquad E^\prime(-5, -1.5)\end{align*}

Notice that the coordinate points in Image B (the dilation image) are \begin{align*}\frac{1}{2}\end{align*} that found in Image A. Therefore the Image A undergoes a dilation about the origin of scale factor \begin{align*}\frac{1}{2}\end{align*}. To write a mapping rule for this dilation you would write: \begin{align*}(x, y) \rightarrow \left(\frac{1}{2}x, \frac{1}{2}y\right)\end{align*}.

Examples

Example A

The mapping rule for the dilation applied to the triangle below is \begin{align*}(x, y) \rightarrow (1.5x, 1.5y)\end{align*}. Draw the dilation image.

With a scale factor of 1.5, each coordinate point will be multiplied by 1.5.

\begin{align*}& \text{Image} \ A \qquad \qquad \ \ \ A(3, 5) \quad \qquad \ \ B(4, 2) \quad \qquad \ \ \ C(1, 1) \\ & \text{Dilation Image} \qquad A^\prime(4.5, 7.5) \qquad B^\prime(6, 3) \qquad C^\prime(1.5, 1.5)\end{align*}

The dilation image looks like the following:

Example B

The mapping rule for the dilation applied to the diagram below is \begin{align*}(x, y) \rightarrow \left(\frac{1}{3}x, \frac{1}{3}y\right)\end{align*}. Draw the dilation image.

With a scale factor of \begin{align*}\frac{1}{3}\end{align*}, each coordinate point will be multiplied by \begin{align*}\frac{1}{3}\end{align*}.

\begin{align*}& \text{Image} \ D \qquad \qquad \ \ \ D(-3, 7) \quad \qquad E(-1, 3) \quad \qquad F(-7, 5) \quad \qquad \ \ G(-5, 1) \\ & \text{Dilation Image} \qquad D^\prime(-1, 2.3) \qquad E^\prime(-0.3, 1) \qquad F^\prime(-2.3, 1.7) \qquad G^\prime(-1.7, 0.3)\end{align*}

The dilation image looks like the following:

Example C

Write the notation that represents the dilation of the preimage A to the dilation image J in the diagram below.

First, pick a point in the diagram to use to see how it has been affected by the dilation.

\begin{align*}C:(-7, 5) \quad C^\prime: (-1.75, 1.25)\end{align*}

Notice how both the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates are multiplied by \begin{align*}\frac{1}{4}\end{align*}. This indicates that the preimage A undergoes a dilation about the origin by a scale factor of \begin{align*}\frac{1}{4}\end{align*} to form the dilation image J. Therefore the mapping notation is \begin{align*}(x, y) \rightarrow \left(\frac{1}{4}x, \frac{1}{4}y\right)\end{align*}.

Vocabulary

Notation Rule
A notation rule has the following form \begin{align*}D_k(x, y)=(kx, ky)\end{align*} and tells you that the preimage has undergone a dilation about the origin by scale factor \begin{align*}k\end{align*}. If \begin{align*}k\end{align*} is greater than one, the dilation image will be larger than the preimage. If \begin{align*}k\end{align*} is between 0 and 1, the dilation image will be smaller than the preimage. If \begin{align*}k\end{align*} is equal to 1, you will have a dilation image that is congruent to the preimage. The mapping rule corresponding to a dilation notation would be: \begin{align*}(x, y) \rightarrow (kx, ky)\end{align*}
Dilations
Dilations in transformations involve changing the size of a preimage to create the dilation image.

Guided Practice

1. Thomas describes a dilation of point \begin{align*}JT\end{align*} with vertices \begin{align*}J(-2, 6)\end{align*} to \begin{align*}T(6, 2)\end{align*} to point \begin{align*}J^\prime T^\prime\end{align*} with vertices \begin{align*}J^\prime(-4, 12)\end{align*} and \begin{align*}T^\prime(12, 4)\end{align*}. Write the notation to describe this dilation for Thomas.

2. Given the points \begin{align*}A(12, 8)\end{align*} and \begin{align*}B(8, 4)\end{align*} on a line undergoing a dilation to produce \begin{align*}A^\prime(6, 4)\end{align*} and \begin{align*}B^\prime(4, 2)\end{align*}, write the notation that represents the dilation.

