
1

Definition
of Reflection Image of point A over reflecting line m 
For a point A, not on
a reflecting line m, the reflection image of A over line m is the
point B if and only if m is the perpendicular bisector of .
Now if point A is on
the reflecting line m, then its reflection is itself.
*Reflection points are always the
same perpendicular distance across the mirror line from each other.

1b 
Definition
of Transformation 
A transformation is a
correspondence between two sets of points such that:
 each point in the
preimage set has a unique image, and
 each point in the
image set has exactly one preimage
*Transformations are
often called mappings. We say the preimage point is mapped onto
the image point.

2

Reflection
Notation 
When working with a reflection
we use a lowercase "r" to refer to it. When discussing
reflections in GENERAL, or when the reflecting or mirror line is
obvious, we write
r(A)=A'
and we say, "The reflection image of A is A'.
Now if we want to emphasize
which line we are reflecting over we write
rm(A)=A'
and we say "The
reflection image of A over line m is A'

3

Reflection
Postulate 
Under a reflection:
 There is a 11 (onetoone)
correspondence between points and their images.
 Collinearity is preserved,
If three points A,B,and C lie on a line, then their images A',
B', and C' are collinear.
 Betweenness is preserved.
If B is between A and C, then the image B' is between the images
A' and C'.
 Orientation is REVERSED.
A polygon and its image, with vertices taken in corresponding
order, have opposite orientations.

4

Figure
Reflection Theorem 
If
a figure is determined by certain points, then its reflection image
is the corresponding figure determined by the reflection images of
those points. 
5

Bouncing
off Surfaces (law of physics) 
When a ball is rolled
without spin against a wall, it bounces off the wall and
travels in a ray that is the REFLECTION IMAGE of the path of the
ball if it had gone straight THROUGH the wall.
Angles "i" and "r" above are always equal in
measure and are referred to as the angle
of incidence (i) and the angle of reflection (r).
Knowing this fact can
help your pool game as well as your miniature golf game.

6

Definition
of Composite Reflections and its notation 
When you reflect more
than one time over more than one line, you get a COMPOSITE
of reflections.
The notation looks like
this:
and is read, " The
reflection over line m following the reflection over line
l of point A."

7

Definition
of Translation or Slide 
A
translation, or slide, is the composite of two reflections
over parallel lines. 
8

TwoReflection
Theorem for Translations 

9

Definition
of Rotation 
A
rotation is the composite of two reflections over intersecting
lines. 
10

TwoReflections
Theorem for Rotations 

11

Definition
of Vector 
A
vector is a quantity that can be characterized by its direction and
magnitude.
Vectors are extremely important in many fields today. Airplanes travel
on vectors that characterize their speed and direction. Forces such
as gravity, and pressure are also vectors because they have a magnitude
and direction. 
12

Definition
of Isometry 
A reflection or a composite
of reflections is called an isometry. So that means that
reflections, translations or slides, and rotations are all isometries.
Also a composite of reflecting and sliding is an isometry known
as a "glide reflection". These are the only isometries
that exist.
The word isometry comes
from the Greek isos meaning "equal" and metron
meaning "measure".

13

Definition
of Congruent Figures 
So reflecting, rotating,
sliding, and glide reflecting produces congruent
figures.
Or put
another way,
two or more plane figures which have the same size and shape are
congruent.

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