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7.1 Rigid Motion in a Plane

- Geometry
- Mrs. Spitz
- Spring 2005

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.

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. Dont 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

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.

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.

Copy this down

Rotation about a point

Reflection in a line

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.

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?

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/

Example 1 Naming transformations

- Name the coordinates of the vertices of the

image. - The cordinates of the vertices of the image,

?ABC, are A(4,1), B(3,5), and C(1,1).

Example 1 Naming transformations

- Is ?ABC congruent to its image?
- Yes ?ABC is congruent to its image ?ABC. 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

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.

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.

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

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.

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.

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.

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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.

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