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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.

$O A^{'} = k \cdot O A \Leftrightarrow k = \frac{O A ^{'}}{O A}$

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 $∠ B A C .$ According to the first property, $A C$ is parallel to $A^{'} C^{'}$ and $A B$ is parallel to $A^{'} B^{'}$

Let $M$ be the intersection point of $A B$ and $A^{'} C^{'},$ and focus on the parallel legs $A C$ and $A^{'} C^{'} .$ According to the Alternate Interior Angles Theorem, angles $∠ M A C$ and $∠ A M A^{'}$ are congruent.

Similarly, since the other legs are also parallel, angles $∠ B^{'} A^{'} M^{'}$ and $∠ A M A^{'}$ are congruent.

Since $∠ A M A^{'}$ is congruent to both $∠ B A C$ 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.

∠ B A C ≅ ∠ B^{'} A^{'} C^{'} ⇓ m ∠ B A C = 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 $A B C D$ 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

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 $A B C D .$ 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.

O A^{'} O B^{'} O C^{'} O D^{'} = 2 \cdot O A = 2 \cdot O B = 2 \cdot O C = 2 \cdot O D

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.

O A^{''} O B^{''} O C^{''} O D^{''} = \frac{1}{2} \cdot O A = \frac{1}{2} \cdot O B = \frac{1}{2} \cdot O C = \frac{1}{2} \cdot O D

These points can be constructed as midpoints between two points.

Example

Dilating a Logo

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 : \frac{8 . 6}{5 . 4} \approx 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

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.

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) B (3, 2) C (2, 0) \to A^{'} (0, 2) \to B^{'} (6, 4) \to 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.

Draw a ray from the center of dilation through the preimage point

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

Measure the distance from the center of dilation to the preimage point

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

Copy the measured distance on the ray

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.

O A^{'} = n \times O A

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.

Draw two rays starting at the center of dilation

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

Copy the same distance on the second ray

n

times

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 $n = 3 .$

Construct parallel lines

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.

Mark the image point

The intersection point of this parallel line with the first ray is the constructed image.

O A^{'} = \frac{1}{n} \times O A

Dilation by a Scale Factor $p / q$

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

Dilation by a scale facor

1 / q

First, construct the image using a scale factor

1 / q .

O T = \frac{1}{q} \times O A

On the diagram

q = 5 .

Dilation by a scale facor

p

Copy the distance of this new point from the center of dilation

p

times, starting at the center of dilation.

O A^{'} = p \times O T O T = \frac{1}{q} \times O A ⟹ O A^{'} = \frac{p}{q} \times O A

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.

Construct the image of the vertices.

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

Connect the image vertices.

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?

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Catch-Up and Review

Proof

Proof

Proof

Answer

Hint

Solution

Answer

Hint

Solution

Answer

Hint

Solution

Dilation by an Integer Scale Factor

Dilation by a Scale Factor 1/n

Dilation by a Scale Factor p/q

Dilation of a Figure

Dilation by a Scale Factor $1 / n$

Dilation by a Scale Factor $p / q$