1. Transformations 3-6, 3-7, & 3-8

2. Transformation Movements of a figure in a plane
May be a SLIDE, FLIP, or TURN
a change in the position, shape, or size of a figure.

3. Image Original Image Slide The symbol ‘ is read “prime”.
The figure you get after a translation A A’
Slide C B C’ B’
Original
Image
To identify the image of point A, use prime notation Al.
You read Al as “A prime”.
The symbol ‘ is read “prime”.
ABC has been moved to A’B’C’.
A’B’C’ is the image of ABC.

4. Translation
a transformation that moves each point of a figure the same distance and in the same direction.
AKA - SLIDE A A’ C B C’ B’

5. Writing a Rule for a Translation
Finding the amount of movement LEFT and RIGHT and UP and DOWN

6. Writing a Rule Right 4 (positive change in x) Down 3 (negative9 8 7 6 5 4 3 2 1
Right 4 (positive change in x) B
Down 3
(negative
change in y) B’ A C A’ C’

7. Rule: (x,y) (x+4, y-3) Writing a Rule Can be written as: R4, D3
(Right 4, Down 3)
Rule: (x,y) (x+4, y-3)

8. Rule (x+6, y-3) Cl = (4, -1) Al = (1, -2) Bl = (5, 1) Translations
Example 1: If triangle ABC below is translated 6 units to the right and 3 units down, what are the coordinates of point Al.
A (-5, 1) B (-1, 4) C (-2, 2)
Rule (x+6, y-3)
-First write the rule and then translate each point.
Cl = (4, -1)
Al = (1, -2)
Bl = (5, 1)
-Now graph both triangles and see if your image points are correct. B A C B’ C’ A’

9. -First graph the triangle and then translate each point.
Example 2: Triangle JKL has vertices J (0, 2), K (3, 4), L (5, 1). Translate the triangle 4 units to the left and 5 units up. What are the new coordinates of Jl?
-First graph the triangle and then translate each point.
Kl = (-1, 9)
Ll = (1, 6)
Jl = (-4, 7)
-You can use arrow notation to describe a translation.
For example: (x, y) (x – 4, y + 5) shows the ordered pair (x, y) and describes a translation to the left 4 unit and up 5 units. K’ J’ L’ K J L

10. Al (-1, 3) Bl (-4, -2) B Al A Bl You try some:
Graph each point and its image after the given translation.
a.) A (1, 3) left 2 units b.) B (-4, 4) down 6 units
Al (-1, 3)
Bl (-4, -2) B Al A Bl

11. Example 3: Write a rule that describes the translation below
Point A (2, -1) Al (-2, 2)
Point B (4, -1) Bl (0, 2)
Point C (4, -4) Cl (0, -1)
Point D (2, -4) Dl (-2, -1)
Rule (x, y) (x – 4, y + 3)
Example 4: Write a rule that describes each translation below.
a.) 3 units left and 5 units up b.) 2 units right and 1 unit down
Rule (x, y) (x – 3, y + 5)
Rule (x, y) (x + 2, y – 1)

12. Reflection
Another name for a FLIP A A’ C B B’ C’

13. Used to create SYMMETRY on the coordinate plane
Reflection
Used to create SYMMETRY on the coordinate plane

14. When one side of a figure is a MIRROR IMAGE of the other
Symmetry
When one side of a figure is a MIRROR IMAGE of the other

15. The line you reflect a figure across
Line of Reflection
The line you reflect a figure across
Ex: X or Y axis
X - axis

16. In the diagram to the left you will notice that triangle ABC is reflected over the y-axis and all of the points are the same distance away from the y-axis.
Therefore triangle AlBlCl is a reflection of triangle ABC
Example 1: Draw all lines of reflection for the figures below. This is a line where if you were to fold the two figures over it they would line up. How many does each figure have?
a.) b.) 1 6

17. Example 2: Graph the reflection of each point below over each line of reflection.
a.) A (3, 2) is reflected over the x-axis
b.) B (-2, 1) is reflected over the y-axis A B Bl Al

18. Example 3: Graph the triangle with vertices A(4, 3), B (3, 1), and C (1, 2). Reflect it over the x-axis. Name the new coordinates. C A B
C’ (1,-2)
B’ (3,-1)
A’ (4,-3)

19. Symmetry of the Alphabet
Sort the letters of the alphabet into groups according to their symmetries
Divide letters into two categories:
symmetrical
not symmetrical

