Transformations and the Coordinate Plane

1
Transformations and the Coordinate Plane

• Eleanor Roosevelt High School
• Geometry
• Mr. Chin-Sung Lin

2
The Coordinates of a Point in a Plane
ERHS Math Geometry
Mr. Chin-Sung Lin
3
Coordinate Plane
ERHS Math Geometry
Two intersecting lines determine a plane. The
coordinate plane is determined by a horizontal
line, the x-axis, and a vertical line, the
y-axis, which are perpendicular and intersect at
a point called the origin
Y
X
O
Mr. Chin-Sung Lin
4
Coordinate Plane
ERHS Math Geometry
Every point on a plane can be described by two
numbers, called the coordinates of the point,
usually written as an ordered pair (x, y)
Y
(x, y)
X
O
Mr. Chin-Sung Lin
5
Coordinate Plane
ERHS Math Geometry
The x-coordinate or the abscissa, is the distance
from the point to the y-axis. The y-coordinate or
the ordinate is the distance from the point to
the x-axis. Point O, the origin, has the
coordinates (0, 0)
Y
(x, y)
y
X
O (0, 0)
x
Mr. Chin-Sung Lin
6
Postulates of Coordinate Plane
ERHS Math Geometry
Two points are on the same horizontal line if and
only if they have the same y-coordinates
Y
(x2, y)
(x1, y)
X
O
Mr. Chin-Sung Lin
7
Postulates of Coordinate Plane
ERHS Math Geometry
The length of a horizontal line segment is the
absolute value of the difference of the
x-coordinates d x2 x1
Y
(x2, y)
(x1, y)
X
O
Mr. Chin-Sung Lin
8
Postulates of Coordinate Plane
ERHS Math Geometry
Two points are on the same vertical line if and
only if they have the same x-coordinates
Y
(x, y2)
(x, y1)
X
O
Mr. Chin-Sung Lin
9
Postulates of Coordinate Plane
ERHS Math Geometry
The length of a vertical line segment is the
absolute value of the difference of the
y-coordinates d y2 y1
Y
(x, y2)
(x, y1)
X
O
Mr. Chin-Sung Lin
10
Postulates of Coordinate Plane
ERHS Math Geometry
Each vertical line is perpendicular to each
horizontal line
Y
X
O
Mr. Chin-Sung Lin
11
Locating a Point in the Coordinate Plane
ERHS Math Geometry

1. From the origin, move to the right if the
   x-coordinate is positive or to the left if the
   x-coordinate is negative. If it is 0, there is no
   movement
2. From the point on the x-axis, move up if the
   y-coordinate is positive or down if the
   y-coordinate is negative. If it is 0, there is no
   movement

Y
(x, y)
y
X
O
x
Mr. Chin-Sung Lin
12
Finding the Coordinates of a Point
ERHS Math Geometry

1. From the point, move along a vertical line to the
   x-axis.The number on the x-axis is the
   x-coordinate of the point
2. From the point, move along a horizontal line to
   the y-axis.The number on the y-axis is the
   y-coordinate of the point

Y
(x, y)
y
X
O
x
Mr. Chin-Sung Lin
13
Graphing on the Coordinate Plane
ERHS Math Geometry
Graph the following points A(4, 1), B(1, 5),
C(-2,1). Then draw ? ABC and find its area
Y
X
O
Mr. Chin-Sung Lin
14
Graphing on the Coordinate Plane
ERHS Math Geometry
Graph the following points A(4, 1), B(1, 5),
C(-2,1). Then draw ? ABC and find its area
Y
X
O
Mr. Chin-Sung Lin
15
Graphing on the Coordinate Plane
ERHS Math Geometry
Graph the following points A(4, 1), B(1, 5),
C(-2,1). Then draw ? ABC and find its area AC
4 (-2) 6 BD 5 1 4 Area
½ (AC)(BD) ½ (6)(4) 12
Y
X
O
Mr. Chin-Sung Lin
16
Line Reflections
ERHS Math Geometry
Mr. Chin-Sung Lin
17
Line Reflections
ERHS Math Geometry
Mr. Chin-Sung Lin
18
Line Reflections
ERHS Math Geometry
Line Reflection (Object Image)
Y
Line of Reflection
Mr. Chin-Sung Lin
19
Transformation
ERHS Math Geometry
A one-to-one correspondence between two sets of
points, S and S, such that every point in set S
corresponds to one and only one point in set S,
called its image, and every point in S is the
image of one and only one point in S, called its
preimage
S
S
Mr. Chin-Sung Lin
20
A Reflection in Line k
ERHS Math Geometry

