OBJECTIVES: TO IDENTIFY ISOMETRIES TO FIND TRANSLATION IMAGES OF FIGURES Chapter 9 Section Translations

Transformation -> a change in the position, shape, or size of a geometric figure. Can be done by flipping, sliding, or turning the figure. Pre-image -> the original figure. Image -> the figure after the transformation. Isometry -> transformation in which the pre-image and image are congruent.

Ex: Does the transformation appear to be an isometry? Preimage Image Preimage Image Preimage Image Yes, congruent by a flip Yes, congruent by a flip and slide No, this involves a change in size

A transformation maps a figure onto its image and may be described with arrow (  ) notation. Prime (´) notation is sometimes used to identify image points. In the diagram below, K´ is the image of K (K  K´) J K Q K´K´ J´J´ Q´Q´ ΔJKQ  ΔJ´K´Q´

Translation (aka slide) -> an isometry that maps all points of a figure the same distance in the same direction. The figure below shows a translation of the black rectangle by 4 units right and 2 units down. This can be written as: the original figure (x, y) is mapped to (x + 4, y – 2) A B C D A´A´B´B´ C´C´D´D´

Ex: Write a rule to describe the translation PQRS  P´Q ´R ´S ´ Q R S P Q ´R ´ S ´ P ´ 2 4

Composition -> a combination of two or more transformations. In a composition, each transformation is performed on the image of the preceding transformation. The red arrow is the composition of the two black arrows.

Homework #12 Due Wed/Thurs (Feb 20/21) Page 473 – 474  #1 – 22 all

Section Reflections Objectives:  To find reflection images of figures

Reflection (aka flip) -> an isometry in which a figure and its image have opposite orientations. Think of how a reflected image in a mirror appears “backwards” U B G U´ G´ B´

The following rules can be used to reflect a figure across a line r :  If a point A is on line r, then the image of A is A itself (that is, A´ = A)  If a point B is not on line r, then r is the perpendicular bisector of BB´ r B´ C´ C B A = A´

Ex: Drawing Reflection images.  Given points A(-3, 4), B (0, 1), and C(2, 3), draw ΔABC and its reflection image across each axis (x and y).  Draw the same reflection image across x = 3

Section Rotations Objectives: To draw and identify rotation images of figures

Rotation -> an isometry in which exactly one point is its own image, the center of rotation.  In order to describe a rotation, you must know: 1. The center of rotation (a point) 2. The angle of rotation (positive number of degrees) 3. Whether the rotation is clockwise or counterclockwise.

The following rules can be used to rotate a figure through x° about a point R:  1. The image of R is itself (that is R´ = R)  2. For any point V, RV´ = RV and m<VRV´ = x R V x° R´ V´

Homework # 13 Due Monday (Feb 25) Page 480 – 481  #1 – 21 all

Section Symmetry Objectives: To identify the type of symmetry in a figure

Symmetry  exists in a figure if there is an isometry that maps the figure onto itself Reflectional Symmetry/Line Symmetry  one half of the figure is a mirror image of its other half. If the image were folded across the line of symmetry, the halves would match up exactly.

Rotational Symmetry  a figure that is its own image for some rotation of 180° or less. If you rotate the figure about an axis a certain degree, it will look the same as before the rotation. Point Symmetry  a figure that has 180° rotational symmetry is said to have “point symmetry”

Ex: Draw all the lines of symmetry for a regular hexagon. Draw all the lines of symmetry for a rectangle.

Homework #14 Due Tuesday (Feb 26) Page 494 – 495  #2 – 18 even  #25 – 32 all

Section Dilations Objectives: To locate dilation images of figures

Dilation  a transformation whose pre-image and image are similar. Thus, a dilation is a similarity transformation, not an isometry.  Every dilation has: 1. A center 2. A scale factor n  The scale factor describes the size change from the original figure to the image Enlargement  scale factor is greater than 1 Reduction  scale factor is between 0 and 1

B C D A = A´ B´ C´ D´ Enlargement Center A Scale Factor E FG H F´ G´ H´ E´ ·C

Ex: Finding a Scale Factor The blue triangle is a dilation image of the red triangle. Describe the dilation. (center/dilation factor/type of dilation) X = X´ R T R´T´ 4 8

Homework #15 Due Wednesday (Feb 27) Page 500 – 501  #1 – 22 all Quiz Thurs/Fri (9.1 – 9.5)

Section 9.6 – Compositions of Reflections Objectives: To use a composition of reflections To identify glide reflections

Theorem 9.1  A translation or rotation is a composition of two reflections. Theorem 9.2  A composition of reflections across two parallel lines is a translation. Theorem 9.3  A composition of reflections across two intersecting lines is a rotation.

Theorem 9.4  In a plane, one of two congruent figures can be mapped onto the other by a composition of at most three reflections. Theorem 9.5  There are only four isometries. They are the following: Reflection Translation Rotation Glide Reflection R R R R R R

Glide Reflection  the composition of a glide (translation) and a reflection across a line parallel to the direction of translation. RR

Ex: Find the image of R for a reflection across line l followed by a reflection across line m. Describe the resulting translation. R l m

Ex: Find the image of ΔTEX [T(-5, 2), E(-1, 3), X(-2, 1)] for a glide reflection where the translation is (x, y)  (x, y-5) and the reflection line is x = 0. Use ΔTEX from the above example.  Find the image of ΔTEX under a glide reflection where the translation is (x, y)  (x+1, y) and the reflection line is y = -2  Would the reflection be the same if you reflection ΔTEX first, and then translated it? Explain.

Homework #16 Due Tuesday (March 05) Page 509 – 510  # 1 – 23 all

Section Tessellations Objectives: To identify transformations in tessellations, and figures that will tessellate. To identify symmetries in tessellations.

Tessellation (aka tiling)  a repeating pattern of figures that completely covers a plane, without gaps or overlaps.  Tessellations can be created with translations, rotations, and reflections. They can be found in art, nature (cells in a honeycomb), and everyday life (tiled floors).

Identify the transformation and the repeating figures in these tessellations.

Theorem 9.6  Every triangle tessellates Theorem 9.7  Every quadrilateral tessellates

It is also possible to identify symmetries in tessellations. There are four possible symmetries that can be identified in tessellations.  Reflectional Symmetry  can be reflected about some vector and remain the same.  Rotational Symmetry  can be rotated by some angle about some point and remain unchanged.  Translational Symmetry  can be translated by some vector and remain unchanged.  Glide Reflectional Symmetry  can be translated by some vector and then reflected and remain unchanged.

Identify the type of symmetry in each tessellation.

Homework #17 Due Wednesday (March 06) Page 518 – 519  # 1 – 14 all  # 18 – 28 even