RIGID MOTION IN A PLANE Chapter 9 Section 1 Slide 1.

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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