1. In geometry, a transformation is a way to change the position of a figure.

2. translations, rotations, and reflections
In some transformations, the figure retains its size and only its position is changed.
Examples of this type of transformation are:
translations, rotations, and reflections
In other transformations, such as dilations, the size of the figure will change.

3. TRANSLATION

4. TRANSLATION
A translation is a transformation that slides a figure across a plane or through space.
With translation all points of a figure move the same distance and the same direction.

5. Basically, translation means that a figure has moved.
A TRANSLATION IS A CHANGE IN LOCATION.
A translation is usually specified by a direction and a distance.

6. What does a translation look like?
original
image x y
Translate from x to y
A TRANSLATION IS A CHANGE IN LOCATION.

7. TRANSLATION
In the example below triangle A is translated to become triangle B. A B
Triangle A is slide directly to the right.
Describe the translation.

8. TRANSLATION
In the example below arrow A is translated to become arrow B. A B
Arrow A is slide down and to the right.
Describe the translation.

9. ROTATION

10. Basically, rotation means to spin a shape.
A rotation is a transformation that turns a figure about (around) a point or a line.
Basically, rotation means to spin a shape.
The point a figure turns around is called the center of rotation.
The center of rotation can be on or outside the shape.

11. What does a rotation look like?
center of rotation
A ROTATION MEANS TO TURN A FIGURE

12. This is another way rotation looks
The triangle was rotated around the point.
This is another way rotation looks
center of rotation
A ROTATION MEANS TO TURN A FIGURE

13. All the way around ROTATION
If a shape spins 360, how far does it spin?
360
All the way around
This is called one full turn.

14. Half of the way around ROTATION
If a shape spins 180, how far does it spin?
Rotating a shape 180 turns a shape upside down.
Half of the way around
180
This is called a ½ turn.

15. One-quarter of the way around
ROTATION
If a shape spins 90, how far does it spin?
One-quarter of the way around
90
This is called a ¼ turn.

16. ROTATION
Describe how the triangle A was transformed to make triangle B A B
Triangle A was rotated right 90
Describe the translation.

17. ROTATION Describe how the arrow A was transformed to make arrow B B A
Arrow A was rotated right 180
Describe the translation.

18. to spin a shape the exact same ROTATION
When some shapes are rotated they create a special situation called rotational symmetry.
to spin a shape
the exact same

19. ROTATIONAL SYMMETRY
A shape has rotational symmetry if, after you rotate less than one full turn, it is the same as the original shape.
Here is an example…
As this shape is rotated 360, is it ever the same before the shape returns to its original direction?
90
Yes, when it is rotated 90 it is the same as it was in the beginning.
So this shape is said to have rotational symmetry.

20. Here is another example…
ROTATIONAL SYMMETRY
A shape has rotational symmetry if, after you rotate less than one full turn, it is the same as the original shape.
Here is another example…
As this shape is rotated 360, is it ever the same before the shape returns to its original direction?
Yes, when it is rotated 180 it is the same as it was in the beginning.
So this shape is said to have rotational symmetry.
180

21. Here is another example…
ROTATIONAL SYMMETRY
A shape has rotational symmetry if, after you rotate less than one full turn, it is the same as the original shape.
Here is another example…
As this shape is rotated 360, is it ever the same before the shape returns to its original direction?
No, when it is rotated 360 it is never the same.
So this shape does NOT have rotational symmetry.

22. Does this shape have rotational symmetry?
ROTATION SYMMETRY
Does this shape have rotational symmetry?
Yes, when the shape is rotated 120 it is the same. Since 120  is less than 360, this shape HAS rotational symmetry
120

23. REFLECTION
REFLECTION

24. A reflection is a transformation that flips a figure across a line.
A REFLECTION IS FLIPPED OVER A LINE.

25. After a shape is reflected, it looks like a mirror image of itself.
REFLECTION
Remember, it is the same, but it is backwards
After a shape is reflected, it looks like a mirror image of itself.
A REFLECTION IS FLIPPED OVER A LINE.

26. The line that a shape is flipped over is called a line of reflection.
Notice, the shapes are exactly the same distance from the line of reflection on both sides.
The line of reflection can be on the shape or it can be outside the shape.
The line that a shape is flipped over is called a line of reflection.
Line of reflection
A REFLECTION IS FLIPPED OVER A LINE.

27. A REFLECTION IS FLIPPED OVER A LINE.
Determine if each set of figures shows a reflection or a translation. A C B C’ B’ A’
A REFLECTION IS FLIPPED OVER A LINE.

28. reflectional symmetry.
Sometimes, a figure has
reflectional symmetry.
This means that it can be folded along a line of reflection within itself so that the two halves of the figure match exactly, point by point.
Basically, if you can fold a shape in half and it matches up exactly, it has reflectional symmetry.

29. REFLECTIONAL SYMMETRY
An easy way to understand reflectional symmetry is to think about folding.
What happens when you unfold the piece of paper?
Do you remember folding a piece of paper, drawing half of a heart, and then cutting it out?

