Similarity, Right Triangles,
and Trigonometry
Clusters:
1. Understand similarity in terms of
similarity transformations
2. Prove theorems involving similarity.
3. Define trigonometric ratios and solve
problems involving right triangles.
4. Apply trigonometry to general triangles.
Learning Target
1. I can define dilation.
2. I can perform a dilation with a given
center and scale factor on a figure in the
coordinate plane.
Connection to previews lesson…
• Previously, we studied rigid transformations, in
which the image and preimage of a figure are
congruent. In this lesson, you will study a type of
nonrigid transformation called a dilation, in
which the image and preimage of a figure are
similar.
Dilations
A dilation is a type of
transformation that enlarges
or reduces a figure but the
shape stays the same.
The dilation is described by a
scale factor and a center of
dilation.
Dilations
The scale factor k is the ratio of the
length of any side in the image to
the length of its corresponding
side in the preimage. It describes
how much the figure is enlarged
or reduced.
New Side
k
Original Side
The dilation is a reduction if k < 1 and it is an enlargement if k > 1.
P
P´
6
P´
2
3
C
Q
Q´
•
5
P
C
R
•
R´
Q
R´
Reduction: k =
PQR ~
CP
3
1
= =
CP
6
2
Q´
R
Enlargement: k =
CP
5
=
CP
2
P´Q´R´, P´Q´ is equal to the scale factor of the
PQ
dilation.
Constructing a Dilation
Examples of constructed a dilation of a
triangle.
Steps in constructing a dilation
Step 1: Construct ABC on a coordinate
plane with A(3, 6), B(7, 6), and C(7, 3).
18
16
14
12
10
8
A
B
6
4
C
2
O
5
10
15
20
25
30
35
Steps in constructing a dilation
Step 2: Draw rays from the origin O through
A, B, and C. O is the center of dilation.
18
16
14
12
10
8
A
B
6
4
C
2
O
5
10
15
20
25
30
35
Steps in constructing a dilation
Step 3: With your compass, measure the
distance OA. In other words, put the point
of the compass on O and your pencil on A.
Transfer this distance twice along OA so
that you find point A’ such that OA’ =
3(OA). That is, put your point on A and
make a mark on OA. Finally, put your
point on the new mark and make one last
mark on OA. This is A’.
Steps in constructing a dilation
Step 3:
A'
18
16
14
12
10
8
A
B
6
4
C
2
O
5
10
15
20
25
30
35
Steps in constructing a dilation
Step 4: Repeat Step 3 with points B and C.
That is, use your compass to find points B’
and C’ such that OB’ = 3(OB) and OC’ =
3(OC).
Steps in constructing a dilation
Step 4:
A'
B'
18
16
14
12
10
C'
8
A
B
6
4
C
2
O
5
10
15
20
25
30
35
Steps in constructing a dilation
You have now located three points, A’, B’,
and C’, that are each 3 times as far from
point O as the original three points of the
triangle.
Step 5: Draw triangle A’B’C’.
A’B’C’ is the image of ABC under a
dilation with center O and a scale factor of
3. Are these images similar?
Steps in constructing a dilation
Step 5:
A'
B'
18
16
14
12
10
C'
8
A
B
6
4
C
2
O
5
10
15
20
25
30
35
Questions/ Observations:
Step 6: What are the lengths of AB and A’B’?
BC and B’C’? What is the scale factor?
A'
B'
18
16
14
12
10
•AB = 4
•A’B’= 12
•BC = 3
•B’C’= 9
C'
8
A
B
6
4
C
2
O
5
10
15
20
25
30
35
Questions/ Observations:
Step 7: Measure the coordinates of A’, B’,
and C’.
A'
B'
18
16
14
12
10
C'
Image
A´(9, 18)
B´(21, 18)
C´(21, 9)
8
A
B
6
4
C
2
O
5
10
15
20
25
30
35
Questions/ Observations:
Step 8: How do they compare to the original
coordinates?
A'
P(x, y)  P´(kx, ky)
B'
18
16
14
12
10
C'
Pre-image
A(3, 6) 
B(7, 6) 
C(7, 3) 
8
A
B
6
4
C
2
O
5
10
15
20
25
30
Image
A´(9, 18)
B´(21, 18)
C´(21, 9)
35
In a coordinate plane, dilations whose centers are the origin
have the property that the image of P(x, y) is P´(kx, ky).
Draw a dilation of rectangle ABCD with A(2, 2), B(6, 2), C(6, 4), and D(2, 4).
Use the origin as the center and use a scale factor of 1 . How does the perimeter of
the preimage compare to the perimeter of the image? 2
SOLUTION
Because the center of the dilation is the
origin, you can find the image of each
vertex by multiplying its coordinates by
the scale factor.
A(2, 2)  A´(1, 1)
B(6, 2)  B´(3, 1)
C(6, 4)  C´(3, 2)
D(2, 4)  D´(1, 2)
y
D
D´ A
C´
B
A´
1
O
C
•
B´
1
x
In a coordinate plane, dilations whose centers are the origin
have the property that the image of P(x, y) is P´(kx, ky).
Draw a dilation of rectangle ABCD with A(2, 2), B(6, 2), C(6,
1 use a scale
4), and D(2, 4). Use the origin as the center and
2
factor of . How does the perimeter of the preimage
compare
to the perimeter of the image?
SOLUTION
From the graph, you can see that the
preimage has a perimeter of 12 and the
image has a perimeter of 6.
A preimage and its image after a
dilation are similar figures.
Therefore, the ratio of the perimeters of
a preimage and its image is equal to the
scale factor of the dilation.
y
D
D´ A
C´
B
A´
1
O
C
•
B´
1
x
Example 3
Determine if ABCD
and A’B’C’D’ are
similar figures. If
so, identify the
scale factor of the
dilation that maps
ABCD onto A’B’C’D’
as well as the
center of dilation.
Is this a reduction
or an enlargement?
Assignment/Homework
Work with a partner
in the classwork on
“Constructing
Dilation”
Homework:
Answer Guided
Practice page 510
#12 to 15.