Transcript Chapter 9 Section 9.1

Chapter 9
Section 9.1 - Translations
OBJECTIVES:
TO IDENTIFY ISOMETRIES
TO FIND TRANSLATION IMAGES OF FIGURES
 Transformation -> a change in the position, shape,
or size of a geometric figure. Can be done by
flipping, sliding, or turning the figure.
 Pre-image -> the original figure.
 Image -> the figure after the transformation.
 Isometry -> transformation in which the pre-image
and image are congruent.
 Ex: Does the transformation appear to be an isometry?
Preimage
Image
Preimage
Image
Yes, congruent
by a flip
Yes, congruent by
a flip and slide
Preimage
Image
No, this involves a
change in size
 A transformation maps a figure onto its image and
may be described with arrow () notation. Prime (´)
notation is sometimes used to identify image points.
In the diagram below, K´ is the image of K (K  K´)
J
J´
K
K´
ΔJKQ  ΔJ´K´Q´
Q
Q´
 Translation (aka slide) -> an isometry that maps all
points of a figure the same distance in the same
direction.
 The figure below shows a translation of the black
rectangle by 4 units right and 2 units down. This can be
written as: the original figure (x, y) is mapped to
(x + 4, y – 2)
A
D
B
C
A´
B´
D´
C´
 Ex: Write a rule to describe the translation
PQRS  P´Q ´R ´S ´
Q´
R´
4
2
-6 P ´
-4
-2
Q
R
2
P
S´
4
6
S
 Composition -> a combination of two or more
transformations. In a composition, each
transformation is performed on the image of the
preceding transformation.
The red arrow is the
composition of the two
black arrows.
Homework #12
Due Wed/Thurs (Feb 20/21)
Page 473 – 474
#1
– 22 all
Section 9.2 - Reflections
 Objectives:
 To
find reflection images of figures
 Reflection (aka flip) -> an isometry in which a figure
and its image have opposite orientations. Think of
how a reflected image in a mirror appears
“backwards”
U
U´
G
G´
B
B´
 The following rules can be used to reflect a figure
across a line r :


If a point A is on line r, then the image of A is A itself
(that is, A´ = A)
If a point B is not on line r, then r is the perpendicular
bisector of BB´
C´
r
B´
A = A´
C
B
 Ex: Drawing Reflection images.
 Given
points A(-3, 4), B (0, 1), and C(2, 3),
draw ΔABC and its reflection image across each
axis (x and y).
 Draw
the same reflection image across x = 3
Section 9.3 - Rotations
 Objectives:
To draw and identify rotation images of figures
 Rotation -> an isometry in which exactly one point is
its own image, the center of rotation.

In order to describe a rotation, you must know:
1. The center of rotation (a point)
2. The angle of rotation (positive number of degrees)
3. Whether the rotation is clockwise or counterclockwise.
 The following rules can be used to rotate a figure
through x° about a point R:


1. The image of R is itself (that is R´ = R)
2. For any point V, RV´ = RV and m<VRV´ = x
V´
R´
R
x°
V
Homework # 13
Due Monday (Feb 25)
Page 480 – 481
#1
– 21 all
Section 9.4 - Symmetry
 Objectives:
To identify the type of symmetry in a figure
 Symmetry  exists in a figure if there is an isometry
that maps the figure onto itself
 Reflectional Symmetry/Line Symmetry  one half of
the figure is a mirror image of its other half. If the
image were folded across the line of symmetry, the
halves would match up exactly.
 Rotational Symmetry  a figure that is its own
image for some rotation of 180° or less. If you rotate
the figure about an axis a certain degree, it will look
the same as before the rotation.
 Point Symmetry  a figure that has 180° rotational
symmetry is said to have “point symmetry”
 Ex:
Draw all the lines of symmetry for a regular
hexagon.
Draw all the lines of symmetry for a rectangle.
Homework #14
Due Tuesday (Feb 26)
Page 494 – 495
#2
– 18 even
#25 – 32 all
Section 9.5 - Dilations
 Objectives:
To locate dilation images of figures
 Dilation  a transformation whose pre-image and
image are similar. Thus, a dilation is a similarity
transformation, not an isometry.

