7.1:Transformations And Symmetry 7.2: Properties of Isometries

Transformation: Moving all the points of a geometric figure according to certain rules to create an image of the original figure.

Pre-Image:original figure Image:after transformation. Use prime notation

Geometric Transformations:

Isometry or Rigid Transformation: The image is congruent to the original figure. It preserves the size and shape. In other words, it preserves distance and angle measures.

Nonrigid Transformation: The Image is not congruent to the original figure. It does not preserve the size and shape. For example, if an image is reduced or enlarge (Dilation), or if the shape changes.

TRANSLATION: A rigid motion that slides each point of a figure the same distance an direction, called the translation vector.

Translations PRESERVE: Size (Distance: Corresponding segments are ≅) Shape (Angles: Corresponding angles are ≅) Orientation (Direction, position)

TRANSLATION VECTOR: Direction Distance

PRE-IMAGE

IMAGE

Example 1

(𝒙 , 𝒚) → (𝒙 + 𝒉 , 𝒚)

(𝒙 , 𝒚) → (𝒙 , 𝒚 + 𝒌)

Vertical and horizontal translation (𝒙 , 𝒚) → (𝒙 + 𝒉 , 𝒚 + 𝒌)

Rotation

Rotation: Is a rigid motion or isometry that turns a figure about a fixed point, called the point of rotation. The figure is rotated for a given angle, called the angle of rotation. The figure is rotated in a given direction: clockwise or counterclockwise.

Point of rotation

clockwise counterclockwise

1 2

3 4

R O T A T I N G W I T H

P A T T Y P A P E R

Rules on a coordinate plane counterclockwise rotations

about the origin

90 (x, y) -> (-y, x)

180 (x, y) -> (-x, -y)

270 (x, y) -> (y, -x)

360 (x, y) -> (x, y)

90 counterclockwise rotation about the origin (x, y) → (-y, x)

A

C B

A’

B’

C’

180 counterclockwise rotation about the origin (x, y) → (-x, -y)

A

B

C

C’

A’

B’

270 counterclockwise rotation about the origin (x, y) → (y, -x)

Original (5,7)

Image (7, -5)

Rotations PRESERVE: Size (Distance: Corresponding segments are ≅) Shape (Angles: Corresponding angles are ≅) Orientation (Direction, position)

Reflection

Reflections: Figures that are mirror images of each other. Reflection: Is a rigid motion that reflects, or “flips,” a figure over a given line called a line of reflection . A line of reflection is a line over which a figure is reflected so that corresponding points are the same distance from the line.

The Basic Property of Reflection

The line of reflection is the perpendicular bisector of every segment joining a point in the original figure with its image.

Properties of

reflections

Reflections PRESERVE : Size (Distance: Corresponding segments are ≅) Shape (Angles: Corresponding angles are ≅) Reflections CHANGE : Orientation (Direction, position) (MIRROR)

Reflection across the x-axis: (x,y) → (x,-y) Change sign of y-coordinate

Reflection across the y-axis: (x,y) → (-x,y) Change sign of x coordinate

Reflection across line y = x (x,y) → (y,x)

C (3, -1) → C’ ( -1, 3)

Reflection across line y = -x (x,y) → ( -y,-x)

A (-3,0) → A’ ( 0,3)

A

A’

Reflectional Symmetry: If a figure can be reflected over a line in such a way that the resulting image coincide with the original. The Reflection line is called the Line of Symmetry

More about symmetry

Rotational Symmetry: If a figure can be rotated about a point in such a way that its rotated image coincides with the original figure before turning a full 360 degrees

Point Symmetry: Is a two-fold rotational symmetry

To know how many folds of rotational symmetry a figure has, count the # of times the copy and the original coincide until the copy is back in its original position.

Say whether the transformations are rigid or non-rigid, Explain how do you know. If rigid, name and describe as much as possible.

10. A regular polygon of n sides has ___reflectional symmetries and ___rotational symmetries. 11. Which letters of the English alphabet have rotational symmetry?

Identify the type of transformation. State the rule. The pink figure is the image.

Homework: Due next class Workbook 7.1: 1-8 7.2: 1-3,6-8

Gizmos: Due Friday 10/2, odd periods Monday 10/5, even periods