1. If point P is not on k, then the image of P is P
   where k is the perpendicular bisector of PP
2. If point P is on k, the image of P is P

Mr. Chin-Sung Lin
21
Theorem of Line Reflection - Distance
ERHS Math Geometry
Under a line reflection, distance is
preserved Given Under a reflection in line k,
the image of A is A and the image of B is
B Prove AB AB
A
A
B
B
k
Mr. Chin-Sung Lin
22
Theorem of Line Reflection - Distance
ERHS Math Geometry
Under a line reflection, distance is
preserved Given Under a reflection in line k,
the image of A is A and the image of B is
B Prove AB AB
A
A
C
B
B
D
k
Mr. Chin-Sung Lin
23
Theorem of Line Reflection - Distance
ERHS Math Geometry
Under a line reflection, distance is
preserved Given Under a reflection in line k,
the image of A is A and the image of B is
B Prove AB AB
A
A
C
SAS
B
B
D
k
Mr. Chin-Sung Lin
24
Theorem of Line Reflection - Distance
ERHS Math Geometry
Under a line reflection, distance is
preserved Given Under a reflection in line k,
the image of A is A and the image of B is
B Prove AB AB
A
A
C
CPCTC
B
B
D
k
Mr. Chin-Sung Lin
25
Theorem of Line Reflection - Distance
ERHS Math Geometry
Under a line reflection, distance is
preserved Given Under a reflection in line k,
the image of A is A and the image of B is
B Prove AB AB
A
A
C
SAS
B
B
D
k
Mr. Chin-Sung Lin
26
Theorem of Line Reflection - Distance
ERHS Math Geometry
Under a line reflection, distance is
preserved Given Under a reflection in line k,
the image of A is A and the image of B is
B Prove AB AB
A
A
C
CPCTC
B
B
D
k
Mr. Chin-Sung Lin
27
Theorem of Line Reflection - Distance
ERHS Math Geometry
Since distance is preserved under a line
reflection, the image of a triangle is a
congruent triangle
A
A
M
M
SSS
B
B
D
C
C
D
k
Mr. Chin-Sung Lin
28
Corollaries of Line Reflection
ERHS Math Geometry
Under a line reflection, angle measure is
preserved Under a line reflection, collinearity
is preserved Under a line reflection, midpoint is
preserved
A
A
M
M
B
B
D
C
C
D
k
Mr. Chin-Sung Lin
29
Notation of Line Reflection
ERHS Math Geometry
We use rk as a symbol for the image under a
reflection in line k rk (A) A rk (? ABC
) ? ABC
A
A
B
B
C
C
k
Mr. Chin-Sung Lin
30
Construction of Line Reflection
ERHS Math Geometry
If rk (AC) AC, construct AC
A
C
k
Mr. Chin-Sung Lin
31
Construction of Line Reflection
ERHS Math Geometry
Construct the perpendicular line from A to k. Let
the point of intersection be M
M
A
C
k
Mr. Chin-Sung Lin
32
Construction of Line Reflection
ERHS Math Geometry
Construct the perpendicular line from C to k. Let
the point of intersection be N
M
A
N
C
k
Mr. Chin-Sung Lin
33
Construction of Line Reflection
