Slide 1

7.1 Rigid Motion in a Plane

GeometryMrs. Spitz

Spring 2005

Slide 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.

Slide 3

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 4

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 5

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 preimage and the red figure is the image. We will use this color convention throughout the rest of the book.

Slide 6

Copy this down

Reflection in a line Rotation about a point

Translation

Slide 7

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 8

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

4

2

-2

-4

-5 5 10

C'

B'

A'A

B

C

Slide 9

Example 1: Naming transformations

a. 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/

6

4

2

-2

-4

-5 5 10

C'

B'

A'A

B

C

Slide 10

Example 1: Naming transformations

b. 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).

6

4

2

-2

-4

-5 5 10

C'

B'

A'A

B

C

Slide 11

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

4

2

-2

-4

-5 5 10

C'

B'

A'A

B

C

Slide 12

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 13

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.

ImagePreimage

Slide 14

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 15

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.

PREIMAGE

IMAGE

Slide 16

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.

FD

E

A

B

C

Slide 17

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.

P

Q

R

3

Z

X

Y

35°

Slide 18

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°.

P

Q

R

3

Z

X

Y

35°