Properties of Dilations
Essential Question?
How do you describe the properties
of dilations?
8.G.4
Common Core Standard:
8.G ─Understand congruence and similarity using physical models,
transparencies, or geometry software.
4. Understand that a two-dimensional figure is similar to another
if the second can be obtained from the first by a sequence of
rotations, reflections, translations, and dilations; given two
similar two-dimensional figures, describe a sequence that
exhibits the similarity between them.
Objectives:
• To describe the properties of dilation and their effect on the
similarity and orientation of figures.
Curriculum Vocabulary
Center of Dilation (centro de dilatación):
The point of intersection of lines through each pair of
corresponding vertices in a dilation.
Dilation (dilatación):
A transformation that moves each point along the ray
through the point emanating from a fixed center, and
multiplies distances from the center by a common scale
factor.
Ratio (razón):
A comparison of two quantities by division.
Scale (escala):
The ratio between two sets of measurements.
Curriculum Vocabulary
Enlargement (agrandamiento):
An increase in the size of all dimensions in the same
proportions.
Reduction (reducción):
A decrease in the size of all dimensions in the same
proportions.
Scale Factor (factor de escala):
The ratio used to enlarge or reduce similar figures.
Similar (similar / semejantes):
Figures with the same shape but not necessarily the same size.
• Ratios of corresponding sides are proportional
• Measures of corresponding angles are equal
Properties of Dilations
We often see a scale model or a toy that replicates a real object.
A replica is a transformation called a DILATION.
Unlike the other transformations you have studied:
• Translations • Rotations
• Reflections
DILATIONS CHANGE THE SIZE (BUT NOT THE SHAPE)
OF A FIGURE.
ORIENTATION IS PRESERVED.
Every dilation has a fixed point called the
CENTER OF DILATION
located where the lines connecting corresponding
parts of figures intersect.
Dilations
Let’s look at some dilations:
The dilation of
this tiger made
it bigger.
The dilation of
this smiley
made it smaller.
IMAGE
PREIMAGE
In all dilations, the image and
pre-image look exactly the
same. The only thing that
changes is the size.
Dilations
IMAGE
PREIMAGE
CENTER
OF
DILATION
In this drawing, △R´S´T´ is a dilation of △RST.
Point C is the center of dilation.
Let’s work together to complete the worksheet.
Fill in the following information:
Center of Dilation: ___________
Image: ___________
Preimage: ___________
Use a ruler to measure each of the line segments to the nearest centimeter:
𝑅𝑆 =
𝑅´𝑆´ =
𝑆𝑇 =
𝑆´𝑇´ =
𝑅𝑇 =
𝑅´𝑇´ =
Now use your answers to find the following RATIOS:
(SIMPLIFY EACH FRACTION!)
𝑅´𝑆´
𝑅𝑆
𝑆´𝑇´
𝑆𝑇
𝑅´𝑇´
𝑅𝑇
=
=
=
Use a protractor to measure each of the angles:
mRST 
mRS T  
mSTR 
mS T R 
mTRS 
mT RS  
Based on the information you found in PART 3, what can you conclude about the ratios of
corresponding sides?
Based on the information you found in PART 4, what can you conclude about the measures of
corresponding angles?
Let’s see if we can draw some conclusions:
image
Notice that all the ratios in part 3 were found by using the
preimage
You found that the ratios of corresponding sides were _________________________________ .
When two fractions are equivalent, we say that they are _______________________________.
You have now discovered that the ratios of corresponding sides are _______________________
and the measures of corresponding angles are _______________________________________ .
We can now conclude that △RST and △R´S´T´ are _____________________________________.
Each line segment in the image is ______ times its corresponding line segment in the preimage.
Recall that the ratio used to enlarge or reduce similar figures is called the __________________.
The dilation of △RST to △R´S´T´ uses a ______________________ of _______.
Dilations
Now lets examine a dilation on a coordinate plane.
Which is the image and
which is the preimage?
What is the center of dilation?
Fill out the table:
Vertex
x
y
Vertex
A´
A
B´
B
C´
C
D´
D
x
y
Vertex
x
y
Vertex
x
y
A´
−1
2.5
A
−2
5
B´
2
1
B
4
2
C´
2
−2
C
4
−4
D´
−2
−2
D
−4
−4
Use the information in your table
to answer the following:
First using only the
x-coordinates, then
using only the
y-coordinates,
what is the ratio of
the
𝑖𝑚𝑎𝑔𝑒
𝑝𝑟𝑒𝑖𝑚𝑎𝑔𝑒
Ratios for x-coordinates
Ratios for y-coordinates
A´ & A
A´ & A
B´ & B
B´ & B
C´ & C
C´ & C
D´ & D
D´ & D
Ratios for x-coordinates
−1 1
A´ & A
=
−2 2
2 1
B´ & B
=
4 2
2 1
C´ & C
=
4 2
−2 1
D´ & D
=
−4 2
Ratios for y-coordinates
2.5 1
A´ & A
=
5
2
1 1
B´ & B
=
2 2
−2 1
C´ & C
=
−4 2
−2 1
D´ & D
=
−4 2
Use the information in your table to answer the following:
What can you say about the ratio of the coordinates
𝑖𝑚𝑎𝑔𝑒
of the
?
𝑝𝑟𝑒𝑖𝑚𝑎𝑔𝑒
IMAGE
PREIMAGE
IMAGE
PREIMAGE
CENTER OF DILATION
In our first example, △R´S´T´ is BIGGER than △RST.
In our second example, A´B´C´D´ is SMALLER than ABCD.
When the image is BIGGER than the preimage it is an
ENLARGEMENT
When the image is SMALLER than the preimage it is a
REDUCTION
SCALE FACTOR
We already know that
SCALE FACTOR
is the ratio used to enlarge or reduce similar figures.
With an ENLARGEMENT, the SCALE FACTOR is
ALWAYS a number GREATER THAN 1.
With a REDUCTION, the SCALE FACTOR is
ALWAYS a number BETWEEN 0 and 1.
SCALE FACTOR
We already know how to find the scale factor.
What did we use in both our enlargement to find the RATIO?
𝑖𝑚𝑎𝑔𝑒
SCALE FACTOR =
𝑝𝑟𝑒𝑖𝑚𝑎𝑔𝑒
When you know the LENGTH of a line segment
of the image and preimage,
you can determine the scale factor.
Find the SCALE FACTOR of the dilation:
Is the dilation an
enlargement or
reduction?
How do you know?