Translation: slide

Reflection: flip (mirror)

Rotation: turn

Dialation: enlarge or reduce

Geometric Transformations:

Pre-Image: original figure

Image: after transformation. Use prime notation

Notation:

A

A’

B

B’

C

C ’

Isometry

AKA: congruence transformation

a transformation in which an original figure and its image are congruent.

Theorems about isometries

FUNDAMENTAL THEOREM OF ISOMETRIESAny two congruent figures in a plane can be

mapped onto one another by at most 3 reflections

ISOMETRY CLASSIFICATION THEOREMThere are only 4 isometries. They are:

TRANSLATION:

moves all points in a plane

a given direction

a fixed distance

TRANSLATION VECTOR:(also known as translation rule)

DirectionMagnitude

PRE-IMAGE

IMAGE

Translate by the vector (rule) <x, y>

x moves horizontaly moves vertical

Translate by <3, 4>

Different notationT(x, y) -> (x+3, y+4)

Translations PRESERVE:

SizeShape

Orientation

Reflectionover a line (mirror)

line l is a line of reflection

C'

D'

E'

A'

B'

B

A

E

D

C

Properties of reflections

PRESERVE• Size (area, length, perimeter…)• Shape

CHANGE orientation (flipped)

Reflect x-axis: (a, b) -> (a,-b)Change sign y-coordinate

Reflect y-axis: (a, b) -> (-a, b)Change sign on x coordinate

6

4

2

-2

-4

-10 -5 5 10

A: (2, 5)

6

4

2

-2

-4

-10 -5 5 10

A': (2, -5)

A: (2, 5)

X-axis reflection

Y-axis reflection

6

4

2

-2

-4

-10 -5 5 10

A': (-2, 5) A: (2, 5)

Rotations have:

Center of rotation

Angle of rotation:

CENTER of rotation

Rotated 90 degrees counterclockwise

C

A

B

F

C'

A'

B'

C

A

B

F

mC'FC = 90.00

C'

A'

B'

C

A

B

F

ROTATIONS PRESERVE

SIZE– Length of sides– Measure of angles– Area– Perimeter

SHAPE

ORIENTATION

Rotations on a coordinate plane about

the origin90 (a, b) -> (-b, a)

180 (a, b) -> (-a, -b)

270 (a, b) -> (b, -a)

360 (a, b) -> (a, b)

Coordinate Geometry rules

Reflectionsx axis (a, b) -> (a, -b)y axis (a, b) -> (-a, b)y=x (a, b) -> (b, a)

Rotations about the origin

90 (a, b) -> (-b, a)180 (a, b) -> (-a, -b)270 (a, b) -> (b, -a)360 (a, b) -> (a, b)

GLIDE REFLECTIONS

You can combine different Geometric Transformations…

Practice: Reflect over y = x then translate by the vector <2, -3>

After Reflection…

After Reflection and translation…

SymmetryLine Symmetry

If a figure can be reflected onto itself over a line.

Rotational SymmetryIf a figure can be rotated about some point onto itself through a rotation between 0 and 360 degrees

What kinds of symmetry do each of the following have?

What kinds of symmetry do each of the following have?

Rotational (180) Point Symmetry

Rotational (90, 180, 270)Point Symmetry

Rotational (60, 120, 180, 240, 300)Point Symmetry

DilationsA dilation is a transformation (notation   ) that produces an image that is the same shape as the original, but is a different size. A dilation stretches or shrinks the original figure.   

The description of a dilation includes the scale factor (orratio) and the center of the dilation.   The center of dilation is a fixed point in the plane about which all points are expanded or contracted.  

Dilations & Scale Factor

A dilation of scale factor k whose center of dilation is the origin may be written:  Dk (x, y) = (kx, ky).

If the scale factor, k, is greater than 1, the image is an enlargement (a stretch).

If the scale factor is between 0 and 1, the image is a reduction (a shrink).

(It is possible, but not usual, that the scale factor is 1, thus creating congruent figures.) 

Dilations PreserveProperties Preserved (invariant) under a dilation:

1.  angle measures (remain the same)2.  parallelism (parallel lines remain parallel)3.  colinearity (points stay on the same lines)4.  midpoint (midpoints remain the same in each figure)5.  orientation (lettering order remains the same)---------------------------------------------------------------6.  distance is NOT preserved (NOT an isometry)     (lengths of segments are NOT the same in all cases

      except a scale factor of 1)

Dilations Create Similar Figures