3. Janet was playing around with a drawing program on her computer. She created the following diagrams and then wanted to determine the transformations. Write the notation rule that represents the transformation of the preimage A to the image A'.

1.

Since the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates are each multiplied by 2, the scale factor is 2. The mapping notation is: \begin{align*}(x, y)\rightarrow (2x, 2y)\end{align*}

2. In order to write the notation to describe the dilation, choose one point on the preimage and then the corresponding point on the dilation image to see how the point has moved. Notice that point \begin{align*}EA\end{align*} is:

\begin{align*}A(12, 8) \rightarrow A^\prime(6, 4)\end{align*}

Since both \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates are multiplied by \begin{align*}\frac{1}{2}\end{align*}, the dilation is about the origin has a scale factor of \begin{align*}\frac{1}{2}\end{align*}. The notation for this dilation would be: \begin{align*}(x, y) \rightarrow \left(\frac{1}{2}x, \frac{1}{2}y \right)\end{align*}.

3. In order to write the notation to describe the dilation, choose one point on the preimage A and then the corresponding point on the dilation image \begin{align*}A^\prime\end{align*} to see how the point has changed. Notice that point \begin{align*}E\end{align*} is shown in the diagram:

\begin{align*}E(-5, -3) \rightarrow E^\prime(-1, -0.6)\end{align*}

Since both \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates are multiplied by \begin{align*}\frac{1}{5}\end{align*}, the dilation is about the origin has a scale factor of \begin{align*}\frac{1}{5}\end{align*}. The notation for this dilation would be: \begin{align*}(x, y) \rightarrow \left(\frac{1}{5}x, \frac{1}{5}y \right)\end{align*}.

Summary

In this concept of lesson Dilations of Geometric Shapes you have learned to describe a dilation in notation form. Notations for dilations can allow you to quickly determine what the scale factor is for the preimage to undergo a dilation. The notation for a dilation of scale factor \begin{align*}k\end{align*} is \begin{align*}D_k(x, y)=(kx, ky)\end{align*}.

Scale Factor, \begin{align*}k\end{align*} Size change for preimage
\begin{align*}k>1\end{align*} Dilation image is larger than preimage
\begin{align*}0 < k < 1\end{align*} Dilation image is smaller than preimage
\begin{align*}k=1\end{align*} Dilation image is the same size as the preimage

Mapping notation for dilations follows the form \begin{align*}(x, y) \rightarrow (kx, ky)\end{align*} and also allows you to quickly determine the scale factor in images as the dilation image can simply be determine by multiplying the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates by \begin{align*}k\end{align*}.

Problem Set

Complete the following table:

Starting Point Scale factor = 2 Scale factor = 5 Scale factor = \begin{align*}\frac{1}{2}\end{align*} Scale factor = \begin{align*}\frac{3}{4}\end{align*}
1. (1, 4)
2. (4, 2)
3. (2, 0)
4. (–1, 2)
5. (–2, –3)

Write the notation that represents the dilation of the preimage A to the dilation image in the diagrams below.

## The Properties of a Dilation

Concept Content

As you learned earlier, dilations involve increasing or decreasing the size of a preimage to create a dilation image. In this last concept you will look at the properties of dilations. Dilations result when an image is multiplied by a scale factor with respect to a center point. You have seen in the examples of dilations from the previous three concepts where a dilation occurs on the preimage across the origin or from a point (called the center point) and with a specified scale factor. Therefore rotated images have the same length and angle measurements as their preimages. As well, points of the rotated image are collinear if they are collinear in the preimage. In other words, if \begin{align*}B\end{align*} is between \begin{align*}A\end{align*} and \begin{align*}C\end{align*}, then \begin{align*}B^\prime\end{align*} will be between \begin{align*}A^\prime\end{align*} and \begin{align*}C^\prime\end{align*}.