20. Symmetry of the Alphabet
Symmetrical: A, B, C, D, E, H, I, K, M, N, O, S, T, U, V, W, X, Y, Z
Not Symmetrical: F, G, J, L, P, Q, R

21. Rotation
Another name for a TURN C’ B’ B A’ C A

22. A transformation that turns about a fixed point
Rotation
A transformation that turns about a fixed point

23. Center of Rotation
The fixed point C’ B’ B A’ C A
(0,0)

24. Measuring the degrees of rotation
Rotating a Figure
Measuring the degrees of rotation C’ B’ B A’
90 degrees C A

25. Rotations in a Coordinate Plane
In a coordinate plane, sketch the quadrilateral whose vertices are A(2, –2), B(4, 1), C(5, 1), and D(5, –1). Then, rotate ABCD 90º counterclockwise about the origin and name the coordinates of the new vertices. Describe any patterns you see in the coordinates.
SOLUTION
Plot the points, as shown in blue. Use a protractor, a compass, and a straightedge to find the rotated vertices. The coordinates of the preimage and image are listed below.
In the list, the x-coordinate of the image is the opposite of the y-coordinate of the preimage. The y-coordinate of the image is the x-coordinate of the preimage.
Figure ABCD
Figure A'B'C'D'
A(2, –2)
A '(2, 2)
B(4, 1)
B '(–1, 4)
This transformation can be described as (x, y) (–y, x).
C(5, 1)
C '(–1, 5)
D(5, –1)
D '(1, 5)

26. Rotational symmetry can be found in many objects that rotate about a centerpoint.
Determine the angle of rotation for each hubcap. Explain how you found the angle.
Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.

27. Hubcap 1
Determine the angle of rotation for each hubcap. Explain how you found the angle.
Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.

28. There are 5 lines of symmetry in this design.
Hubcap 1
There are 5 lines of symmetry in this design.
360 degrees divided by 5 =

29. The angle of rotation is 72º.
Hubcap 1
72º
The angle of rotation is 72º.

30. There are NO lines of symmetry in this design.
Hubcap 2
There are NO lines of symmetry in this design.

31. The angle of rotation is 120º.
Hubcap 2
120º
The angle of rotation is 120º.
(360 / 3)
There are NO lines of symmetry in this design.

32. Hubcap 3
Determine the angle of rotation for each hubcap. Explain how you found the angle.
Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.

33. There are 10 lines of symmetry in this design.
Hubcap 3
There are 10 lines of symmetry in this design.
360 / 10 = 36
However to make it look exactly the same you need to rotate it 2 angles.
36 x 2 = 72

34. The angle of rotation is 36º.
Hubcap 3
36º
The angle of rotation is 36º.
There are 10 lines of symmetry in this design.

35. Hubcap 4
Determine the angle of rotation for each hubcap. Explain how you found the angle.
Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.

36. There are 9 lines of symmetry in this design.
Hubcap 4 .
There are 9 lines of symmetry in this design.

37. The angle of rotation is 40º.
Hubcap 4
40º
The angle of rotation is 40º.
There are 9 lines of symmetry in this design.

38. Think About it: Is there a way to determine the angle of rotation for a particular design without actually measuring it?

39. When there are lines of symmetry 360 ÷ number of lines of symmetry = angle of rotation When there are no lines of symmetry: 360 ÷ number of possible rotations around the circle.
5 lines of symmetry
3 points to rotate it to

40. Homework
Pg 138 #8, 12, 18, & 22
Pg 143 #8, 10, 16, & 18
Pg 148 #6, 8, 10

42. A design that covers a plane with NO GAPS and NO OVERLAPS
Tessellation
A design that covers a plane with NO GAPS and NO OVERLAPS

43. Formed by a combination of TRANSLATIONS, REFLECTIONS, and ROTATIONS
Tessellation
Formed by a combination of TRANSLATIONS, REFLECTIONS, and ROTATIONS

44. A tessellation that uses only ONE shape
Pure Tessellation
A tessellation that uses only ONE shape

45. Pure Tessellation

46. Pure Tessellation

47. Semiregular Tessellation
A design that covers a plane using more than one shape

48. Semiregular Tessellation

49. Semiregular Tessellation

50. Semiregular Tessellation

51. Semiregular Tessellation

52. Used famously in artwork by M.C. Escher
Tessellation
Used famously in artwork by M.C. Escher

55. Group Activity Choose a letter (other than R) with no symmetries
On a piece of paper perform the following tasks on the chosen letter:
rotation
translation
Reflection