ERHS Math Geometry
Construct A on AM such that AM AM Construct
C on CN such that CN CN
A
M
A
N
C
C
k
Mr. Chin-Sung Lin
34
Construction of Line Reflection
ERHS Math Geometry
Draw AC
A
M
A
N
C
C
k
Mr. Chin-Sung Lin
35
Line Symmetry in Nature
ERHS Math Geometry
Mr. Chin-Sung Lin
36
Line Symmetry
ERHS Math Geometry
A figure has line symmetry when the figure is its
own image under a line reflection This line of
reflection is a line of symmetry, or an axis of
symmetry
Mr. Chin-Sung Lin
37
Line Symmetry
ERHS Math Geometry
It is possible for a figure to have more than one
axis of symmetry
Mr. Chin-Sung Lin
38
Line Reflections in the Coordinate Plane
ERHS Math Geometry
Mr. Chin-Sung Lin
39
Reflection in the y-axis
ERHS Math Geometry
Under a reflection in the y-axis, the image of
P(a, b) is P(-a, b)
y
Q(0, b)
P(a, b)
P(-a, b)
x
O
Mr. Chin-Sung Lin
40
Reflection in the y-axis
ERHS Math Geometry
If ? ABC is reflected in the y-axis, where A(-3,
3), B(-4, 1), and C(-1, 1), draw ry-axis (? ABC )
? ABC
A(-3, 3)
y
B(-4, 1)
C(-1, 1)
x
O
Mr. Chin-Sung Lin
41
Reflection in the y-axis
ERHS Math Geometry
If ? ABC is reflected in the y-axis, where A(-3,
3), B(-4, 1), and C(-1, 1), draw ry-axis (? ABC )
? ABC
A(3, 3)
A(-3, 3)
y
B(4, 1)
B(-4, 1)
C(-1, 1)
C(1, 1)
x
O
Mr. Chin-Sung Lin
42
Reflection in the x-axis
ERHS Math Geometry
Under a reflection in the x-axis, the image of
P(a, b) is P(a, -b)
y
P(a, b)
Q(a, 0)
x
O
P(a, -b)
Mr. Chin-Sung Lin
43
Reflection in the x-axis
ERHS Math Geometry
If ? ABC is reflected in the x-axis, where A(3,
3), B(4, 1), and C(1, 1), draw rx-axis (? ABC )
? ABC
y
A(3, 3)
B(4, 1)
C(1, 1)
x
O
Mr. Chin-Sung Lin
44
Reflection in the x-axis
ERHS Math Geometry
If ? ABC is reflected in the x-axis, where A(3,
3), B(4, 1), and C(1, 1), draw rx-axis (? ABC )
? ABC
y
A(3, 3)
B(4, 1)
C(1, 1)
x
C(1, -1)
B(4, -1)
O
A(3, -3)
Mr. Chin-Sung Lin
45
Reflection in the Line y x
ERHS Math Geometry
Under a reflection in the y x, the image of
P(a, b) is P(b, a)
P(a, b)
y
R(b, b)
P(b, a)
Q(a, a)
O
x
Mr. Chin-Sung Lin
46
Reflection in the Line y x
ERHS Math Geometry
If ? ABC is reflected in the x-axis, where A(2,
2), B(1, 4), and C(-1, 1), draw ryx (? ABC ) ?
ABC
B(1, 4)
y
A(2, 2)
C(-1, 1)
O
x
Mr. Chin-Sung Lin
47
Reflection in the Line y x
ERHS Math Geometry
If ? ABC is reflected in the x-axis, where A(2,
2), B(1, 4), and C(-1, 1), draw ryx (? ABC ) ?
ABC Point A is a fixed point since it
is on the line of reflection
B(1, 4)
y
A(2, 2)A(2, 2)
C(-1, 1)
B(4, 1)
O
x
C(1, -1)
Mr. Chin-Sung Lin
48
Point Reflections in the Coordinate Plane
ERHS Math Geometry
Mr. Chin-Sung Lin
49
A Point Reflection in P
ERHS Math Geometry

