Jurgensen Unit 2 Mind Map

Jurgensen Unit 2 Mind Map

1. Mapping and Functions

   • Preimage
     • A line, point, etc. before a transformation
   • Image
     • A point, line, etc after a transformation
   • Mapping
     • A correspondence between a set of points
   • One-to-one Mapping
     • A mapping (or function) from set A to set B
   • Theorem 14-1
     • An isometry maps a triangle to a congruent triangle
       • An isometry maps an angle to a congruent angle
       • An isometry maps a polygon to a polygon with the same area

2. Translations and Glide Reflections

   • Translation
     • A transformation that glides all point of the plane in the same direction
     • The transformation applies the same distance with all points on the plane
       • Ex. Point A has the same distance with A' as B and B'
     • Theorem 14-3
       • A translation is an isometry
   • A glide reflection is a combination of a translation and a reflection
     • If the glide (translation) is parallel to the reflection line, the order of the transformations do not matter

3. Reflections

   • Reflections map any point (P) to its image (P')
     • Such that the reflection line (m) is the perpendicular bisector of line PP' if point P is not on line m
     • If point P is on line m, then P = P'
   • Theorem 14-2
     • A reflection in a line is an isometry

4. Dilations

   • Notations
     • A dilation around the origin, by a factor of 2 is notated as Do, 2
     • A negative factor results in the image being flipped over
   • A dilation is an expansion or contraction of a figure on the plane
     • If a figure is dilated by a value <1, it is contracted, and vice versa
   • Theorem 14-5
     • A dilation maps any triangle to a similar triangle
       • A dilation maps an angle to a congruent angle
       • A dilation Do,k maps any segment to a parallel segment [k] times long
       • A dilation Do,k maps any polygon to a similar polygon whose area is k^2 times as large

5. Rotations

   • Notations
     • For the rotation of 90° around the origin, it would be Ro, 90
     • A rotation of 180° around the origin can be notated into Ho
   • A rotation is positive if it is a counterclockwise rotation about the point of rotation
   • Theorem 14-4
     • A rotation is an isometry
   • Rotations are not isometries, unless the factors are 1 or -1

6. Composites of Mappings

   • The order of the composites of two mappings affects the final image
     • If the composite is (S * T), you apply T first
   • Theorem 14-6
     • The composite of two isometry is an isometry
   • Theorem 14-7
     • A composite of reflections in to parallel lines is a translation. The translation glides all points through twice the distance from the first line of reflection to the second
   • Theorem 14-8
     • A composite of reflections in two intersecting lines is a rotation about the point of intersection of the two lines. The measure of the angle of rotation is twice the measure of the angle from the first line of reflection to the second

7. Symmetry

   • Two types: Point and Line symmetry
     • Point Symmetry is symmetry of a figure if it was rotated around a point in its center
     • Line symmetry is symmetry of a figure on a line (can have multiple lines