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1. Dilation and Scale Factor
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Chapter 2
1. 

Dilation and Scale Factor

Dilation is a transformation that alters an object's size without changing its shape. Through the use of scale factors, one can either enlarge or reduce figures. This process, when combined with similarity principles, offers a method to understand and compare the proportionality of shapes. Grasping dilations and scale factors enables professionals, students, and educators to analyze geometric patterns more effectively, aiding in tasks ranging from design to theorem derivation. A solid foundation in these concepts provides an avenue to more advanced geometrical studies and applications.

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Student Learning Objectives:
  • Understand and draw dilations
  • Identify the scale factor of a dilation
13 Theory slides
9 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Dilation and Scale Factor
Slide of 13
This is the first lesson on dilations and similarities. In this lesson the properties of dilations will be explored to build the foundations to investigate similarities later.

Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.

Although not directly used, it can be useful to recall the properties of reflections, rotations, and translations to help identify the differences and similarities of rigid motions and dilations.


Explore

Investigating Dilations in the Real World

On the image you can see a photograph taken in Tennoji Park in Osaka, Japan.

  • Move the sliders to move the orange figure on the left.
  • Suppose you were to draw this image, standing in a different position than this image's photographer. How could you draw the image so that the two figures' heights always appear to be the same?
External credits: Laitche
Discussion

Definition of a Dilation

When the vertical slider is moved on the previous applet, the transformation applied to the figure is called a dilation.

Concept

Dilation

A dilation is a transformation that changes the size of a figure while keeping its shape the same. This transformation involves enlarging or reducing the figure by a certain length scale factor k from a fixed point O called the center of dilation. For example, the image of every point on a leaf lies on the ray that starts at the center of the dilation and passes through its preimage.

As shown in the diagram, O is the center of the dilation, k is the scale factor, A is the preimage, and A' is the image point. By definition, the scale factor k can also be defined as the ratio of a length in the image to the corresponding length in the preimage.

OA'=k * OA ⇔ k = OA'/OA

When the scale factor is greater than 1, the dilation is called an enlargement because the image is larger than the preimage. When the scale factor is between 0 and 1, the dilation is called a reduction because the image is smaller than the preimage.
Explore

Properties of a Dilation

The following applet shows the images of points on a straight line. There are two modes in the applet, Setup and Mark image — complete the first to reach the next.

  • In setup mode, set the scale factor, move the line, and slide the center of dilation as you please.
  • Once done with the setup, move the point on the line to a different position. Then, mark the image point on the board. Mark multiple points.
  • What do you notice about the image points?
Explore

More Properties of a Dilation

This applet shows the distance of two points (middle blue points) and the corresponding image points (further right green points).

  • Move the center (furthest left red point), move the two middle points around, and slide the bar to change the scale factor.
  • Is there a relationship between the two distances?
Discussion

Summary of a Dilation's Properties

The following is a list of a few essential properties of dilations.

  • The dilation of a line is a line.
    • If the line does not pass through the center of dilation, the image line is parallel to the preimage.
    • If the line passes through the center of dilation, the image line and the preimage line are the same.

Proof
The formal proof of this property is beyond the scope of this course. The applet below, however, gives an experimental verification.

  • Move the line, the center of dilation, and set the scale factor.
  • Move a point along the line and verify that the image points indeed form a line parallel to the preimage.




  • The dilation of a line segment is a line segment. The image is longer or shorter than the preimage in the ratio given by the scale factor.

Proof
The formal proof of this property is beyond the scope of this course. The applet below, however, gives an experimental verification.

  • Move the segment, the center of dilation, and set the scale factor.
  • Move a point along the segment and verify that the image points indeed form a segment parallel to the preimage.
  • Check also that the ratio of the length of the image and the length of the preimage is indeed the scale factor. Note that the displayed lengths are only approximate, so the ratio will only be approximately equal to the scale factor.




  • Dilations preserve angle measures.

Proof
Let ∠ B'A'C' be the dilated image of ∠ BAC. According to the first property, AC is parallel to A'C' and AB is parallel to A'B'

Let M be the intersection point of AB and A'C', and focus on the parallel legs AC and A'C'. According to the Alternate Interior Angles Theorem, angles ∠ MAC and ∠ AMA' are congruent.

Similarly, since the other legs are also parallel, angles ∠ B'A'M' and ∠ AMA' are congruent.

Since ∠ AMA' is congruent to both ∠ BAC and ∠ B'A'C', the transitive property of congruence implies that these two angles are congruent.

By the definition of congruence, this completes the proof that dilation preserves angle measures.

∠ BAC ≅ ∠ B'A'C' ⇓ m∠ BAC = m∠ B'A'C'

The properties guarantee that the dilated image has the same shape as the preimage. The size is different if the scale factor is not 1. If the scale factor is less than 1, then the image is reduced. If the scale factor is greater than 1, then the image is enlarged.
Illustration

Dilating a Drawing

On the applet below you can experiment with dilations.