1. If point A is not point P, then the image of A is
   A and P the midpoint of AA
2. The point P is its own image

y
A
P
x
O
A
Mr. Chin-Sung Lin
50
Theorem of Point Reflections
ERHS Math Geometry
Under a point reflection, distance is preserved
y
B
A
P
x
O
A
B
Mr. Chin-Sung Lin
51
Theorem of Point Reflections
ERHS Math Geometry
Given Under a reflection in point P, the image
of A is A and the image of B is B Prove AB
AB
y
B
A
P
x
O
A
B
Mr. Chin-Sung Lin
52
Theorem of Point Reflections
ERHS Math Geometry
Given Under a reflection in point P, the image
of A is A and the image of B is B Prove AB
AB
y
B
SAS
A
P
x
O
A
B
Mr. Chin-Sung Lin
53
Theorem of Point Reflections
ERHS Math Geometry
Given Under a reflection in point P, the image
of A is A and the image of B is B Prove AB
AB
y
B
CPCTC
A
P
x
O
A
B
Mr. Chin-Sung Lin
54
Properties of Point Reflections
ERHS Math Geometry

1. Under a point reflection, angle measure is
   preserved
2. Under a point reflection, collinearity is
   preserved
3. Under a point reflection, midpoint is preserved

y
A
P
x
O
A
Mr. Chin-Sung Lin
55
Notation of Point Reflections
ERHS Math Geometry

• We use Rp as a symbol for the image under a
  reflection in point P
• Rp (A) B means The image of A under a
  reflection in point P is B.
• R(1,2) (A) A means The image of A under a
  reflection in point (1, 2) is A.

Mr. Chin-Sung Lin
56
Point Symmetry
ERHS Math Geometry
A figure has point symmetry if the figure is its
own image under a reflection in a point
Mr. Chin-Sung Lin
57
Point Symmetry
ERHS Math Geometry
Other examples of figures that have point
symmetry are letters such as S and N and numbers
such as 8
8
S
N
Mr. Chin-Sung Lin
58
Reflection in the Origin
ERHS Math Geometry
Under a reflection in the origin, the image of
P(a, b) is P(-a, -b) RO (a, b) (-a, -b)
y
P(a, b)
x
O
P(-a, -b)
Mr. Chin-Sung Lin
59
Reflection in the Origin
ERHS Math Geometry
If ? ABC is reflected in the origin, where A(-3,
3), B(-4, 1), and C(-1, 1), draw RO (? ABC ) ?
ABC
A(-3, 3)
y
B(-4, 1)
C(-1, 1)
x
O
Mr. Chin-Sung Lin
60
Reflection in the Origin
ERHS Math Geometry
If ? ABC is reflected in the origin, where A(-3,
3), B(-4, 1), and C(-1, 1), draw RO (? ABC ) ?
ABC
A(-3, 3)
y
B(-4, 1)
C(-1, 1)
x
B(4, -1)
C(1, -1)
O
Mr. Chin-Sung Lin
A(3, -3)
61
Reflection in the point
ERHS Math Geometry

1. What are the coordinates of B, the image of A(-3,
   2) under a reflection in the origin?
2. What are the coordinates of C, the image of
   A(-3, 2) under a reflection in the x-axis?
3. What are the coordinates of D, the image of C
   under a reflection in the y-axis?
4. Does a reflection in the origin give the same
   result as a reflection in the x-axis followed by
   a reflection in the y-axis? Justify your answer.

