1. Geometric Transformations:
Translation: slide Reflection: mirror Rotation: turn Dialation: enlarge or reduce

2. Notation:
Pre-Image: original figure Image: after transformation. Use prime notation A’ C
C ’ B B’ A

3. Isometry AKA: congruence transformation
a transformation in which an original figure and its image are congruent.

4. Theorems about isometries
FUNDAMENTAL THEOREM OF ISOMETRIES
Any any two congruent figures in a plane can be mapped onto one another by at most 3 reflections
ISOMETRY CLASSIFICATION THEOREM
There are only 4 isometries. They are:

5. TRANSLATION: moves all points in a plane a given direction a fixed distance

6. TRANSLATION VECTOR: Direction Magnitude
PRE-IMAGE
IMAGE

7. Translate by the vector <x, y>

8. x moves horizontal y moves vertical
Translate by <3, 4>

9. Different notation T(x, y) -> (x+3, y+4)

10. Translations PRESERVE: Size Shape Orientation

11. Reflection over a line (mirror)

12. Properties of reflections
PRESERVE
Size (area, length, perimeter…)
Shape
CHANGE
orientation (flipped)

13. Reflect x-axis: (a, b) -> (a,-b) Change sign y-coordinate

14. Reflect y-axis: (a, b) -> (-a, b) Change sign on x coordinate

16. X-axis reflection

17. Y-axis reflection

18. PARTNER SWAP: Part I: (Live under my rules)
Use sketchpad to graph & label any three points
Graph & Reflect them over the line y = x
Graph->Plot new function->x->OK
Construct two points on the line and connect them
Mark this line segment as your mirror.
WRITE a conjecture about how (a, b) will be changed after reflecting over y = x. Explain.
Repeat by reflecting over the line y = -x. Write a conjecture.

19. Starter:
Find one vector which would accomplish the same thing as translating (3, -1) by <3, 8> then applying the transformation T(x, y)->(x-4, y+9)
Find coordinates of (7, 6) reflected over:
a.) the y-axis
b.) the x-axis
c.) the line y = x
d.) the line x = -3
3. HW Check & Peer edit

20. Rotations have:
Center of rotation
Angle of rotation:

21. Example: Rotate Triangle ABC 60 degrees clockwise about “its center”
Find the image of A after a 120 degree rotation
Find the image of A after a 180 degree rotation
Find the image of A after a 240 degree rotation
Find the image of A after a 300 degree rotation
Find the image of A after a 360 degree rotation

22. Rotated 90 degrees counterclockwise

23. ROTATIONS PRESERVE SIZE SHAPE ORIENTATION Length of sides
Measure of angles
Area
Perimeter
SHAPE
ORIENTATION

24. PARTNER SWAP: Part II: (Live under new rules)
Use sketchpad to graph & label any three points. Connect them and construct triangle interior.
Rotate your pre-image about the origin 90
Rotate the pre-image about the origin 180
Rotate the pre-image about the origin 270
Rotate the pre-image about the origin 360
WRITE A CONJECTURE: What are the coordinates of (a, b) after a 90, 180, and 270 degree rotation about the origin?

25. Rotations on a coordinate plane about the origin
90 (a, b) -> (-b, a)
180 (a, b) -> (-a, -b)
270 (a, b) -> (b, -a)
360 (a, b) -> (a, b)

26. DEBRIEFING: Find the coordinates of (2, 5) Reflected over the x-axis
Reflected over the y-axis
Reflected over the line x = 3
Reflected over the line y = -2
Reflected over the line y = x
Rotated about the origin 180
Rotated about the origin 270 
Rotated about the origin 360 

27. Review the rules for coordinate geometry transformations
Which two transformations would accomplish the same thing as a 90 degree rotation about the origin?
Use sketchpad to justify your answer

28. Coordinate Geometry rules
Reflections
x axis (a, b) -> (a, -b)
y axis (a, b) -> (-a, b)
y=x (a, b) -> (b, a)
Rotations about the origin
90 (a, b) -> (-b, a)
180 (a, b) -> (-a, -b)
270 (a, b) -> (b, -a)
360 (a, b) -> (a, b)

29. GLIDE REFLECTIONS You can combine different Geometric Transformations…

30. Practice: Reflect over y = x then translate by the vector <2, -3>

31. After Reflection…

32. After Reflection and translation…

33. Santucci’s Starter:
Complete the following transformations on (6, 1) and list coordinates of the image:
a. Reflect over the x-axis
b. Reflect over the y-axis
c. Rotate 90 about the origin
d. Rotate 180 about the origin
e. Rotate 270 about the origin
EXPLAIN in writing: what two transformations would accomplish the same thing as a 90 degree rotation about the origin?