30. REFLECTIONAL SYMMETRY
Line of Symmetry
Reflectional Symmetry
means that a shape can be folded along a line of reflection so the two haves of the figure match exactly, point by point.
The two halves are exactly the same…
They are symmetrical.
The line of reflection in a figure with reflectional symmetry is called a line of symmetry.
The two halves make a whole heart.

31. REFLECTIONAL SYMMETRY
The line created by the fold is the line of symmetry.
How can I fold this shape so that it matches exactly?
A shape can have more than one line of symmetry.
Where is the line of symmetry for this shape?
I CAN THIS WAY
NOT THIS WAY
Line of Symmetry

32. REFLECTIONAL SYMMETRY
How many lines of symmetry does each shape have? 3 4 5
Do you see a pattern?

33. REFLECTIONAL SYMMETRY
Which of these flags have reflectional symmetry?
United States of America
Canada No
England No
Mexico

34. If a figure can be mapped onto itself using a reflection, it has a line of symmetry.
Line p is a line of
symmetry.
Line q is not a line
of symmetry. q

35. Are these lines of symmetry? a a b b a a c b b

36. Draw all lines of symmetry.

37. CONCLUSION FLIP REFLECTION SLIDE TRANSLATION TURN ROTATION
We just discussed three types of transformations.
See if you can match the action with the appropriate transformation.
FLIP
REFLECTION
SLIDE
TRANSLATION
TURN
ROTATION

38. Translation, Rotation, and Reflection all change the position of a shape, while the size remains the same.
The fourth transformation that we are going to discuss is called dilation.

39. Dilation changes the size of the shape without changing the shape.
When you go to the eye doctor, they dilate you eyes. Let’s try it by turning off the lights.
When you enlarge a photograph or use a copy machine to reduce a map, you are making dilations.

40. Enlarge means to make a shape bigger.
DILATION
Enlarge means to make a shape bigger.
Reduce means to make a shape smaller.
The scale factor tells you how much something is enlarged or reduced.

41. DILATION 50% 200% REDUCE ENLARGE
Notice each time the shape transforms the shape stays the same and only the size changes.
50%
200%
REDUCE
ENLARGE

42. Look at the pictures below
DILATION
Look at the pictures below
Dilate the image with a scale factor of 75%
Dilate the image with a scale factor of 150%

43. Look at the pictures below
DILATION
Look at the pictures below
Dilate the image with a scale factor of 100%
Why is a dilation of 75% smaller, a dilation of 150% bigger, and a dilation of 100% the same?

44. Lets try to make sense of all of this
TRANSFORMATIONS
CHANGE THE POSTION OF A SHAPE
CHANGE THE SIZE OF A SHAPE
TRANSLATION
ROTATION
REFLECTION
DILATION
Change in location
Turn around a point
Flip over a line
Change size of a shape

45. See if you can identify the transformation that created the new shapes
TRANSLATION

46. See if you can identify the transformation that created the new shapes
Where is the line of reflection?
REFLECTION

47. See if you can identify the transformation that created the new shapes
DILATION

48. See if you can identify the transformation that created the new shapes
ROTATION

49. See if you can identify the transformation in these pictures?
REFLECTION

50. See if you can identify the transformation in these pictures?
ROTATION

51. See if you can identify the transformation in these pictures?
TRANSLATION

52. See if you can identify the transformation in these pictures?
DILATION

53. See if you can identify the transformation in these pictures?
REFLECTION

54. Reflections

55. Reflections in the Coordinate Plane
Identify the coordinates of
the points.
A ( _____, _____ )
B ( _____, _____ )
C ( _____, _____ )
D ( _____, _____ ) y x A B C D
– –4 –3 –

56. Reflections in the Coordinate Plane
Coordinates of the original points.
A ( _____, _____ )
B ( _____, _____ )
C ( _____, _____ )
D ( _____, _____ )
Reflect each point in the x-axis.
Identify the new coordinates.
A’ ( _____, _____ )
B’ ( _____, _____ )
C’ ( _____, _____ )
D’ ( _____, _____ )
– –4 –3 – y B’ C’ A –2 D’ x D – A’ C – B
What is the rule for reflecting a point in the x-axis?
x-axis: (x, y)  ( _______, _______ )
x –y

57. Reflections in the Coordinate Plane
Coordinates of the original points.
A ( _____, _____ )
B ( _____, _____ )
C ( _____, _____ )
D ( _____, _____ )
Reflect each point in the y-axis.
Identify the new coordinates.
A’ ( _____, _____ )
B’ ( _____, _____ )
C’ ( _____, _____ )
D’ ( _____, _____ )
– –4 –3 – y x A B C D A’ – D’ –4 –3 C’ B’
What is the rule for reflecting a point in the y-axis?
y-axis: (x, y)  ( _______, _______ )
– x y