Mr. Chin-Sung Lin
62
Translations in the Coordinate Plane
ERHS Math Geometry
Mr. Chin-Sung Lin
63
Translation
ERHS Math Geometry
A translation is a transformation of the plane
that moves every point in the plane the same
distance in the same direction
A
y
B
C
A
x
O
B
C
Mr. Chin-Sung Lin
64
Translation
ERHS Math Geometry
In the coordinate plane, the distance is given in
terms of horizontal distance (change in the
x-coordinates) and vertical distance (change in
the y-coordinates)
A
y
B
C
A
x
y-coor.
x-coor.
O
B
C
Mr. Chin-Sung Lin
65
Translation
ERHS Math Geometry
A translation of a units in the horizontal
direction and b units in the vertical direction
is a transformation of the plane such that the
image of P(x, y) is P(x a, y b)
P(x a, y b)
y
b
a
P(x, y)
x
Mr. Chin-Sung Lin
66
Translation
ERHS Math Geometry
The image of P(x, y) is P(x a, y b), if
the translation moves a point to the right, a gt
0 if the translation moves a point to the
left, a lt 0 if the translation moves a point
up, b gt 0 if the translation moves a point
down, b lt 0
P(x a, y b)
y
b
a
P(x, y)
x
Mr. Chin-Sung Lin
67
Theorem of Translation
ERHS Math Geometry
Under a translation, distance is preserved
A
y
B
A
x
O
B
Mr. Chin-Sung Lin
68
Theorem of Translation
ERHS Math Geometry
Given A translation in which the image of
A(x1,y1) is A(x1a, y1b) and the image of B(x2,
y2) is B(x2a, y2b) Prove AB AB
A (x1a, y1b)
y
B (x2a, y2b)
A (x1, y1)
x
O
B (x2, y2)
Mr. Chin-Sung Lin
69
Theorem of Translation
ERHS Math Geometry
Given A translation in which the image of
A(x1,y1) is A(x1a, y1b) and the image of B(x2,
y2) is B(x2a, y2b) Prove AB AB
A (x1a, y1b)
y
B (x2a, y2b)
A (x1, y1)
x
O
B (x2, y2)
Mr. Chin-Sung Lin
70
Theorem of Translation
ERHS Math Geometry
Given A translation in which the image of
A(x1,y1) is A(x1a, y1b) and the image of B(x2,
y2) is B(x2a, y2b) Prove AB AB
A (x1a, y1b)
y
y1-y2
B (x2a, y2b)
A (x1, y1)
x1-x2
y1-y2
x
O
B (x2, y2)
x1-x2
Mr. Chin-Sung Lin
71
Theorem of Translation
ERHS Math Geometry
Given A translation in which the image of
A(x1,y1) is A(x1a, y1b) and the image of B(x2,
y2) is B(x2a, y2b) Prove AB AB
A (x1a, y1b)
y
y1-y2
B (x2a, y2b)
A (x1, y1)
x1-x2
y1-y2
x
O
B (x2, y2)
SAS CPCTC
x1-x2
Mr. Chin-Sung Lin
72
Properties of Translation
ERHS Math Geometry

1. Under a translation, angle measure is preserved
2. Under a translation, collinearity is preserved
3. Under a translation, midpoint is preserved

A (x1a, y1b)
y
B (x2a, y2b)
A (x1, y1)
x
O
B (x2, y2)
Mr. Chin-Sung Lin
73
Notation of Translation
ERHS Math Geometry

• We use Ta, b as a symbol for the image under a
  translation of a units in the horizontal
  direction and b units in the vertical direction
• Ta, b (x, y) (x a, y b)

Mr. Chin-Sung Lin
74
Translation
ERHS Math Geometry
If ? ABC has vertices A(-3, 3), B(-4, 1), and
C(-1, 1), find the coordinates of the vertices of
? ABC, the image of ? ABC under T7,1
A(-3, 3)
y
B(-4, 1)
C(-1, 1)
x
O
Mr. Chin-Sung Lin
75
Translation
ERHS Math Geometry
If ? ABC has vertices A(-3, 3), B(-4, 1), and
C(-1, 1), find the coordinates of the vertices of
? ABC, the image of ? ABC under T7,1
A(4, 4)
A(-3, 3)
y
B(3, 2)
C(6, 2)
B(-4, 1)
C(-1, 1)
x
O
Mr. Chin-Sung Lin
76
Translational Symmetry
ERHS Math Geometry
A figure has translational symmetry if the image
of every point of the figure is a point of the
figure
Mr. Chin-Sung Lin
77
Translational Symmetry
ERHS Math Geometry
Mr. Chin-Sung Lin
78
Rotations in the Coordinate Plane
ERHS Math Geometry
Mr. Chin-Sung Lin
79
Rotation
ERHS Math Geometry

• A rotation is a transformation of a plane about a
  fixed point P through an angle of d degrees such
  that
• For A, a point that is not the fixed point P, if
  the image of A is A, then PA PA and m ?APA
  d
• The image of the center of rotation P is P

