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Chapter 11: Extending Geometry
11.1
Transformations
11.1.1. Translations, Rotations, and Reflections
11.1.1.1. Slides or Translations
11.1.1.1.1. If each point P in a plane corresponds to a unique point in
the plane, P’, such that directed segments PP” is congruent and
parallel to the directed segment AB, then the correspondence is
called the translation associated with the directed segment AB and
is written TAB
11.1.1.1.2. TAB(P) = P’
A
C
D
A'
C'
D'
B
B'
11.1.1.1.3.
11.1.1.2. Turns or Rotations
11.1.1.2.1. Consider a point C and an angle with a measure  from 180 to 180. If each point P in the plane corresponds to a unique
point in the plane, P’, such that mPCP’ =  and PC = P’C, then
the correspondence is called a rotation, with center C and angle ,
and is written RC,
11.1.1.2.2. RC,(P) = P’
E
F
E'
H
H'
F'
G
11.1.1.2.3.
RG,-45o
K
J
K'
L
L'
J'
I
I'
M
11.1.1.2.4.
RM,-45o
N
O
N'
Q
Q'
R
O'
P'
P
RR,-60o
11.1.1.2.5.
11.1.1.3. Flips and Reflections
11.1.1.3.1. If each point on line l corresponds to itself, and each other
point P in the plane corresponds to a unique point P’ in the plane,
such that l is the perpendicular bisector of PP' , then the
correspondence is called the reflection in the line l, and is written
Ml
11.1.1.3.2. Ml (P) = P’
U
T
V
T'
U'
V'
S
S'
11.1.1.3.3.
11.1.1.4. Identifying transformations – see example 11.1 p. 598
11.1.1.5. Combinations of motions
11.1.1.5.1. glide-reflection
11.1.1.5.1.1.
Slide + Flip
11.1.1.5.1.2.
Translation + Reflection
X
W
Z
Y
W''
Y ''
Z''
X''
11.1.1.5.1.3.
11.1.1.6. Transformations and congruence
11.1.1.6.1. no change in size or shape
11.1.1.6.2. all objects translated are congruent
11.1.1.6.3. congruence transformation
11.1.1.6.3.1.
preserves distance
11.1.1.6.3.2.
other characteristics
11.1.1.6.3.3.
also called an isometry
11.1.1.6.4. Two figures are congruent if and only if there exists a
translation, rotation, reflection, or glide-reflection that sets up a
correspondence of one figure as the image of the other
11.1.2. Connecting Transformations and Symmetry
11.1.2.1. Symmetry in the plane
11.1.2.1.1. Reflectional symmetry
11.1.2.1.2. line of symmetry
11.1.2.1.3. n rotational symmetry
11.1.2.1.4. center of rotational symmetry
11.1.3. Transformations that Change Size
11.1.3.1. size transformations
11.1.3.1.1. If point O corresponds to itself, and each other point P in the
plane corresponds to a unique point on OP such that OP’ = r(OP)
for r > 0, then the correspondence is called the size transformation
associated with center O and scale factor r, and can be written S O, r
11.1.3.1.2. SO, r (P) = P’
11.1.3.2. Size transformations and similarity
11.1.3.2.1. Two figures are similar if and only if there exists a
combination of an isometry and a size transformation that
generates one figure as the image of the other
11.1.3.2.2.
11.1.4. Transformations that Change Both Size and Shape
11.1.4.1. Topological transformations
11.1.4.1.1. shrinking
11.1.4.1.2. bending
11.1.4.1.3. stretching
11.1.4.1.4. If one figure can be transformed to another by a topological
transformation then they are topologically equivalent
11.1.5. Problems and Exercises p. 669
11.1.5.1. Home work: 7-10, 11, 12, 16, 17, 20, 22, 23b, 25, 37