Every dilation has:
1. A center
2. A scale factor n

The scale factor describes the size change from the original
figure to the image
Enlargement  scale factor is greater than 1
Reduction  scale factor is between 0 and 1
B´
C´
F
4
B
C
G
F´
2
G´
·C
E´
A = A´
D
Enlargement
Center A
Scale Factor 2
D´
E
Reduction
Center C
1
Scale Factor 4
H´
H
 Ex: Finding a Scale Factor
The blue triangle is a dilation image of the red triangle. Describe the
dilation. (center/dilation factor/type of dilation)
R´
T´
8
R
T
4
X = X´
 Ex: Scale Models

The packaging of a model car lists its length as 7.6 cm. It also
gives the scale as 1 : 63. What is the length of the actual car?

The height of a tractor-trailer truck is 4.2 m. The scale factor
1
for a model of the truck is . Find the height of the model to
54
the nearest centimeter.
Homework #15
Due Wednesday (Feb 27)
Page 500 – 501
#1
– 22 all
Quiz Thurs/Fri (9.1 – 9.5)
Section 9.6 – Compositions of Reflections
 Objectives:
To use a composition of reflections
To identify glide reflections
 Theorem 9.1
 A translation or rotation is a composition of two reflections.
 Theorem 9.2
 A composition of reflections across two parallel lines is a
translation.
 Theorem 9.3
 A composition of reflections across two intersecting lines is a
rotation.
 Theorem 9.4
 In a plane, one of two congruent figures can be mapped onto
the other by a composition of at most three reflections.
 Theorem 9.5
 There are only four isometries. They are the following:
Reflection
R
Translation
R
Rotation
R
R
Glide Reflection
R
R
 Glide Reflection  the composition of a glide
(translation) and a reflection across a line parallel to
the direction of translation.
R
R
 Ex: Find the image of R for a reflection across line l
followed by a reflection across line m. Describe the
resulting translation.
l
R
m
 Ex: Find the image of ΔTEX [T(-5, 2), E(-1, 3),
X(-2, 1)] for a glide reflection where the translation is
(x, y)  (x, y-5) and the reflection line is x = 0.
 Use ΔTEX from the above example.
 Find the image of ΔTEX under a glide reflection where the
translation is (x, y)  (x+1, y) and the reflection line is y = -2
 Would the reflection be the same if you reflection ΔTEX first,
and then translated it? Explain.
Homework #16
Due Tuesday (March 05)
Page 509 – 510
#
1 – 23 all
Section 9.7 - Tessellations
 Objectives:
To identify transformations in tessellations, and
figures that will tessellate.
To identify symmetries in tessellations.
 Tessellation (aka tiling)  a repeating
pattern of figures that completely covers a
plane, without gaps or overlaps.
 Tessellations
can be created with translations,
rotations, and reflections. They can be found in
art, nature (cells in a honeycomb), and everyday
life (tiled floors).
Identify the transformation and the
repeating figures in these tessellations.
 Because the figures in a tessellation do not overlap or
leave gaps, the sum of the measures of the angles
around any vertex must be 360°. If the angles
around a vertex are all congruent, then the measure
of each angle must be a factor of 360.
 Ex: Determine whether a regular 18-gon tessellates a
plane.
a=
180(𝑛 −2)
𝑛
 Theorem 9.6
 Every triangle tessellates
 Theorem 9.7
 Every quadrilateral tessellates
 It is also possible to identify symmetries in
tessellations. There are four possible symmetries that
can be identified in tessellations.




Reflectional Symmetry  can be reflected about some vector
and remain the same.
Rotational Symmetry  can be rotated by some angle about
some point and remain unchanged.
Translational Symmetry  can be translated by some vector
and remain unchanged.
Glide Reflectional Symmetry  can be translated by some
vector and then reflected and remain unchanged.
Identify the type of symmetry in each tessellation.
Homework #17
Due Wednesday (March 06)
Page 518 – 519
#
1 – 14 all
# 18 – 28 even