1. Transformations
Chapter 4

2. Translations
I can translate a figure in a coordinate plane.

3. Translations Vocabulary (page 94 in Student Journal)
vector: a quantity that has both direction and magnitude (size), and is represented by an arrow drawn from 1 point to another
initial point: starting point of the vector
terminal point:

4. Translations terminal point: ending point of the vector
vertical component: up and down
horizontal component: left to right
component form: combines the horizontal and vertical component

5. Translations
transformation: a change in the position, shape, or size of a geometric figure
image: the resulting figure after the transformation
preimage: the original figure

6. Translations
translation: sliding all points of a figure the same distance and direction
rigid motion (isometry): a transformation that preserves distance and angle measures
composition of transformations: a combination of 2 or more transformations

7. Translations Examples (space on page 95 in Student Journal)
a) Name the vector in the diagram and name its component.

8. Translations
Solutions
a) Vector PQ, <-4, 5>

9. Translations
b) Translate triangle ABC using the vector <-1, -2> if the vertices of the triangle are A(0, 3), B(2, 4) and C(1, 0).

10. Translations
Solution
b) A’(-1, 1), B’(1, 2), C’(0, -2)

11. Translations
c) Write a rule for the translation of triangle ABC to the triangle A’B’C’ based on the graph.

12. Translations
Solution
c) (x, y)  (x + 2, y – 1)

13. Translations
d) Graph segment RS with endpoint R(-8, 5) and S(-6, 8). Graph its image after the composition of translations below.
(x, y)  (x – 1, y + 4)
(x, y)  (x + 4, y – 6)

14. Translations
Solution d)

15. Reflections
I can reflect a figure in a coordinate plane.

16. Reflections Vocabulary (page 99 in Student Journal)
reflection: a transformation that uses a line line a mirror to reflect a figure
line of reflection: the mirror type line

17. Reflections
glide reflection: a transformation involving a translation followed by a reflection
line symmetry: a plane figure that can be mapped onto itself by a reflection in a line
line of symmetry: the line of reflection

18. Reflections Core Concepts (page 100 in Student Journal)
Coordinate Rules for Reflections
Preimage
Line of reflection
Image
(a, b)
x-axis
(a, -b)
y-axis
(-a, b)
y = x
(b, a)
y = -x
(-b, -a)

19. Reflections Reflection Postulate (Postulate 4.2)
A reflection is a rigid motion.

20. Reflections Examples (space on pages 99 and 100 in Student Journal)
Graph triangle ABC with vertices A(1, 3), B(5, 2), C(2, 1) and its image after the described reflection.
in the line x = -1
in the line y = 3

21. Reflections
Solutions
a) b)

22. Reflections
Graph segment AB with endpoints A(3, -1) and B(3, 2) and its image after the described reflection.
c) in the line y = x
d) in the line y = -x

23. Reflections
Solutions
c) d)

24. Reflections
e) Graph triangle ABC with vertices A(3, 2), B(6, 3) and C(7, 1) and its image after the glide reflection below.
(x, y)  (x, y – 6), reflection in the y-axis

25. Reflections
Solution e)

26. Reflections
f) How many lines of symmetry does the triangle in the diagram have?

27. Reflections
Solution
f) 3

28. Rotations
I can rotate a figure in a coordinate plane.

29. Rotations Vocabulary (page 104 in Student Journal)
rotation: a transformation in which a figure is turned about a fixed point
center of rotation: the fixed point the figure is rotated about

30. Rotations
angle of rotation: the angle formed by rays drawn from the center of rotation to a point and its image
rotational symmetry: a plane figure that can be mapped onto itself by a rotation of 180 degrees or less about its center
center of symmetry:

31. Rotations
center of symmetry: the center of the figure

32. Rotations Core Concepts (page 105 in Student Journal)
Coordinate Rules for Rotations about the Origin
Preimage
Angle of Rotation
Image
(a, b)
90 degrees counterclockwise
(-b, a)
180 degrees counterclockwise
(-a, -b)
270 degrees counterclockwise
(b, -a)

33. Rotations Rotation Postulate (Postulate 4.3)
A rotation is a rigid motion.

34. Rotations Examples (space on pages 104 and 105 in Student Journal)
a) Draw a 60 degree rotation for triangle ABC about point P.