In summary, dilations are considered to have the following properties:

• angle measures in a figure remain the same (angles)
• same orientation (lettering remains the same)
• points that are on each line remain on the line for the dilation image (collinear)
• parallelism (parallel lines remain parallel)

Guidance

The triangle \begin{align*}ABC\end{align*} is drawn such that the vertices are at \begin{align*}A(1, 1), B(5, 5)\end{align*} and \begin{align*}C(-2, 4)\end{align*}. Triangle \begin{align*}A^\prime B^\prime C^\prime\end{align*} is a result of multiplying the preimage by a scale factor of 2.

a) What are the coordinates of Triangle \begin{align*}A^\prime B^\prime C^\prime\end{align*}?

b) Measure each angle either with a protractor or using geometry software. What angles are congruent?

c) Can you conclude that the triangle \begin{align*}A^\prime B^\prime C^\prime\end{align*} is a dilation of triangle \begin{align*}ABC\end{align*}?

a) The coordinates of \begin{align*}A^\prime B^\prime C^\prime\end{align*} after a dilation of triangle \begin{align*}ABC\end{align*} would be:

\begin{align*}& \text{Preimage} \ ABC \qquad \qquad \quad \ \ A(1, 1) \qquad \ B(5, 5) \quad \qquad \ C(-2, 4) \\ & \text{Reflected Image} \ A^\prime B^\prime C^\prime \qquad A^\prime(2, 2) \qquad B^\prime(10, 10) \qquad C^\prime(-4, 8) \end{align*}

b) The measures of each angle are shown in the diagram.

Congruent angles have the same measure. Since \begin{align*}\angle A = \angle A^\prime, \angle B = \angle B^\prime\end{align*}, and \begin{align*}\angle C = \angle C^\prime\end{align*}, the angles are congruent.

c) Since the angle measures are the same the triangles are congruent and each coordinate point of the preimage has been multiplied by the scale factor to produce the dilation image. Therefore, you can conclude that triangle \begin{align*}ABC\end{align*} has undergone a dilation to produce triangle \begin{align*}A^\prime B^\prime C^\prime\end{align*}.

Examples

Example A

Graph the following two quadrilaterals to determine if Quadrilateral A undergoes a dilation to form Quadrilateral B. Show that this graph represents a dilation. Determine the scale factor.

Quadrilateral A Vertices: \begin{align*}W(2, -2), X(5, 0), Y(5, 3),\end{align*} and \begin{align*}Z(4, 3)\end{align*}.

Quadrilateral B Vertices: \begin{align*}W^\prime(1, -1), X^\prime(2.5, 0), Y^\prime(2.5, 1.5),\end{align*} and \begin{align*}Z^\prime(2, 1.5)\end{align*}.

The angle lengths are the same \begin{align*}(\angle X = \angle X^\prime, \angle Y = \angle Y^\prime, \angle W = \angle W^\prime\end{align*}, and \begin{align*}\angle Z = \angle Z^\prime)\end{align*}.

Since the angle measures are the same Quadrilateral A is congruent to Quadrilateral B. Each coordinate point on \begin{align*}ABCD\end{align*} is multiplied by \begin{align*}\frac{1}{2}\end{align*} to produce Quadrilateral B. With all of this, you can conclude that Quadrilateral B is a dilation image of the preimage Quadrilateral A. The dilation is about the origin with a scale factor of \begin{align*}\frac{1}{2}\end{align*}.

You can also test the scale factor by looking at the side lengths and using the formula:

\begin{align*}& scale \ factor = \frac{dilation \ image \ length}{preimage \ length}\\ & scale \ factor = \frac{1.50}{3.00} \\ & scale \ factor = \frac{1}{2} \end{align*}

Example B

Describe how you would know if image \begin{align*}T\end{align*} has undergone a dilation to form image \begin{align*}T^\prime\end{align*}.

As indicated on the graph below the angle measures are all the same. Image \begin{align*}T\end{align*} is congruent to Image \begin{align*}T^\prime\end{align*}. Each coordinate point is multiplied by \begin{align*}\frac{3}{2}\end{align*}. With all of this, you can conclude that the Image \begin{align*}T\end{align*} undergoes a dilation of scale factor \begin{align*}\frac{3}{2}\end{align*} about the origin to form Image \begin{align*}T^\prime\end{align*}.