58. Reflection in the…
What is the rule for reflecting a point in the x-axis?
x-axis: (x, y)  ( _______, _______ )
x –y
What is the rule for reflecting a point in the y-axis?
y-axis: (x, y)  ( _______, _______ )
– x y

59. Reflections in the Coordinate Plane
Reflect ABC over the x-axis.
Also known as y = 0.
Write the new coordinates.
A’ ( _____, _____ )
B’ ( _____, _____ )
C’ ( _____, _____ )
Reflect ABC over the y-axis.
Also known as x = 0. y x A B C –1 –2 –3 A’ B’ C’ A’ B’ C’ – – –
What are the coordinates of ABC?
A (_____, _____ )
B ( _____, _____ )
C (_____, _____ )

60. Translation Find point A and Translate ABC 6 units to the right.
Find point B and
Find point C and
6 Units A A’
count 6 units to the right. Plot point A’. B’ B C C’
count 6 units to the right. Plot point B’.
count 6 units to the right. Plot point C’.

61. Translation Rules To translate a figure to the right,
add the amount to the x-coordinate of each point.
To translate a figure a units to the left, subtract the amount from the x-coordinate of each point.
Translate point P (3, 2) 9 units to the right.
Since we are going to the right, we add 9 to the x-coordinate = 12, so the new coordinates of P’ are (12, 2)
Translate point P (3, 2) 6 units to the left.
Since we are going up, we subtract 6 to the x-coordinate = -3, so the new coordinates of P’ are (-3, 2)

62. Translation Rules
To translate a figure a units up, increase the y-coordinate of each point by a amount.
Translate point P (3, 2) 9 units up.
To translate a figure a units down, decrease the y-coordinate of each point by a amount.
Since we are going up, we add 9 to the y-coordinate = 11, so the new coordinates of P’ are (3, 11)
Translate point P (3, 2) 6 units down.
Since we are going down, we subtract 6 to the y-coordinate = -4, so the new coordinates of P’ are (3, -4)

63. Practice Point P (5, 8). Translate 2 to the left and 6 up. P’ (3, 14)
Point Z (-3, -6). Translate 5 to the right and 9 down.
Translate LMN, whose coordinates are (3, 6), (5, 9), and (7, 12), 9 units left and 14 units up.
P’ (3, 14)
Z’ (2, -15)
L’M’N’ (-6, 20), (-4, 23), (-2, 26)

64. Rotation in a Coordinate Plane
For a Rotation, you need;
An angle or degree of turn
Ex: 90° or a Quarter Turn
Ex: 180 ° or a Half Turn
A direction
Clockwise
Counterclockwise
A Center of Rotation
A point around which Object rotates

65. A Rotation of 90° Counterclockwise about (0,0)y x 1 2 3 4 5 6 7 8
–7 –6 –5 –4 –3 –2 –1 -1 -2 -3 -4 -5 -6
A Rotation of 90° Counterclockwise about (0,0)
(x, y)→(-y, x) x x x
C(3,5) x
B’(-2,4)
C’(-5,3)
B(4,2)
A’(-1,2)
A(2,1) x

66. y x 1 2 3 4 5 6 7 8
–7 –6 –5 –4 –3 –2 –1 -1 -2 -3 -4 -5 -6
A Rotation of 180° about (0,0)
(x, y)→(-x, -y) x x x x
C(3,5) x
B(4,2) x
A(2,1) x x
A’(-2,-1)
B’(-4,-2)
C’(-3,-5)

67. A Rotation of 270° counterclockwise (or 90º clockwise) about (0,0)y x 1 2 3 4 5 6 7 8
–7 –6 –5 –4 –3 –2 –1 -1 -2 -3 -4 -5 -6
A Rotation of 270° counterclockwise (or 90º clockwise) about (0,0)
(x, y)→(-x, -y) x x x x
C(3,5) x
B(4,2)
A(2,1) x x x
A’(1,-2)
C (5,-3)
B’(2,-4)

68. Rotation in a Coordinate Plane

69. Rotations in a Coordinate Plane
Example 4
Rotations in a Coordinate Plane
Sketch the quadrilateral with vertices A(2, –2), B(4, 1), C(5, 1), and D(5, –1). Rotate it 90° counterclockwise about the origin and name the coordinates of the new vertices.
SOLUTION
Plot the points, as shown in blue.
Remember a 90° counterclockwise rotation means (x,y)→(y, -x)
The coordinates of the vertices of the image
are A'(2, 2),
B'(–1, 4),
C'(–1, 5),
and D'(1, 5).

70. Rotations in a Coordinate Plane
Checkpoint
Rotations in a Coordinate Plane
Sketch the triangle with vertices A(0, 0), B(3, 0), and C(3, 4). Rotate ∆ABC 90° counterclockwise about the origin. Name the coordinates of the new vertices A', B', and C'. 4.
A'(0, 0), B'(0, 3), C'(–4, 3)
ANSWER