34. Starter:
Find the coordinates of pre-image (3, 4) after the following transformations (do without graphing…)
reflect over y-axis
reflect over x-axis
reflect over y=x
reflect over y=-x
translate <-2, 6>
rotate 90 about origin
rotate 180 about origin
rotate 270 about origin
rotate 360 about origin

35. PAIRS Sketchpad Exploration:
Rotate (3, 4) 90 degrees about the point (1, 6). What two transformations will produce the same result?
Try it again by rotating (3, 4) 90 degrees about (-2, 5).
Rotate (2, -6) 90 degrees about (1, 7)
Describe OR LIST STEPS FOR how you can find the image of any point after a 90 rotation about (a, b).
Try it again with a 180 rotation about (a,b). How can you find the image?
Try it again with a 270 rotation about (a,b). How can you find the image?

36. Starter HW Peer edit Practice 12-5
Reflectional symmetry
Both rotational and Reflectional symmetry
See key
No lines of symmetry
Line symmetry (5 lines) and 72 degree rotational symmetry
Line symmetry (1 line)
Line symmetry (4 lines) and 90 degree rotational symmetry
Line symmetry (8 lines) and 45 degree rotational symmetry
180 degree rotational symmetry
Line symmetry (1 line) #17-21 see key

37. Symmetry Line Symmetry Rotational Symmetry
If a figure can be reflected onto itself over a line.
Rotational Symmetry
If a figure can be rotated about some point onto itself through a rotation between 0 and 360 degrees

38. What kinds of symmetry do each of the following have?

39. What kinds of symmetry do each of the following have?
Rotational (180) Point Symmetry
Rotational (90, 180, 270)
Point Symmetry
Rotational (60, 120, 180, 240, 300)
Point Symmetry

40. NOTE: TEST WILL BE END OF NEXT WEEK!!!
Isometry Wrap Up…
Sketchpad Activitiy # 6 Symmetry in Regular Polygons
Dilations Exploration
NOTE: TEST WILL BE END OF NEXT WEEK!!!

41. Dilations
Plot any 5 points to make a convex polygon and fill in its interior red.
Mark the origin as center.
Make the polygon larger by a scale factor of 2 and fill it in green.
Make the polygon smaller by a scale factor of 1/3. Fill it in red.
Measure your coordinates and Explain how you can find coordinates of a dilation image.
Try marking a new center and dilating a few points. What is the “center” of a dilation? How does it change the measurements?

42. Tessellations web-quest
VISIT:
Explore & read information underTessellations:
What are they
The beginnings
Symmetry & MC Escher
The galleries
Solid Stuff
Answer the following questions:
1. What is symmetry and list the types discussed.
2. What are the Polya’ symmetries?
3. How many Polya’ symmetries are there?
4. What are the Rhomboid possibilities?
5. What is the difference between a periodic and aperiodic tiling?

43. TO-DO Complete Tessellations Sketchpad explorations, # 8, 9
Read rubric and write questions. Begin design

44. INDIRECT PROOF If ~q then ~p
Assume that the conclusion is FALSE.
Reason to a contradiction.
If n>6 then the regular polygon will not tessellate.
ASSUME: The polygon tessellates
SHOW: n can not be >6

45. Indirect proof Regular polygons with n>6 sides will not tessellate
Assume a polygon with n>6 sides will tessellate.
This means that n*one interior <measure will equal 360
IF n = 3 there are 6 angles about center point
IF n = 4 there are 4 angles about center point
IF n = 6 there are 3 angles about center point
Therefore, if n>6 then there must be fewer than 3 angles about the center point. In other words, there must be 2 or fewer. If there are 2 angles about the center point then each angle must measure 180 to sum to 360
But no regular polygon exists whose interior angle measures 180 (int. < sum must be LESS than 180). Therefore, the polygon can not tessellate.

46. Santucci’s Starter
Determine if the following will tessellate & provide proof:
Isosceles triangle
Kite
Regular pentagon
Regular hexagon
Regular heptagon
Regular octagon
Regular nonagon
Regular decagon

47. Review practice
Find the image of A(-1, 4) reflected over the x-axis then over the y-axis (two intersecting lines). What one transformation would accomplish the same result?
Find the image of B(6, -2) reflected over x=3 then over x=-5 (two parallel lines). What one transformation would accomplish the same result?
List all the rotational symmetries of a regular decagon.
Draw a regular octagon with all its lines of symmetry (on sketchpad).

48. Problem from
HSPA test

49. Coordinate Transformations MOAT game
Groups of “3”
Write answer on white board and send one “runner” to stand facing the class with representatives from all other groups (hold board face down). When MOAT is called flip answer so all members seated can see answer.
1st group correct = +3 points
2nd group correct = +2 points
3rd group correct = +1 points
Group with HIGHEST # points +3 on quiz
Group with 2nd highest # points +2 on quiz
Group with 3rd highest # points +1 on quiz

50. HW Answers p. 650 12-4 10. H 4. F translate twice the distance 11. M
13. Segment BC
14. A
15. Segment LM
16. I
17. K
34. a.) B(-2, 5)
b.) C(-5, -2)
c.) D(2, -5)
d.) Square: 4 congruent sides & angles
12-4
4. F translate twice the distance
Translate T across m twice the distance between l and m
V rotated 145
Peer edit
opp; reflection
same; translation
same; 270 rotation
Glide <-2, -2>, reflect over y = x – 1
28. Glide <0, 4>, reflect over y = 0 (x-axis)