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7.1 Rigid Motion in a Plane. Images and Preimages. Figures in a plane can be changed to produce a new figure or image. In this unit, we will be investigating 3 different methods for making new figures (images) from old ones. Later in the course we will investigate a fourth method.

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Images and preimages
Images and Preimages

Figures in a plane can be changed to produce a new figure or image. In this unit, we will be investigating 3 different methods for making new figures (images) from old ones. Later in the course we will investigate a fourth method.

The new figure is called the image. The original figure is called the preimage (the one that came before).


Transformations and isometry
Transformations and Isometry

The operation that maps or moves the preimage onto the image is called a transformation (change).

An isometry is a transformation that preserves length, it is considered a rigid transformation or motion. The new figure or image is congruent to the original figure or preimage.


One way to use notation
One Way to Use Notation

The image or new figure can be named with different letters.


A second way to use notation
A Second Way to Use Notation

Sometimes we use the same letter to represent the preimage and the image so that it is clear to the reader that they aren’t just totally different images, but that they have a relationship to one another. Namely the image was produced by the preimage. Of course the letters can’t be exactly the same, or we won’t know which is which, like using the same name for all of your children. (Which one is George?) Using this type of notation- if the preimage is A, then the image is A’, which is read “A prime”. (Like George and George Jr)


What are the 3 transformations
What are the 3 Transformations?

1. Translations are accomplished by sliding the figure across the plane. (Hint-the “sl” in translation can remind you of the “sl” in slide.)

2. Reflections are accomplished by flipping the figure across a line of reflection, often the x or y axis, in the plane. (Hint- the “fl” in reflections can remind you of the “fl” in flip.)

3. Rotations are accomplished by turning the figure about a point, often the origin, in the plane. (Hint-you probably already know that a rotation is a turn.)

Note: Keep in mind that these transformations are rigid. They do not change the size and shape of the figure, but merely relocate the figure in the plane and/or change its orientation.


Illustrations of the 3 transformations
Illustrations of the 3 Transformations

  • 1. Translations ( or slides): this moves the figure to a new location with no change to the looks of the figure.

  • 2. Reflections: these are like mirror images as seen across a line or a point.

  • 3. Rotations: this turns the figure clockwise or counter-clockwise but doesn’t change the figure.

(HCPS)


More illustrations
More Illustrations

As you look at the image determine the type of translation.

Art work

Floor tiling

ROTATION

TRANSLATION

http://library.thinkquest.org/16661/escher/tessellations.11.html

http://www.camaraslate.com/images/flooring/E42601.jpg


More images
More images

Art work

Stained Glass Window

TRANSLATION

REFLECTION

http://www.easyboo.com/images/tb/IMG_3814a.JPG

http://library.thinkquest.org/16661/escher/tessellations.11.html


More images1
More images

Art work

Quilting

ROTATION

REFLECTION

http://library.thinkquest.org/16661/escher/tessellations.11.html

www.lisettescountryfabriccreations.com



Examples
Examples

Does the transformation appear to be an isometry.

1.

2.

3.

YES

NO

YES


More examples
More Examples

Name the type of transformation that maps the unshaded right triangle (preimage) onto the shaded right triangle (image).

4.

5.

REFLECTION

ROTATION

6.

TRANSLATION


More examples1

7. HIJK

8. Rotation

9.

10.

11. (9,2)

12. See Sketch

More Examples

Complete each statement.


Last example
Last Example

Find the value of the variables given the transformation is an isometry

13.

3a + 1 = 10

3a = 9

a = 3

b = 4


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