Geometry Legal Reasons
Geometry Legal Reasons
Level 4
Reflections, Translations, Rotations, Vectors, Isometries, Congruent Figures
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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 .
Reflection points are always the same perpendicular distance across the mirror line from each other.

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:

  1. each point in the preimage set has a unique image, and
  2. 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.

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


and we say, "The reflection image of A is A'.

Now if we want to emphasize which line we are reflecting over we write


and we say "The reflection image of A over line m is A'

Reflection Postulate

Under a reflection:

  1. There is a 1-1 (one-to-one) correspondence between points and their images.
  2. Collinearity is preserved, If three points A,B,and C lie on a line, then their images A', B', and C' are collinear.
  3. Betweenness is preserved. If B is between A and C, then the image B' is between the images A' and C'.

  4. Orientation is REVERSED. A polygon and its image, with vertices taken in corresponding order, have opposite orientations.
Notice all points are reversed in order on the opposite or reflected side.
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.
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.
angle i is = angle r
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.

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:

Do the right hand reflection first with this notation.

and is read, " The reflection over line m following the reflection over line l of point A."

Definition of Translation or Slide A translation, or slide, is the composite of two reflections over parallel lines.
Two-Reflection Theorem for Translations
Definition of Rotation A rotation is the composite of two reflections over intersecting lines.
Two-Reflections Theorem for Rotations
Definition of Vector A vector is a quantity that can be characterized by its direction and magnitude.
vectors have 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.
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".

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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