1. Chapter 7
Transformations

2. Chapter Objectives Identify different types of transformations
Define isometry
Identify reflection
Identify rotations
Identify translations
Describe composition transformations

3. Lesson 7.1
Rigid Motion in a Plane

4. Lesson 7.1 Objectives Identify basic rigid transformations
Define isometry

5. Definition of Transformation
A transformation is any operation that maps, or moves, an object to another location or orientation.

6. Transformation Terms
When performing a transformation, the original figure is called the pre-image.
The new figure is called the image.
Many transformations involve labels
The image is named after the pre-image, by adding a prime symbol (apostrophe)
A A’ A’’
We say it as “A prime”

7. Types of Transformations
Reflection
Rotation
Translation
Characteristics
Orientation
Pictures
Flips object over line of reflection
Turns object using a fixed point as center or rotation
Slides object through a plane
Order in which object is drawn is reversed
Stays same just tilted
Stays same and stays upright

8. Definition of Isometry
An isometry is a transformation that preserves length.
Isometry also preserve angle measures, parallel lines, and distances between points.
If you look at the meaning of the two parts of the word, iso- means same, and metry- means meter or measure.
So simply stated, isometry preserves size.
Any transformation that is an isometry is called a Rigid Transformation.

9. Homework 7.1 1-33, 36-39 In Class – 9, 13, 27, 33 Due Tomorrowp
In Class – 9, 13, 27, 33
Due Tomorrow

10. Lesson 7.2
Reflections

11. Lesson 7.2 Objectives Utilize reflections in a plane
Define line symmetry
Derive formulas for specific reflections in the plane

12. Reflections
A transformation that uses a line like a mirror is called a reflection.
The line that acts like a mirror is called the line of reflection.
When you talk of a reflection, you must include your line of reflection
A reflection in a line m is a transformation that maps every point P in the plane to a point P’, so that the following is true
If P is not on line m, then m is the perpendicular bisector of PP’.
If P is on line m, then P=P’.

13. Theorem 7.1: Reflection Theorem
A reflection is an isometry.
That means a reflection does not change the shape or size of an object! m

14. Line of Symmetry
A figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in a line.
What that means is a line can be drawn through an object so that each side reflects onto itself.
There can be more than one line of symmetry, in fact a circle has infinitely many around.

15. Homework 7.2a
1-11, 22-29 p
In Class – 7, 23
Due Tomorrow

16. Reflection Formula There is a formula to all reflections.
It depends on which type of a line are you reflecting in.
vertical
horizontal
y = x
Vertical:
y-axis
x = a
Horizontal:
x-axis
y = a
y = x
( x , y)
( -x + 2a , y)
( x , -y + 2a)
( y , x)

17. Homework 7.2b
12-14, 18-21, 50-51 p
In Class – 19
Due Tomorrow

18. Lesson 7.3
Rotations

19. Lesson 7.3 Objectives Utilize a rotation in a plane
Define rotational symmetry
Observe any patterns for rotations about the origin

20. Definitions of Rotations
A rotation is a transformation in which a figure is turned about a fixed point.
The fixed point is called the center of rotation.
The amount that the object is turned is the angle of rotation.
A clockwise rotation will have a negative measurement.
A counterclockwise rotation will have a positive measurement.
clockwise or
negative (-) Q

21. Theorem 7.2: Rotation Theorem
A rotation is an isometry. A B A’ B’ P

22. Rotational Symmetry
A figure in a plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180° or less.
A square has rotational symmetry because it maps onto itself with a 90° rotation, which is less than 180°.
A rectangle has rotational symmetry because it maps onto itself with a 180° rotation.

23. Homework 7.3a
1-19
p416
In Class – 6, 11, 13
Due Tomorrow

24. Rotating About the Origin
Rotating about the origin in 90o turns is like reflecting in the line y = x and in an axis at the same time!
So that means to switch the positions of x and y.
(x,y)  (y,x)
Then the original x-value changes sign, no matter where it is flipped to.
So overall the transformation can be described by
(x,y)  (-y,x)
Every time you 90o you repeat the process.
So going 180o means you do the process twice!