  • Move the pencil to draw a curve.
  • The red point is the center of dilation.
  • Use the slider to set the scale factor.
Example

Dilating a Quadrilateral

Consider the quadrilateral ABCD and the point O in the interior.

a

Use O as a center and dilate the quadrilateral with scale factor 2.

b

Use O as a center and dilate the quadrilateral with scale factor 0.5.

Answer
a

b

Hint
Start with drawing rays connecting the center of dilation with the vertices of the quadrilateral.

Solution
For both parts, the properties of dilation guarantee that the image is a quadrilateral, where the vertices are the images of the vertices of a quadrilateral ABCD. These image vertices are on the rays connecting the center of dilation with vertices A, B, C, and D.

a

For scale factor 2 the image points are twice as far from the center of dilation as the preimage.

OA'&=2* OA OB'&=2* OB OC'&=2* OC OD'&=2* OD These points can be constructed using a compass. Copy the distance from the center to a vertex once on the ray beyond the original point.

B

For scale factor 0.5 the image points are half the distance from the center of dilation as the preimage.

OA''&=1/2* OA OB''&=1/2* OB OC''&=1/2* OC OD''&=1/2* OD These points can be constructed as midpoints between two points.

Example

The larger logo is an enlarged image of the smaller one.

Find the center and the scale factor of the dilation.

Answer

Scale factor: 1.6

Hint
Look for pairs of corresponding points on the two logos.

Solution
The center of dilation, a point, and the image of this point are on a straight line. Therefore, the center of dilation can be determined to be the intersection point of any two lines connecting a point and its image.

A ruler can be used to find the distance of any point and its image from the center of dilation.

The distance from the center of dilation to the lower-left corner of the letter M on the preimage is 5.4 centimeters. The distance from the center of dilation to the corresponding image point is 8.6 centimeters. The scale factor is the ratio of these. Scale factor:8.6/5.4≈ 1.6
Pop Quiz

Practice Dilations

On the applet the preimage is the shaded (blue) triangle, the image is the green triangle. Find the center of dilation and the scale factor. The scale factor is one of the following numbers: 0.25, 0.5, 0.75, 2, 3, 4

Example

Dilating a Triangle in a Cooordinate Plane

Use the origin as a center to dilate the triangle by a scale factor 2.

Triangle in the coordinate system with vertices at A(0,1), B(3,2), and C(2,0).


Answer
Two triangles in the coordinate system with vertices at A(0,1), B(3,2), C(2,0) and A'(0,2), B'(6,4), C'(0,4).

Hint
Use horizontal and vertical movements to copy distances.

Solution
The points on the coordinate axes stay on the coordinate axes while their distances, from the origin, double.

When moving the third vertex, notice that moving three units to the right and two units up from the origin gives the position of B. The same movement starting at B will end on the ray connecting the origin with B. This image point is twice as far from the origin as the preimage.

Connecting the images of the vertices gives the dilated triangle.

Two triangles in the coordinate system with vertices at A(0,1), B(3,2), C(2,0) and A'(0,2), B'(6,4), C'(0,4).

Since the center of the dilation is the origin and the scale factor is 2, the coordinates of the image point are double the coordinates of the preimage points. A(0,1) &→ A'(0,2) B(3,2) &→ B'(6,4) C(2,0) &→ C'(4,0)
Discussion

Dilation Using a Straightedge and a Compass

The methods to construct the image of a point depend on the scale factor.

Dilation by an Integer Scale Factor

When the scale factor is an integer, a compass can be used to copy the distance between the center of dilation and the preimage point to find the position of the image.

1
Draw a ray from the center of dilation through the preimage point
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Use a straightedge to draw a ray from the center of dilation through the preimage point.


2
Measure the distance from the center of dilation to the preimage point
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Use a compass to measure the distance from the center of dilation to the preimage point.

3
Copy the measured distance on the ray
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Keep the same opening of the compass and copy the distance on the ray. If the scale factor is n, copy it n-1 times. OA'=n* OA The diagram illustrates the construction for scale factor 3, so the distance is copied twice from the preimage point.

Dilation by a Scale Factor 1/n

To construct the image when the scale factor is 1/n, the properties of dilation can be used.

1
Draw two rays starting at the center of dilation
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Use a straightedge to connect the center of dilation with the preimage point and draw a second ray starting at the center of dilation.


2
Copy the same distance on the second ray n times
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Use a compass to copy the same distance n times on the second ray, starting at the center of dilation. The diagram illustrates the process when n=3.


3
Construct parallel lines
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Mark the first and last point on the second ray. Connect the last point with the preimage point and construct a parallel line through the first point.


