Warm-Up

1. Warm-Up Which of the following does not belong?

2. 4.8 Congruence Transformations Objectives: • To define transformations • To view tessellations as an application of transformations • To perform transformations in the coordinate plane using coordinate notation

3. Transformations A transformationis 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. C C’ Pre-image Image Notice the labeling Transformation A A’ B B’

4. Congruence Transformations 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…

5. Isometries Which of the following transformations is not an isometry?

6. Tessellations 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.

7. Tessellations

8. Tessellations

9. 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?

10. 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?

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

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

13. Transformation Coordinate Ruleswill help us answer these questions: What are the new coordinates of the point (x, y) under each of the following transformations? • Translation under the vector <a, b> • Reflection across the x-axis Reflection across the y-axis • Reflection across the line y = x Reflection across the line y = −x • Rotation of 90° counterclockwise around the origin

14. Transformation Coordinate Rules Coordinate Notation for a Translation You can describe a translation of the point (x, y) under the vector <a, b> by the notation:

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

16. Reflection Coordinate Rules Coordinate Notation for a Reflection

17. Reflection Coordinate Rules

18. Reflection Coordinate Rules

19. Rotation Coordinate Rules Coordinate Notation for a Rotation

20. Rotation Coordinate Rules Rotate 900 counterclockwise around(about) the origin

21. Coordinate Rules in a nutshell Click on the button below

22. Example 2: copy and complete in your notebook

23. Example 3: answer in notebook Draw and label ΔABC after each of the following transformations: • Reflection across the x-axis • Reflection across the y-axis • Translation under the vector <−3, 5>

24. 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›

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

26. Example 5 Draw the image of ABC after it has been rotated 90° counterclockwise around the origin. Notice the red lines make a 900 angle

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

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

29. Example 6c Does the order matter when you perform multiple transformations in a row? Only look at the pre-image and final image. YES!

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

31. 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.

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

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

34. Example 7 Draw and label ΔABC after the following glide reflection: • Translation under the vector <4, −2> • Reflection across the line y = x Draw answer on graph then tape or glue into your notebook.