Transcript 6 7 Similarity Transformations

6.7: Similarity Transformations
Objectives:
1. To define and
perform dilations
Assignment:
• P. 412-415: 1, 2, 4-12
even, 13-20
• Challenge Problems
Objective 1
You will be able to
define and perform
dilations
Warm-Up
Which of the following does not belong?
Transformations
A transformation is an
operation that changes
some aspect of the
geometric figure to
produce a new figure.
A'
The original figure is
called the pre-image
A
The transformed figure
is called the image
Transformations
Transformations
Rigid Motion
Rotation
Translation
Reflection
Preserves
size and
shape
Non-Rigid
Motion
Stretch
Shrink
Dilate
Example 1
Use the coordinate
mapping (x, y) →
(x + 8, y + 3) to
translate ΔSAM to
create ΔS’A’M’.
Dilations
A dilation is a type of
transformation that enlarges
or reduces a figure.
Scale Factor
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.
New Side
k
Original Side
Investigation 1
Complete the following investigation to
construct a dilation of a triangle.
Investigation 1
Step 1: Construct ΔABC on a coordinate
plane with A(3, 6), B(7, 6), and C(7, 3).
Investigation 1
Step 2: Draw rays from the origin O through
A, B, and C. O is the center of dilation.
Investigation 1
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’.
Investigation 1
Step 3:
Investigation 1
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).
Investigation 1
Step 4:
Investigation 1
Step 5: 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. Draw triangle A’B’C’. ΔA’B’C’
is the image of ABC under a dilation with
center O and a scale factor of 3. Are these
images similar?
Investigation 1
Step 5:
Investigation 1
Step 6: What are the lengths of AB and A’B’?
BC and B’C’? What is the scale factor?
Investigation 1
Step 7: Measure the coordinates of A’, B’,
and C’.
Investigation 1
Step 8: How do they compare to the original
coordinates?
Similarity Transformations
Transformations in which the pre-image and
image are similar are called
similarity transformations.
Translation
Reflection
Rotation
Dilation
Example 2
What happens to
any point (x, y)
under a dilation
centered at the
origin with a
scale factor of k?
Dilations in the Coordinate Plane
You can describe
a dilation with
respect to the
origin with the
notation
(𝑥, 𝑦) → (𝑘𝑥, 𝑘𝑦),
where k is the
scale factor.
Enlargement: 𝑘 > 1
Dilations in the Coordinate Plane
You can describe
a dilation with
respect to the
origin with the
notation
(𝑥, 𝑦) → (𝑘𝑥, 𝑘𝑦),
where k is the
scale factor.
Reduction: 0 < 𝑘 < 1
Example 3
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?
Example 4
A graph shows PQR with vertices P(2, 4),
Q(8, 6), and R(6, 2), and segment ST with
endpoints S(5, 10) and T(15, 5). At what
coordinate would vertex U be placed to
create ΔSUT, a dilation of ΔPQR?
Example 5
Figure J’K’L’M’N’ is
a dilation of figure
JKLMN. Find the
coordinates of J’
and M’.
Not the Origin
If the center of
dilation is not the
origin, you must
do the following:
1. Translate preimage so that the
center of dilation
is at the origin
Not the Origin
If the center of
dilation is not the
origin, you must
do the following:
2. Perform the
dilation
Not the Origin
If the center of
dilation is not the
origin, you must
do the following:
3. Translate image
so that the
center of dilation
is in its original
position
Example 6
Write a composition
of transformations
that maps JKLMN
to J’K’L’M’N’.
Dilation: Formal Definition
Given a point 𝑂 and
a positive real
number 𝑘, the
dilation centered at
𝑂 with scale factor 𝑘
maps point 𝑃 to 𝑃’
such that 𝑃’ is on
𝑂𝑃 and 𝑂𝑃′ = 𝑘 ∙ 𝑂𝑃
𝐷𝑂,𝑘 𝑃 = 𝑃′
6.7: Similarity Transformations
Objectives:
1. To define and
perform dilations
Assignment:
• P. 412-415: 1, 2, 4-12
even, 13-20
• Challenge Problems