35. Rotations
Solution a)

36. Rotations
b) Graph triangle ABC with vertices A(3, 1), B(3, 4) and C(1, 1) and its image after a 180 degree rotation about the origin.

37. Rotations
Solution b)

38. Rotations
c) Graph segment RS with endpoints R(1, -3) and S(2, -6) and its image after the composition below.
180 degree rotation about the origin and a reflection in the y-axis

39. Rotations
Solution c)

40. Rotations
Does the figure have rotational symmetry? If so, describe the rotation that maps the figure onto itself.
d) e) f)

41. Rotations
Solutions
d) yes, 120 degrees
e) no
f) yes, 180 degrees

42. Congruence and Transformations
I can identify congruent figures and describe congruence transformations.

43. Congruence and Transformations
Vocabulary (page 109 in Student Journal)
congruent figures: figures that have the same shape and size
congruence transformation: another name for a rigid motion because the preimage and the image are congruent

44. Congruence and Transformations
Core Concepts (page 109 in Student Journal)
Reflections in Parallel Lines Theorem (Theorem 4.2)
If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is the same as a translation.

45. Congruence and Transformations
Reflections in Intersecting Lines (Theorem 4.3)
If lines k and m intersect at point P, then a reflection in line k followed by a reflection in line m is the same as a rotation about point P.

46. Congruence and Transformations
Example (space on page 109 in Student Journal)
a) Identify any congruent figures in the diagram.

47. Congruence and Transformations
Solution
a) Triangle ABC is congruent to triangle PQR

48. Congruence and Transformations
b) Describe a congruence transformation that maps ABCD to PQRS.

49. Congruence and Transformations
Solution
b) 90 degree rotation about the origin

50. Congruence and Transformations
In the diagram, PJ = 3 and LP’’ = 8.
c) Name any segments congruent to each other.
d) Find PP’’.

51. Congruence and Transformations
Solutions
c) segments PQ, P’Q’ and P’’Q’’, segments PJ and P’J, segments P’L and P’’L, segments QK and Q’K, segments Q’M and Q’’M, segments JK and LM, segments JL and KM
d) 22 units

52. Dilations
I can identify and perform dilations.

53. Dilations Vocabulary (page 114 in Student Journal)
dilation: a transformation in which a figure is enlarged or reduced with respect to a fixed point
center of dilation: the fixed point for an enlargement or reduction

54. Dilations
scale factor: the ratio of the lengths of the corresponding sides of the image and the preimage
enlargement: occurs when the scale factor is greater than 1
reduction: occurs when the scale factor is less than 1

55. Dilations
reduction: occurs when the scale factor is less than 1

56. Dilations Core Concepts (page 115 in Student Journal)
Coordinate Rules for Dilations
If P(x, y) is the preimage of a point, then its image after a dilation centered at the origin (0,0) with scale factor k is the point P’(kx, ky).

57. Dilations Examples (space on pages 114 and 115 in Student Journal)
Find the scale factor. Then determine is the dilation is an enlargement or reduction.
a) b)

58. Dilations
Solutions
1/3, reduction
5/2, enlargement

59. Dilations
c) Graph triangle PQR with vertices P(0, 2), Q(1, 0) and R(2, 2) and its image after a dilation with a scale factor of 3.

60. Dilations
Solution c)

61. Dilations
d) Graph triangle PQR with vertices P(4, 6), Q(-4, 2) and R(2, -6) and its image after a dilation with a scale factor of 0.5.

62. Dilations
Solution d)

63. Similarity and Transformations
I can perform and describe similarity transformations and prove figures are similar.

64. Similarity and Transformations
Vocabulary (page 119 in Student Journal)
similarity transformation: a dilation or a composition of rigid motions and dilations
similar figures: have the same shape, but different sizes

65. Similarity and Transformations
Example (space on page 119 in Student Journal)
a) Graph segment AB with endpoints A(12, -6) and B(0, -3) and its image after the similarity transformation below.
reflection in the y-axis and (x, y)  (1/3x, 1/3y)

66. Similarity and Transformations
Solution a)

67. Similarity and Transformations
b) Describe a similarity transformation that maps WXYZ to PQRS.

68. Similarity and Transformations
Solution
reflection in the x-axis and (x, y)  (2x, 2y)