You can also test the scale factor by looking at the side lengths and using the formula:

\begin{align*}& scale \ factor = \frac{dilation \ image \ length}{preimage \ length}\\ & scale \ factor = \frac{7.66}{5.11} \\ & scale \ factor = \frac{3}{2} \end{align*}

Example C

Describe how you would know if image \begin{align*}S\end{align*} has undergone a dilation to form the blue image \begin{align*}S^\prime\end{align*}.

As indicated on the graph below the arc angle measures are all the same. Image \begin{align*}S\end{align*} is congruent to Image \begin{align*}S^\prime\end{align*}. Each coordinate point is multiplied by 2. With all of this, you can conclude that the Image \begin{align*}S\end{align*} undergoes a dilation of scale factor 2 about the origin to form Image \begin{align*}S^\prime\end{align*}.

You can also test the scale factor by looking at the side lengths and using the formula:

\begin{align*}& scale \ factor = \frac{dilation \ image \ length}{preimage \ length} \\ & scale \ factor = \frac{10.0}{5.00} \\ & scale \ factor = 2 \end{align*}

Vocabulary

Congruent Angles
Congruent angles have the same measure.
Dilations
Dilations in transformations involve changing the size of a preimage to create the dilation image.

Guided Practice

Describe how you would know if the preimage has undergone a dilation to produce the dilation image.

1.

2. Describe how you would know if image T is a dilation of preimage S.

3. Is the image T a dilation image of preimage A? How do you know?

1. As indicated on the graph below the angle measures are all the same. The preimage is congruent to the dilation image. Each coordinate point is multiplied by 3. With all of this, you can conclude that the preimage undergoes a dilation of scale factor 3 about the origin to form the dilation image.

You can also test the scale factor by looking at the side lengths and using the formula:

\begin{align*}& scale \ factor = \frac{dilation \ image \ length}{preimage \ length}\\ & scale \ factor = \frac{7.51}{2.50} \\ & scale \ factor = 3 \end{align*}

2.

As indicated on the graph above the angle measures are all the same. The preimage S is congruent to the dilation image T. Each coordinate point is multiplied by \begin{align*}\frac{3}{4}\end{align*}. You also see that the points \begin{align*}BDE\end{align*} are collinear in the preimage S and \begin{align*}B^\prime D^\prime E^\prime\end{align*} are collinear in the dilation image T. With all of this, you can conclude that the preimage S undergoes a dilation of scale factor \begin{align*}\frac{3}{4}\end{align*} about the origin to form the dilation image T. You can also test the scale factor by looking at any congruent pair of side lengths and using the formula:

\begin{align*}& scale \ factor = \frac{dilation \ image \ length}{preimage \ length}\\ & scale \ factor = \frac{6.75}{9.00} \\ & scale \ factor = \frac{3}{4} \end{align*}

3.

As indicated on the graph above the angle measures are all the same. The preimage A is congruent to the dilation image T. Each coordinate point is multiplied by a scale factor but since the center point is \begin{align*}C\end{align*} and not the origin, the scale factor must be determined using the formula:

\begin{align*}& scale \ factor = \frac{dilation \ image \ length}{preimage \ length} \\ & scale \ factor = \frac{10.62}{3.54} \\ & scale \ factor = 3 \end{align*}

With all of this, you can conclude that the preimage A undergoes a dilation of scale factor 3 about the center point C to form the dilation image T.

Summary

In the previous three lessons you worked with dilations. A dilation is a transformation of a figure that involves increasing or decreasing the size of the preimage along the Cartesian plane. In this lesson you looked more closely at the properties of dilations. These properties include:

• line segments are the same length (distance)
• same orientation (lettering remains the same)
• points that are on each line remain on the line for the reflected image (collinear)
• parallelism (parallel lines remain parallel)

Through the use of geometry software, it is easy to measure both the angles to determine if they are congruent, thus satisfying the first property. If the points are moving in the same direction, you can be sure that the preimage is undergoing a dilation to form the new image.

Problem Set

Describe how you would know if image S undergoes a dilation to produce the second image in each of the following diagrams.