y
A
d
A
O
x
Mr. Chin-Sung Lin
P
80
Theorem of Rotation
ERHS Math Geometry
Distance is preserved under a rotation about a
fixed point
A
B
B
A
P
Mr. Chin-Sung Lin
81
Theorem of Rotation
ERHS Math Geometry
Given P is the center of rotation. If A is
rotated about P to A, and B is rotated the same
number of degrees to B Prove AB AB
A
B
B
d
A
d
P
Mr. Chin-Sung Lin
82
Theorem of Rotation
ERHS Math Geometry
Given P is the center of rotation. If A is
rotated about P to A, and B is rotated the same
number of degrees to B Prove AB AB
A
m?APA m?BPB
B
B
m?APB m?APB
A
P
Mr. Chin-Sung Lin
83
Theorem of Rotation
ERHS Math Geometry
Given P is the center of rotation. If A is
rotated about P to A, and B is rotated the same
number of degrees to B Prove AB AB
A
B
B
SAS
A
P
Mr. Chin-Sung Lin
84
Theorem of Rotation
ERHS Math Geometry
Given P is the center of rotation. If A is
rotated about P to A, and B is rotated the same
number of degrees to B Prove AB AB
A
B
B
CPCTC
A
P
Mr. Chin-Sung Lin
85
Properties of Rotation
ERHS Math Geometry

1. Under a rotation, angle measure is preserved
2. Under a rotation, collinearity is preserved
3. Under a rotation, midpoint is preserved

A
B
B
A
P
Mr. Chin-Sung Lin
86
Notation of Rotation
ERHS Math Geometry

• We use RP, d as a symbol for the image under a
  rotation of d degrees about point P
• A rotation in the counterclockwise direction is
  called a positive rotation
• A rotation in the clockwise direction is called a
  negative rotation
• RO, 30o (A) B the image of A under a
  rotation of 30 degrees about the origin
  is B

Mr. Chin-Sung Lin
87
Notation of Rotation
ERHS Math Geometry

• The symbol R is used to designate both a point
  reflection and a rotation
• When the symbol R is followed by a letter that
  designates a point, it represents a reflection in
  that point (e.g., RP)
• When the symbol R is followed by both a letter
  that designates a point and the number of
  degrees, it represents a rotation of the given
  number of degrees about the given point (e.g.,
  RO, 30o)
• When the symbol R is followed by the number of
  degrees, it represents a rotation of the given
  number of degrees about the origin (e.g., R90o)

Mr. Chin-Sung Lin
88
Theorem of Rotation
ERHS Math Geometry
Under a counterclockwise rotation of 90 about
the origin, the image of P(a, b) is P(b,
a) RO,90(x, y) (-y, x) or R 90(x, y) (-y,
x)
Mr. Chin-Sung Lin
89
Rotation
ERHS Math Geometry
If ? ABC has vertices A(-3, 3), B(-4, 1), and
C(-1, 1), find the coordinates of the vertices of
? ABC, the image of ? ABC under RO,90o
A(-3, 3)
y
B(-4, 1)
C(-1, 1)
x
O
Mr. Chin-Sung Lin
90
Rotation
ERHS Math Geometry
If ? ABC has vertices A(-3, 3), B(-4, 1), and
C(-1, 1), find the coordinates of the vertices of
? ABC, the image of ? ABC under RO,90o
A(-3, 3)
y
B(-4, 1)
C(-1, 1)
x
C(-1, -1)
O
A(-3, -3)
Mr. Chin-Sung Lin
B(-1, -4)
91
Rotation
ERHS Math Geometry
If ? ABC has vertices A(-3, 3), B(-4, 1), and
C(-1, 1), find the coordinates of the vertices of
? ABC, the image of ? ABC under RO,180o
A(-3, 3)
y
B(-4, 1)
C(-1, 1)
x
Mr. Chin-Sung Lin
92
Rotation
ERHS Math Geometry
If ? ABC has vertices A(-3, 3), B(-4, 1), and
C(-1, 1), find the coordinates of the vertices of
? ABC, the image of ? ABC under RO,180o
A(-3, 3)
y
B(-4, 1)
C(-1, 1)
x
C(-1, -1)
C(1, -1)
O
B(4, -1)
A(-3, -3)
Mr. Chin-Sung Lin
B(-1, -4)
A(3, -3)
93
Rotation
ERHS Math Geometry
If ? ABC has vertices A(-3, 3), B(-4, 1), and
C(-1, 1), find the coordinates of the vertices of
? ABC, the image of ? ABC under RO,180o
A(-3, 3)
y
B(-4, 1)
C(-1, 1)
x
C(1, -1)
O
B(4, -1)
Mr. Chin-Sung Lin
A(3, -3)
94
Rotation 180o Point Reflection
ERHS Math Geometry
? ABC, the image of ? ABC under RO,180o is the
same as the image of ? ABC under point reflection
RO
A(-3, 3)
y
B(-4, 1)
C(-1, 1)
x
C(1, -1)
O
B(4, -1)
Mr. Chin-Sung Lin
A(3, -3)
95
Rotational Symmetry
ERHS Math Geometry
A figure is said to have rotational symmetry if
the figure is its own image under a rotation and
the center of rotation is the only fixed point
Mr. Chin-Sung Lin
96
Rotational Symmetry
ERHS Math Geometry
Many letters, as well as designs in the shapes of
wheels, stars, and polygons, have rotational
symmetry
S
Z
H
8
N
Mr. Chin-Sung Lin
97
Glide Reflections in the Coordinate Plane
ERHS Math Geometry
Mr. Chin-Sung Lin
98
Composition of Transformations
ERHS Math Geometry
When two transformations are performed, one
following the other, we have a composition of
transformations
Mr. Chin-Sung Lin
99
Glide Reflection
ERHS Math Geometry
A glide reflection is a composition of
transformations of the plane that consists of a
line reflection and a translation in the
direction of the line of reflection performed in
either order
A
y
A
B
B
C
C
A
x
B
C
Mr. Chin-Sung Lin
100
Glide Reflection
ERHS Math Geometry
A glide reflection is a composition of
transformations of the plane that consists of a
line reflection and a translation in the
direction of the line of reflection performed in
either order
y
A
B
C
A
A
x
B
B
C
C
Mr. Chin-Sung Lin
101
Theorem of Glide Reflection
ERHS Math Geometry
Under a glide reflection, distance is preserved
A
y
A
B
B
C
C
A
x
B
C
Mr. Chin-Sung Lin
102
Properties of Glide Reflection
ERHS Math Geometry

