Warm-Up Which of the following does not belong?

4.8 Congruence Transformations 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

Transformations transformation image pre-image A transformation is an operation that changes some aspect of the geometric figure to produce a new figure. The new figure is called the image, and the original figure is called the pre-image. Transformation Pre-image A B C Image A’ B’ C’ Notice the labeling

Congruence Transformations congruence transformation isometry A congruence transformation, or isometry, 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…

Isometries Which of the following transformations is not an isometry?

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

Tessellations

Tessellations

Example 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 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?

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

Vectors

Investigation 1 For the following Investigation, we will discover properties of transformations on the coordinate plane. Then we will write a coordinate rule for each transformation.

Transformation Coordinate Rules will help us answer these questions: What are the new coordinates of the point ( x, y ) under each of the following transformations? 1.Translation under the vector 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 Rules Coordinate Notation for a Translation You can describe a translation of the point ( x, y ) under the vector by the notation:

Translation Coordinate Rules (3,-6) (33,34)

Reflection Coordinate Rules Coordinate Notation for a Reflection

Reflection Coordinate Rules

Rotation Coordinate Rules Coordinate Notation for a Rotation

Rotation Coordinate Rules Rotate 90 0 counterclockwise around(about) the origin

Coordinate Rules in a nutshell Click on the button below

Example 2: copy and complete in your notebook TransformationCoordinate Notation Image of (8, − 13) Translation under vector 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 notebook Draw and label Δ ABC after each of the following transformations: 1.Reflection across the x -axis 2.Reflection across the y -axis 3.Translation under the vector

Example 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 notation Translate under the vector ‹10,-2›

Example 5 Draw the image of ABC after it has been rotated 90° counterclockwise around the origin.

Example 5 Draw the image of ABC after it has been rotated 90° counterclockwise around the origin. Notice the red lines make a 90 0 angle

Example 6a Does the order matter when you perform multiple transformations in a row? 1.Translation under Translation under 2.Translation under Translation under Only look at the pre-image and final image. NO!

Example 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 6c Does the order matter when you perform multiple transformations in a row? 1.Translation under Reflection across y -axis 2.Reflection across y -axis Translation under Only look at the pre-image and final image. YES!

Composition of Transformations composite transformation Two or more transformations can be combined to make a single transformation called a composite transformation.

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

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

Example 7 Draw and label Δ ABC after the following glide reflection: 1.Translation under the vector 2.Reflection across the line y = x Draw answer on graph then tape or glue into your notebook.

Assignment P : 1-5, odd, 20, odd, 41, 43, 44, 45 P. 602: 12, 13, 14 P. 611: 6, 8, 11 P. 618: Challenge Problems