7.1 Rigid Motion in a Plane
Geometry
Mrs. Spitz
Spring 2005
Slide 1
Standard/Objectives
Standard:
•
Students will understand geometric concepts and
applications.
Performance Standard:
•
Describe the effect of rigid motions on figures in the
coordinate plane and space that include rotations,
translations, and reflections
Objective:
•
Identify the three basic rigid transformations.
Slide 2
Assignments
• Check your Personal Data Folders and
record your attendance and homework time.
(What we did on Monday. Make sure your
folders get back to where they need to be.
• 7.1 Notes: At least 3 pages long. Don’t
annoy your sub, or you will feel the wrath of
Spitz when she returneth to her den.
• Chapter 7 Definitions (14) on pg. 394
• Chapter 7 Postulates/Theorems
• Worksheet 7.1 A and B
Slide 3
Identifying
Transformations
• Figures in a plane can be
– Reflected
– Rotated
– Translated
• To produce new figures. The new figures is
called the IMAGE. The original figures is
called the PREIMAGE. The operation that
MAPS, or moves the preimage onto the
image is called a transformation.
Slide 4
What will you learn?
•
Three basic transformations:
1.
2.
3.
4.
•
Reflections
Rotations
Translations
And combinations of the three.
For each of the three transformations on the
next slide, the blue figure is the preimage
and the red figure is the image. We will use
this color convention throughout the rest of
the book.
Slide 5
Copy this down
Rotation about a point
Reflection in a line
Translation
Slide 6
Some facts
• Some transformations involve labels.
When you name an image, take the
corresponding point of the preimage
and add a prime symbol. For instance,
if the preimage is A, then the image is
A’, read as “A prime.”
Slide 7
Example 1: Naming
transformations
•
Use the graph of the
transformation at the
right.
a. Name and describe
the transformation.
b. Name the coordinates
of the vertices of the
image.
c. Is ∆ABC congruent to
its image?
6
B
B'
4
2
A
C
C'
A'
-5
5
-2
-4
Slide 8
Example 1: Naming
transformations
a. Name and describe
the transformation.
6
B
B'
4
The transformation is
a reflection in the
y-axis. You can
imagine that the
image was
obtained by flipping
∆ABC over the yaxis/
2
A
C
C'
A'
-5
5
-2
-4
Slide 9
Example 1: Naming
transformations
b. Name the
coordinates of the
vertices of the
image.
6
B
B'
4
2
The cordinates of the
vertices of the
image, ∆A’B’C’, are
A’(4,1), B’(3,5), and
C’(1,1).
A
C
C'
A'
-5
5
-2
-4
Slide 10
Example 1: Naming
transformations
c. Is ∆ABC congruent to
its image?
Yes ∆ABC is congruent to
its image ∆A’B’C’.
One way to show this
would be to use the
DISTANCE
FORMULA to find the
lengths of the sides of
both triangles. Then
use the SSS
Congruence Postulate
6
B
B'
4
2
A
C
C'
A'
-5
5
-2
-4
Slide 11
ISOMETRY
• An ISOMETRY is a transformation the
preserves lengths. Isometries also
preserve angle measures, parallel lines,
and distances between points.
Transformations that are isometries are
called RIGID TRANSFORMATIONS.
Slide 12
Ex. 2: Identifying
Isometries
• Which of the following
appear to be
isometries?
• This transformation
appears to be an
isometry. The blue
parallelogram is
reflected in a line to
produce a congruent
red parallelogram.
Preimage
Image
Slide 13
Ex. 2: Identifying
Isometries
• Which of the
following appear to
be isometries?
• This transformation
is not an
ISOMETRY
because the image
is not congruent to
the preimage
PREIMAGE
IMAGE
Slide 14
Ex. 2: Identifying
Isometries
• Which of the following
appear to be
isometries?
• This transformation
appears to be an
isometry. The blue
parallelogram is rotated
about a point to
produce a congruent
red parallelogram.
IMAGE
PREIMAGE
Slide 15
Mappings
• You can describe the
transformation in the
diagram by writing “∆ABC is
mapped onto ∆DEF.” You
can also use arrow notation
as follows:
B
E
– ∆ABC  ∆DEF
• The order in which the
vertices are listed specifies
the correspondence. Either
of the descriptions implies
that
A
C
F
D
– A  D, B  E, and C  F.
Slide 16
Ex. 3: Preserving
Length and Angle
Measures
• In the diagram
∆PQR is mapped
onto ∆XYZ. The
mapping is a
rotation. Given that
∆PQR  ∆XYZ is an
isometry, find the
length of XY and the
measure of Z.
R
35°
Q
Y
3
P
X
Z
Slide 17
Ex. 3: Preserving
Length and Angle
Measures
• SOLUTION:
• The statement “∆PQR
is mapped onto ∆XYZ”
implies that P  X, Q 
Y, and R  Z. Because
the transformation is an
isometry, the two
triangles are congruent.
So, XY = PQ = 3 and
mZ = mR = 35°.
R
35°
Q
Y
3
P
X
Z
Slide 18