1. Under a glide reflection, angle measure is
   preserved
2. Under a glide reflection, collinearity is
   preserved
3. Under a glide reflection, midpoint is preserved

A
y
A
B
B
C
C
A
x
B
C
Mr. Chin-Sung Lin
103
Isometry
ERHS Math Geometry

• An isometry is a transformation that preserves
  distance
• All five transformations
• line reflection,
• point reflection,
• translation,
• rotation, and
• glide reflection.
• Each of these transformations is called an
  isometry

Mr. Chin-Sung Lin
104
Glide Reflection
ERHS Math Geometry
If ? ABC has vertices A(-3, 3), B(-4, 1), and
C(-1, 1), find the coordinates of the vertices of
? ABC, the image of ? ABC under ry-axis, and ?
ABC, the image of ? ABC under T0, 4
y
A(-3, 3)
B(-4, 1)
C(-1, 1)
x
O
Mr. Chin-Sung Lin
105
Glide Reflection
ERHS Math Geometry
If ? ABC has vertices A(-3, 3), B(-4, 1), and
C(-1, 1), find the coordinates of the vertices of
? ABC, the image of ? ABC under ry-axis, and ?
ABC, the image of ? ABC under T0, 4
y
A(3, 3)
A(-3, 3)
B(-4, 1)
B(4, 1)
C(-1, 1)
C(1, 1)
x
O
Mr. Chin-Sung Lin
106
Glide Reflection
ERHS Math Geometry
If ? ABC has vertices A(-3, 3), B(-4, 1), and
C(-1, 1), find the coordinates of the vertices of
? ABC, the image of ? ABC under ry-axis, and ?
ABC, the image of ? ABC under T0, 4
y
A(3, 3)
A(-3, 3)
B(-4, 1)
B(4, 1)
C(-1, 1)
C(1, 1)
x
O
A(3, 1)
B(4, 3)
Mr. Chin-Sung Lin
C(1, 3)
107
Glide Reflection
ERHS Math Geometry

• The vertices of ?PQR are P(2, 1), Q(4, 1), and
  R(4, 3)
• Find ?PQR, the image of ?PQR under ryx
  followed by T3, 3
• Find ?PQR, the image of ?PQR under T3, 3
  followed by ryx
• Are ?PQR and ?PQR the same triangle?
• Are ryx followed by T3, 3 and T3, 3 followed
  by ryx the same glide reflection? Explain
• Write a rule for this glide reflection

Mr. Chin-Sung Lin
108
Dilations in the Coordinate Plane
ERHS Math Geometry
Mr. Chin-Sung Lin
109
Dilation
ERHS Math Geometry

• A dilation of k is a transformation of the plane
  such that
• The image of point O, the center of dilation, is
  O
• When k is positive and the image of P is P, then
  OP and OP are the same ray and OP kOP
• When k is negative and the image of P is P, then
  OP and OP are opposite rays and OP -kOP.