4
Mark the image point
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The intersection point of this parallel line with the first ray is the constructed image. OA'=1/n* OA


Dilation by a Scale Factor p/q

The combination of the previous two methods gives the dilation of a point by a scale factor p/q.

1
Dilation by a scale facor 1/q
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First, construct the image using a scale factor 1/q. OT = 1/q * OA On the diagram q=5.


2
Dilation by a scale facor p
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Copy the distance of this new point from the center of dilation p times, starting at the center of dilation. OA' = p* OT OT = 1/q* OA ⟹ OA' = p/q* OA The diagram below illustrates the dilation by scale factor 3/5.


Dilation of a Figure

The dilation of a figure is the collection of all dilated points. The dilation of a polygon can be constructed by dilating the vertices and connecting the image points.

1
Construct the image of the vertices.
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Depending on the scale factor, choose the appropriate method to construct the image of the vertices.

2
Connect the image vertices.
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Segments are drawn between the image vertices to create the image polygon.

Closure

Investigating the Effects of Several Dilations

The following applet allows you to investigate the effect of applying two dilations one after the other. Use the sliders to adjust the scale factors and move the centers around.

  • The center of the first dilation is blue, the center of the second dilation is green.
  • The preimage is the orange ellipse. The first dilation maps this to the blue ellipse. The second dilation maps this blue ellipse to the green ellipse.

Notice that the combination of the two dilations can be replaced by one dilation.

  • What is the scale factor of this dilation?
  • Where is the center of this dilation?
  • Is the combination always a dilation?



Level 1
Level 2
Level 3
Dilation and Scale Factor
Exercise 2.1
In the following diagram, the larger triangle is a dilation of the smaller triangle.

Two triangles, one being a dilation of the other
 Find the length of the side marked 2x+9.

Find the measure of the angle marked (3y-34)^(∘).

To determine the value of x, we must equate the side marked (x+ 37) in the original triangle with the side marked (2x+9) in the dilated triangle and then solve for x. However, to do that we must first determine the scale factor of dilation.

Scale Factor of Dilation

The scale factor of dilation can be determined by using the following formula. k= Distance to imageDistance to preimage From the diagram, we know that the distance from the center of dilation to the preimage is 2. The corresponding length from the center of dilation to the image is 7. With this information, we can determine the scale factor. k= 7 2 Any side in the larger triangle is 72 times greater than the corresponding side in the smaller triangle.

Writing the Equation

With this information, we can write an equation. 7 2 (x+3/7)= 2x+9 Let's solve this equation for x.

Calculating the Length

To find the length of the side marked 2x+9, we substitute 5 for x in the expression and evaluate. 2( 5)+9 ⇔ 19 cm

Note that a dilation is a similarity transformation, which means it preserves the shape of the original figure. Therefore, the angle marked (y+18)^(∘) has the same measure as the angle marked (3y-34)^(∘). This means we can equate their measures. 3y-34=y+18 Let's solve for y.

We now want to find the measure of the angle marked (3y-34)^(∘). Let's substitute 26 for y in the expression and evaluate. (3( 26)-34)^(∘) ⇔ 44^(∘)

E
Hint & Answer
S
Solution

△ ABC has vertices A(5,3), B(5,8), and C(9,3). What are the coordinates of C' after a dilation of △ ABC with a center of dilation at (5,0) and a scale factor of 3?

Let's start by graphing △ ABC in a coordinate plane along with the center of dilation, which we will label P.

To dilate this triangle by a factor of 3, the points on the image must be three times as far away from the center of dilation compared to the corresponding points on the preimage. Notice that C needs to be dilated in both the vertical and horizontal directions.

As we can see C' has the coordinates (17,9).

E
Hint & Answer
S
Solution

Is XYZW a dilation of ABCD? Explain your reasoning.

Rectangle XYZW and ABCD in one and the same coordinate plane

We can determine if XYZW is a dilation of ABCD by drawing lines between, what would be, corresponding vertices. If they all converge at one point, that point would be the center of dilation and we have a dilation.

As we can see, the four lines between corresponding vertices do not converge at one point. Therefore, XYZW can not be a dilation of ABCD.

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Hint & Answer
S
Solution

Use a compass to dilate the segment XY with a scale factor of 4 and with X as the center of dilation.

Segment XY and five different regions marked to the right of and some distance away from Y

In which of the five sections will Y' fall?

To dilate the segment, we can measure the distance between X and Y with a compass. Open the compass until it measures the width of XY.

Keeping the compass setting intact, use the compass to copy this distance four times from Y.

Finally, we will use a straightedge to draw the dilated segment.

As we can see, the point falls in section 4.

E
Hint & Answer
S
Solution

Dilation and Scale Factor

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