1. Triangle \begin{align*}ABC\end{align*} with vertices \begin{align*}A(3, 1), B(0, -1)\end{align*} and \begin{align*}C(1, 5)\end{align*} has undergone a dilation of scale factor 4 about the origin to produce triangle \begin{align*}A^\prime B^\prime C^\prime\end{align*}. Draw the diagram representing this dilation and prove the image \begin{align*}A^\prime B^\prime C^\prime\end{align*} is actually a dilation image.
2. Triangle \begin{align*}DEF\end{align*} with vertices \begin{align*}D(0, 1), E(5, 1)\end{align*} and \begin{align*}F(2, 5)\end{align*} has undergone a dilation of scale factor 2 about the origin to produce triangle \begin{align*}A^\prime B^\prime C^\prime\end{align*}. Draw the diagram representing this dilation and prove the image \begin{align*}A^\prime B^\prime C^\prime\end{align*} is actually a dilation image.
3. Quadrilateral \begin{align*}DEFG\end{align*} with vertices \begin{align*}D(0, 1), E(1, 5), F(2, 5)\end{align*}, and \begin{align*}G(5, 1)\end{align*} has undergone a dilation of scale factor \begin{align*}\frac{1}{3}\end{align*} about the origin to produce quadrilateral \begin{align*}D^\prime E^\prime F^\prime G^\prime\end{align*}. Draw the diagram representing this dilation and prove the image \begin{align*}D^\prime E^\prime F^\prime G^\prime\end{align*} is actually a dilation image.
4. Quadrilateral \begin{align*}IJKL\end{align*} with vertices \begin{align*}I(0, 2), J(2, 3), K(3, 6)\end{align*}, and \begin{align*}L(3, 1)\end{align*} has undergone a dilation of scale factor \begin{align*}\frac{2}{5}\end{align*} about the origin to produce quadrilateral \begin{align*}I^\prime J^\prime K^\prime L^\prime\end{align*}. Draw the diagram representing this dilation and prove the image \begin{align*}I^\prime J^\prime K^\prime L^\prime\end{align*} is actually a dilation image.
5. Prove that the preimage triangle 1 undergoes a dilation about the origin to produce triangle 2. Determine the scale factor of triangle 1 to triangle 2

## Summary

In this fourth lesson of Chapter Is it a Slide, a Flip, or a Turn? you have been introduced to dilations. Remember that a dilation involves increasing or decreasing the size of a figure by a scale factor, \begin{align*}k\end{align*}, about a center point. Dilations are an important concept as you notice them probably on a daily basis in everyday life. Making enlargements on photocopiers, enlargements of photo prints, changing the pixel size on your computer, or building models such as model planes or cars all involve using the properties of dilations.

You first learned to describe the dilation using words. So, you learned to look at points in a dilation image and the preimage to determine if the images involve:

• A scale factor greater than 1 and therefore you have an increase in size, or
• A scale factor between 0 and 1 and therefore you have an decrease in size.

You could also have a scale factor equal to 1 and therefore have a congruent image. In this lesson you described the type of dilation using words. Following this you learned to draw a dilation image. You could make this drawing knowing the preimage coordinates and the scale factor if the center point was the origin. It was in this lesson that you were introduced to the formula for finding the scale factor.

\begin{align*}& scale \ factor = \frac{dilation \ image \ length}{preimage \ length}\end{align*}

In the third concept, you learned to write the notation for the dilated image from the preimage. You learned to graph a dilated image and then write the notation for the image. The notation for a dilation is:

\begin{align*}D_k(x, y)=(kx, ky)\end{align*}

with a mapping notation of:

\begin{align*}(x, y) \rightarrow (kx, ky)\end{align*}

Finally in the last concept of this lesson, you learned that dilations have the following properties:

• angle measures in a figure remain the same (angles)
• same orientation (lettering remains the same)
• points that are on each line remain on the line for the reflected image (collinear)
• parallelism (parallel lines remain parallel)

Notice, unlike the other three transformations, dilations do not have sides that are the same length. Remember that dilations involve increasing or decreasing the size of a shape. Remember as well, that the figure itself does not change in shape. You used these tendencies to prove that one image is a dilation of another image by finding the angles of the preimage and the dilation image. You then found where the points of the preimage were multiplied by the scale factor to produce the dilation image. All of these properties proved that the one image was a dilation of the other.

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