Unit 10: Butterflies, Pinwheels and WallPaper Name: ______

Transformation: A geometric operation that relates each point of a figure to an image point. Thetransformations you studied in this Unit—reflections, rotations, and translations—are symmetrytransformations. A symmetrytransformation produces an image that is identical in size and shape to the original figure.

Congruent: Two figures arecongruent when they have the same size and shape. In congruentfigures, all corresponding sides are the same length and all corresponding angles have the same measure.

1.1: Line Reflections

Reflectional Symmetry: A figure or design hasreflectional symmetryif you can draw a line that divides the figure into halves that are mirror images. The design below hasreflectional symmetryabout a vertical line through its center.Reflectional symmetryis sometimes referred to asmirror symmetryorline symmetry.

Line of Symmetry: A line that divides a graph or drawing into two halves that are mirror images of each other.

Line Reflection: A transformation that maps each point of a figure to its mirror image, where a line acts as the mirror. PolygonA’B′C′D′E′below is the image of polygonABCDEunder a reflection in the line. If you drew a line segment from a point to its image, the segment would be perpendicular to, and bisected by, the line of reflection.

1.2: Rotations

Rotational Symmetry: A figure or design hasrotationalsymmetry if it can be rotated less than a full turn about a point to a position in which it looks the same as the original. The design below hasrotationalsymmetry with its center as the center of rotation and a 60° angle of rotation. This means that it can be rotated 60°, or any multiple of 60°, about its center point to produce an image that matches exactly with the original.

Center of Rotation: A fixed point about which a figure rotates.

Angle of Rotation: The number of degrees that a figure rotates.

Example: Find the angle of rotation

1.3: Translations

Translational Symmetry: A design hastranslational symmetryif you can slide it to a position in which it looks exactly the same as it did in its original position. To describetranslational symmetry, you need to specify the distance and direction of the translation. The picture is part of a design that extends infinitely in all directions. This design hastranslational symmetry.

1.4: Properties or Transformations

  • What is true about the angles in the transformations above?
  • What is true about the side lengths in the transformations above?

Therefore, Translations, Rotations and Reflections always create ______figures.

3.1: Coordinate Rules for Reflections

Rule for reflection over x-axis: (x, y) (x, –y)

Rule for reflection over y-axis: (x, y) (–x, y)

Ex:

1)What type of reflection is shown?

2)What is the rule for this reflection?

3)Reflect triangle ADC over the x-axis. Label the coordinates A' D' C'.

A= A'=

D= D'=

C= C'=

4)What is the rule for this reflection?

3.3: Coordinate Rules for Rotations

Rule for 90 rotation counter-clockwise: (x, y) (–y, x)

Rule for 180 rotation: (x, y) (–x, –y)

Ex:

1)What rotation is shown in the picture to the right?

E ( , )E’ ( , )

F ( , )F’ ( , )

G ( , )G’ ( , )

H ( , )H’ ( , )

2)Write the coordinates of ABD

A = C =

D =

3)Apply a 180 rotation. Write the new coordinates below.

A' = C' = D' =

3.2: Coordinate Rules for Translations

Rule for Translation: (x, y) (x ± #, y ± #)

x moves left/right, y moves up/down

Ex:

1)What is the rule for the translation to the right?

2)Write the coordinates of ABCD

A = B =

C = D =

3)Apply the rule (x, y) (x – 2, y – 1). Write the new coordinates below.

A'= B' =

C' = D' =

3.4:All transformations

Describe the transformation or combination of transformations that map…

1)ABE to EFH

2)BCF toIHF

3)HIJ to FEB

4.1: Dilation

Similarity Transformations:A transformation that produces similar figures. The image of a figure under asimilarity transformation, such as a dilation, has the same shape as the original figure, but may be a different size. Asimilarity transformationcan also be a sequence of a rigid motion (reflection, rotation, or translation) and a dilation.

Dilation:A transformation that enlarges or reduces a figure by a scale factor about a center point so that the original figure and its image are similar. If the scale factor is greater than 1, thedilationis an enlargement. If the scale factor is less than 1, thedilationis a reduction.

Similar:Similarfigures have corresponding angles of equal measure and the ratios of each pair of corresponding sides are equivalent.

  • What is true about the angles in the dilation?
  • What is true about the side lengths in the dilation?

Therefore, Dilations always create ______figures.

Rule for Dilations: (x, y) (#x, #y)

# is the scale factor

Ex:

What is the scale factor in the dilation to the right?

What is the rule for the dilation to the right?