P
y
P
k gt 0
x
O
k lt 0
Mr. Chin-Sung Lin
P
110
Notation of Dilations
ERHS Math Geometry
We use Dk as a symbol for the image under a
dilation of k with center at the origin P (x, y)
? P (kx, ky) or Dk (x, y) (kx, ky) D2 (3,
4) (6, 8)
Mr. Chin-Sung Lin
111
Dilation
ERHS Math Geometry
Under a dilation about a fix point, distance is
not preserved, and angle measurement is
preserved Dilation is not an isometry
A
y
B
A
B
x
O
Mr. Chin-Sung Lin
112
Glide Reflection
ERHS Math Geometry
If ? ABC has vertices A(2, 1), B(1, 3), and C(3,
2), find the coordinates of the vertices of ?
ABC, the image of ? ABC under D2
y
B(1, 3)
C(3, 2)
A(2, 1)
x
Mr. Chin-Sung Lin
O
113
Glide Reflection
ERHS Math Geometry
If ? ABC has vertices A(2, 1), B(1, 3), and C(3,
2), find the coordinates of the vertices of ?
ABC, the image of ? ABC under D2
B(2, 6)
y
C(6, 4)
B(1, 3)
A(4, 2)
C(3, 2)
A(2, 1)
x
Mr. Chin-Sung Lin
O
114
Transformations as Functions
ERHS Math Geometry
Mr. Chin-Sung Lin
115
Functions
ERHS Math Geometry
A function is a set of ordered pairs in which no
two pairs have the same first element The set of
first elements is the domain of the function and
the set of second elements is the range
Domain
Range
Mr. Chin-Sung Lin
116
Transformations as Functions
ERHS Math Geometry
Transformation can be viewed as a one-to-one
function
S
S
Mr. Chin-Sung Lin
117
Notations of Functions
ERHS Math Geometry
For example, y x 1 is a function f, it can
represented as y x 1 f(x) x 1 f x -gt x
1 f (x, y) y x 1 y and f(x) both
represent the second element of the ordered pair
Mr. Chin-Sung Lin
118
Composition of Transformations
ERHS Math Geometry
When two transformations are performed, one (f)
following the other (g), we have a composition of
transformations y g( f(x) ) or y g o
f
Mr. Chin-Sung Lin
119
Composition of Transformations
ERHS Math Geometry
A is the image of A(2, 5) under a reflection in
the line y x followed by the translation T2,0,
we can write T2, 0 (ry x (A)) A or T2, 0 o
ry x (A) A A T2, 0 (ry x (2, 5))
T2, 0 o ry x (2, 5) T2, 0 (5, 2) (7, 2)
Mr. Chin-Sung Lin
120
Orientation
ERHS Math Geometry
In a figure, the vertices, when traced from A to
B to C to . are in the clockwise or the
counter-clockwise direction, called the
orientation of the points
A
Clockwise Orientation
B
C
Mr. Chin-Sung Lin
121
Direct Isometry
ERHS Math Geometry

• A direct isometry is a transformation that
  preserves distance and orientation
• The following three transformations
• point reflection,
• translation, and
• rotation
• each of these transformations is direct isometry

Mr. Chin-Sung Lin
122
Opposite Isometry
ERHS Math Geometry

• An opposite isometry is a transformation that
  preserves distance , but changes the orientation
• The following two transformations
• line reflection, and
• glide reflection
• each of these transformations is opposite isometry

Mr. Chin-Sung Lin
123
Q A
ERHS Math Geometry
Mr. Chin-Sung Lin
124
The End
ERHS Math Geometry
Mr. Chin-Sung Lin