Common Core Geometry R
Aim #8: How do we construct a similarity transformation?
Do Now:
A
B
E
D
C
D
A ___
1. Name two congruent figures: ___
B
A ____
2. Name two similar, but not congruent, figures: ____
B ____
D
3. Name two other similar, not congruent, figures: ____
E
C ____
4. Name two figures, neither congruent nor similar. ____
Dilations
A similarity transformation consists of 1 or more transformations ( ___________,
Reflections done in succession.
Translations _______________,
Rotations
______________,
____________)
Rotations , _____________
Translations _________
Reflections )
The three rigid motions (___________,
Measure while
Angle __________,
Length
Side
preserve __________
__________
and __________
Angle dilations (although included in similarity transformations) preserve _________
Measure only.
__________
We use a composition of rigid motions to determine whether two figures are
Congruent
____________,
and we use compositions of rigid motions and dilations, (called
Similarity
_____________________
transformations), to determine whether figures are
Similar
_________________.
What transformations would map Figure 1 onto Figure 2?
1
2
Rotation
Translation
Dilation
Properties of Similarity Transformations:
Point
1. Each point is mapped to a distinct ___________.
Pre­image
2. Each point P‛ in the image has a _______________.
r PQ.
3. Given a scale factor r and points P and Q , then P‛Q‛ = ___
lines rays to ______,
rays line segments to
4. A similarity transformation sends lines to _____,
line segments
parallel lines
_____
_________, and parallel lines to ______________
_______.
equal measure.
5. A similarity transformation sends angles to angles of _________
6. Given a scale factor r, a similarity transformation maps a circle of radius R to a
circle of radius r R
__.
Ex. Under a similarity transformation with scale factor 0.5, a circle of radius
circle
3.5
7 maps onto a __________
with radius____.
Exercise (1)
Similarity transformation G consists of a rotation about the point P by 60°,
followed by a dilation centered at P with scale factor r = 2, and then a reflection
across line l . Perform the similarity transformation and label ΔA'''B'''C'''.
l
A"
C'''
C"
B"
A'
C'
B'''
B'
A
A'''
P
B
C
Exercise (2): A similarity transformation G applied to Triangle ABC consists of a
translation by vector XY, followed by a reflection across line
followed by a dilation centered at P with scale factor r = 2.
m , and then
Using composition notation, this similarity transformation is written:
D
r
T
xy
P,2
____
( ____
(ABC))) = A'''B'''C'''
m ( ______
Construct A'''B'''C'''.
C'''
B'''
Y
X
A
C''
B'
B
A'''
A'
B''
C'
C
P
m
A''
Exercise (3) For the Figure Z, find the image of rl(RP,180° (DP,2(Z)).
Z
P
l
Z'''
Exercise (4)
Given ΔABC, with vertices A(4,-4), B(-2,4), and C(6,0), locate and label the image
of the triangle under the similarity transformation DB',0.5(rx=2(ΔABC)) . x=2
B
B'
C
C'
A'
A
Let's Sum it Up!
• The scale factor associated with any congruence transformation is 1. • The scale factor of a similarity transformation is the product of the scale
factors of all the transformations that compose the similarity.
• All properties of a similarity transformation are passed along to each succeeding transfo