25. Theorem 7.3: Angle of Rotation Theorem
The angle of rotation is twice as big as the angle of intersection.
But the intersection must be the center of rotation.
And the angle of intersection must be acute or right only. m k A B 2x x A’ B’ P

26. Homework 7.3b 25-35, 45-50, 54 In Class – 25, 35 Due Tomorrowp
In Class – 25, 35
Due Tomorrow
Quiz Wednesday
Lessons

27. Translations and Vectors
Lesson 7.4
Translations
and
Vectors

28. Lesson 7.4 Define a translation Identify a translation in a plane
Use vectors to describe a translation
Identify vector notation

29. Translation Definition
A translation is a transformation that maps an object by shifting or sliding the object and all of its parts in a straight light.
A translation must also move the entire object the same distance.

30. Theorem 7.4: Translation Theorem
A translation is an isometry.

31. Theorem 7.5: Distance of Translation Theorem
The distance of the translation is twice the distance between the reflecting lines. x k m P Q P’ Q’
P’’
Q’’ 2x

32. Coordinate form
Every translation has a horizontal movement and a vertical movement.
A translation can be described in coordinate notation.
(x,y)  (x+a , y+b)
Which tells you to move a units horizontal and b units vertical. Q
b units up P
a units to the right

33. Vectors Another way to describe a translation is to use a vector.
A vector is a quantity that shows both direction and magnitude, or size.
It is represented by an arrow pointing from pre-image to image.
The starting point at the pre-image is called the initial point.
The ending point at the image is called the terminal point.

34. Component Form of Vectors
Component form of a vector is a way of combining the individual movements of a vector into a more simple form.
<x , y>
Naming a vector is the same as naming a ray. PQ Q
y units up P
x units to the right

35. Use of Vectors Adding/subtracting vectors
Add/subtract x values and then add y values
<2 , 6> + <3 , -4>
<5 , 2>
Distributive property of vectors
Multiply each component by the constant
5<3 , -4>
<15 , -20>
Length of vector
Pythagorean Theorem
x2 + y2 = lenght2
Direction of vector
Inverse tangent
tan-1 (y/x)

36. Homework 7.4 1-30, 44-47 In Class – 3,7,17,25,45 Due Tomorrowp
In Class – 3,7,17,25,45
Due Tomorrow
Quiz Tomorrow
Lessons

37. Glide Reflections and Compositions
Lesson 7.5
Glide Reflections
and
Compositions

38. Lesson 7.5 Objectives Identify a glide reflection in a plane
Represent transformations as compositions of simpler transformations

39. Glide Reflection Definition
A glide reflection is a transformation in which a reflection and a translation are performed one after another.
The translation must be parallel to the line of reflection.
As long as this is true, then the order in which the transformation is performed does not matter!

40. Compositions of Transformations
When two or more transformations are combined to produce a single transformation, the result is called a composition.
So a glide reflection is a composition.
The order of compositions is important!
A rotation 90o CCW followed by a reflection in the y-axis yields a different result when performed in a different order.

41. Theorem 7.6: Composition Theorem
The composition of two (or more) isometries is an isometry.

42. Homework 7.5 1-8, 9-21, 23-24, 26-30 In Class – 9,13,19 Due Tomorrow
skip 16, 28 p
In Class – 9,13,19
Due Tomorrow

43. Lesson 7.6
Frieze Patterns

44. Lesson 7.6 Objectives Identify a frieze pattern by type
Visualize the different compositions of transformations

45. Frieze Patterns
A frieze pattern is a pattern that extends to the left or right in such a way that the pattern can be mapped onto itself by a horizontal translation.
Also called a border pattern.

46. Classifying Frieze Patterns
The horizontal translation is the minimum that must exist.
However, there are other transformations that can occur.
And they can occur more than once.
Type
Abbreviation
Description
Translation T
Horizontal translation left or right
180o Rotation R
180o Rotation CW or CCW
Reflection in
Horizontal Line
Reflection either up or down
in a horizontal line H
Reflection in
Vertical Line
Reflection either left or right
in a vertical line V
Horizontal
Glide Reflection
Horizontal translation with
reflection in a horizontal line G

47. Examples TR TG TV
THG
TRVG
TRHVG

48. Homework 7.6 2-23 In Class – 9,13,17,21 Due Tomorrow Quiz Tuesdayp
In Class – 9,13,17,21
Due Tomorrow
Quiz Tuesday
Lessons –