Warm-UpWarm-Up

Which of the following does not belong?

4.8 Congruence 4.8 Congruence TransformationsTransformations

Objectives:

1. To define transformations

2. To view tessellations as an application of transformations

3. To perform transformations in the coordinate plane using coordinate notation

TransformationsTransformations

A transformationtransformation is an operation that changes some aspect of the geometric figure to produce a new figure. The new figure is called the imageimage, and the original figure is called the pre-imagepre-image.

TransformationTransformationPre-image

A B

C

Image

A’ B’

C’Notice the labeling

Congruence TransformationsCongruence Transformations

A congruence transformationcongruence transformation, or isometryisometry, is a type of transformation that changes the position of a figure without changing its size or shape.

– In other words, in an isometry, the pre-image is congruent to the image.

– There are three basic isometries…

IsometriesIsometries

Which of the following transformations is not an isometry?

TessellationsTessellations

An interesting application of transformations is a tessellation. A tessellationtessellation is a tiling of a plane with one or more shapes with no gaps or overlaps. They can be created using transformations.

TessellationsTessellations

TessellationsTessellations

Example 1Example 1

Frank is looking to impress his wife by retiling the guest bathroom. Which of the following shapes could he not use to tile the floor?

Example 1Example 1

Frank is looking to impress his wife by retiling the guest bathroom. Which of the following shapes could he not use to tile the floor?

VectorsVectors

Translations are usually done with a vectorvector, which gives a direction and distance to move our shape.

VectorsVectors

Translations are usually done with a vectorvector, which gives a direction and distance to move our shape.

Transformation Coordinate RulesTransformation Coordinate Ruleswill help us answer these will help us answer these questions:questions:What are the new coordinates of the point (x, y)

under each of the following transformations?

1. Translation under the vector <a, b>

2. Reflection across the x-axis

Reflection across the y-axis

3. Reflection across the line y = x

Reflection across the line y = −x

4. Rotation of 90° counterclockwise around the origin

Transformation Coordinate Transformation Coordinate RulesRules

Coordinate Notation for a TranslationCoordinate Notation for a Translation

You can describe a translation of the point (x, y) under the vector <a, b> by the notation:

byaxyx ,,

Translation Coordinate RulesTranslation Coordinate Rules

(3,-6) (33,34)

Reflection Coordinate RulesReflection Coordinate Rules

Coordinate Notation for a ReflectionCoordinate Notation for a Reflection

Reflection Coordinate RulesReflection Coordinate Rules

Reflection Coordinate RulesReflection Coordinate Rules

Rotation Coordinate RulesRotation Coordinate Rules

Coordinate Notation for a RotationCoordinate Notation for a Rotation

Rotation Coordinate RulesRotation Coordinate Rules

Rotate 900 counterclockwise around(about) the origin

Coordinate Rules in a nutshellCoordinate Rules in a nutshell

Click on the button below

Example 2: copy and Example 2: copy and complete complete in your in your notebooknotebookTransformation Coordinate Notation Image of (8, −13)

Translation under vector <−2, 6>

Reflection across x-axis

Reflection across y-axis

Reflection across y = x

Reflection across y = −x

Rotation 90° CC around origin

Rotation 180° CC around origin

Rotation 270° CC around origin

Rotation 360° CC around origin

Example 3: answer in Example 3: answer in notebooknotebook

Draw and label ΔABC after each of the following transformations:

1. Reflection across the x-axis

2. Reflection across the y-axis

3. Translation under the vector <−3, 5>

Example 4Example 4

What translation vector was used to translate ABC to A’B’C’? Write a coordinate rule for the translation.

Coordinate notation(x,y) (x+10,y-2)

Vector notationTranslate under the vector ‹10,-2›

Example 5Example 5

Draw the image of ABC after it has been rotated 90° counterclockwise around the origin.

8

6

4

2

-2

-4

5

A

B

C

Example 5Example 5

Draw the image of ABC after it has been rotated 90° counterclockwise around the origin.

8

6

4

2

-2

-4

5

A'

C'

B'

A

B

C

Notice the red lines make a 900 angle

Example 6aExample 6a

Does the order matter when you perform multiple transformations in a row?

1. Translation under <2, −3> Translation under <−4, −1>

2. Translation under <−4, −1> Translation under <2, −3>

Only look at the pre-image and final image.

NO!

Example 6bExample 6b

Does the order matter when you perform multiple transformations in a row?

1. Reflection across y-axis Reflection across x-axis

2. Reflection across x-axis Reflection across y-axis

Only look at the pre-image and final image.

NO!

Example 6cExample 6c

Does the order matter when you perform multiple transformations in a row?

1. Translation under <2, −3> Reflection across y-axis

2. Reflection across y-axis Translation under <2, −3>

Only look at the pre-image and final image.

YES!

Composition of TransformationsComposition of Transformations

Two or more transformations can be combined to make a single transformation called a composite transformationcomposite transformation.

Composition of TransformationsComposition of Transformations

When the transformations being composed are of different types (like a translation followed by a reflection), then the order of the transformations is usually important.

Glide ReflectionGlide Reflection

A special type of composition of transformations starts with a translation followed by a reflection. This is called a glide reflectionglide reflection.

Glide ReflectionGlide Reflection

A special type of composition of transformations starts with a translation followed by a reflection. This is called a glide reflectionglide reflection.

Example 7Example 7

Draw and label ΔABC after the following glide reflection:

1. Translation under the vector <4, −2>

2. Reflection across the line y = x

Draw answer on graph then tape or glue into your notebook.