1. Transformations involving a reflection or composition of reflections
Isometries
Transformations involving a reflection or composition of reflections

2. Reflections
A line is a line of reflection iff it is the perpendicular bisector of the segment with endpoints of the image and pre-image.
The are two parts to this definition:
P: It is a line of reflection
Q: It is the perpendicular bisector of the segment with endpoints of the image and pre-image.
PQ

3. If you are told that A’B’C’ is the reflection image of ABC over line l , then you know that: l is the perpendicular bisector of A B C A’ B’ C’ l

4. If you are told that l is the perpendicular bisector of , & , then A’B’C’ is the reflection image of ABC over line l A B C l C’ B’ A’

5. The Reflection Postulate
A common non-mathematical way for people to define a reflection is: “It is the same image, except it is backwards.”
This idea that everything is the same, except backwards, describes the properties of the Reflection Postulate.

6. The Reflection Postulate Reflections preserve:
Angle Measure
Betweeness
Collinearity
Distance
These are the mathematical properties that people refer to when they say “everything” stays the same.

7. The reflection postulate also states that orientation is reversed
The reflection postulate also states that orientation is reversed. ABC has clockwise orientation A’B’C’ has counter-clockwise orientation A B C A’ B’ C’ l

8. A little about notation…
You must be able to read and write reflections using function notation.
The image
The pre-image
The subscript tells which line is the line of reflection
The lower case r represents reflection

9. Compositions of Reflections
A composition is an image resulting from two or more transformations.
Compositions can be written in a couple of ways
Both of these mean the same thing:
The reflection over l of the reflection over m of P is P’’

10. Order of compositions
It is important to remember that the first transformation to be performed is the one nearest the pre-image.
Reflect over m first

11. Types of Isometries
Reflection
Translation
Rotation
Glide Reflection

12. Translations
Translations are the result of a composition over two parallel lines or
Translations are the result of slide by a vector

13. Translations
Translations result from an even number (2, 4, 6, 8…) of reflections
Orientation is preserved
The magnitude of the translation is twice the distance between the two parallel lines.

14. Vector Translations Translations can also be expressed with a vector.
A vector consists of a
direction (the way that the arrow points)
&
magnitude (the length of the arrow).
Vectors allow us to express translations without using reflections.

15. Rotations
Rotations are the result of a composition over intersecting lines.
Earlier (chapter 2) we rotated shapes given a:
direction (clockwise or counter-clockwise) &
magnitude (angle of rotation).
Hmmm…it sounds like a vector…

16. Rotations
Rotations result from an even number (2, 4, 6, 8…) of reflections
Orientation is preserved
The magnitude of the rotation is twice the angle between the two intersecting lines.

17. Glide Reflections
Glide Reflections are a composition of a translation and a reflection over a line parallel to the direction of translation.

18. Glide Reflections
Glide reflections result from an odd number (3, 5, 7, 9…) of reflections.
Orientation is not preserved

19. Figures are congruent iff one maps onto the other under an isometry.
Congruency
Figures are congruent iff one maps onto the other under an isometry.

20. Reflections   Notice what the definition of congruency doesn’t say.
It doesn’t say congruent shapes have the same angle measures.
It doesn’t say congruent shapes have the same lengths.
These are true because of the Reflection Postulate

21. What do I expect you to do?
Draw figures by applying the definition of reflection image. (4-1, 4-2)
Draw reflection images of segments, angles, and polygons over a given line. (4-2)
Draw translation and glide-reflection images of figures (4-6, 4-7)
Draw or identify images of figures under composites of two reflections. (4-4, 4-5)
Apply properties of reflections to make conclusions, using one or more of the following justifications: Definition of reflection; Reflection Postulate (ABCD); Figure Reflection Theorem. (4-1, 4-2)
Apply properties of reflections to obtain properties of other isometries. (4-4, 4-5, 4-7)
Apply the Two-Reflection Theorems for Translations and for Rotations. (4-4, 4-5)
Determine the isometry which maps one figure onto another. (4-7)
Use congruence in real situations. (4-8)
Find coordinates of reflection and translation images of points over the coordinate axes. (4-1, 4-2, 4-6)