1. Rigid Motion in a Plane
Geometry

2. 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.

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.

4. What will you learn? Three basic transformations:
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.

5. Copy this down
Rotation about a point
Reflection in a line
Translation

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.”

7. Example 1: Naming transformations
Use the graph of the transformation at the right.
Name and describe the transformation.
Name the coordinates of the vertices of the image.
Is ∆ABC congruent to its image?

8. Example 1: Naming transformations
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/

9. Example 1: Naming transformations
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).

10. Example 1: Naming transformations
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

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.

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.

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

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.

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:
∆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.

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.
35°

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°.
35°