RIGID MOTION IN A PLANEChapter 9
Section 1
Slide 1
Standard/ObjectivesStandard: • 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
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 PRE-IMAGE. The operation that MAPS, or moves the pre-image onto the image is called a transformation.
Slide 3
What will you learn?• Three basic transformations:
1. Reflections2. Rotations3. Translations4. And combinations of the three.
• For each of the three transformations on the next slide, the blue figure is the pre-image and the red figure is the image. We will use this color convention throughout the rest of the book.
Slide 4
Slide 5
Copy this down
Slide 6
Reflection in a line Rotation about a point
Translation
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?
Slide 8
6
4
2
-2
-4
-5 5 10
C'
B'
A'A
B
C
Example 1: Naming transformationsa. Name and describe
the transformation.
The transformation is a reflection in the y-axis. You can imagine that the image was obtained by flipping ∆ABC over the y-axis/
Slide 9
6
4
2
-2
-4
-5 5 10
C'
B'
A'A
B
C
Example 1: Naming transformationsb. Name the
coordinates of the vertices of the image.
The cordinates of the vertices of the image, ∆A’B’C’, are A’(4,1), B’(3,5), and C’(1,1).
Slide 10
6
4
2
-2
-4
-5 5 10
C'
B'
A'A
B
C
Example 1: Naming transformationsc. 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
Slide 11
6
4
2
-2
-4
-5 5 10
C'
B'
A'A
B
C
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.
Slide 13
ImagePreimage
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
Slide 14
PREIMAGE IMAGE
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.
Slide 15
PREIMAGE
IMAGE
Mappings• You can describe the
transformation in the diagram by writing “∆ABC is mapped onto ∆DEF.” You can also use arrow notation as follows:• ∆ABC ∆DEF
• The order in which the vertices are listed specifies the correspondence. Either of the descriptions implies that • A D, B E, and C F.
Slide 16
FD
E
A
B
C
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.
Slide 17
P
Q
R
3
Z
X
Y
35°
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 mZ = mR = 35°.
Slide 18
P
Q
R
3
Z
X
Y
35°
What have you learned?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.
• Three basic transformations:1. Reflections2. Rotations3. Translations
Slide 19
HW• Page 576 • 3-13 all
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