Chapter 7 Transformations

Examples of symmetry

Lines of Symmetry

Lesson 7.1 Warm Up 1. List the pairs of corresponding sides and angles. 2.Quadrilateral ABCD quadrilateral PQRS. The measure of,,. Find 3. Name 3 points that are the same distance from (0, 0) as (2, 5) 4. If B ( -1, 4) is moved right 2 units and down 1 unit, where is its new location?

Section 7.1 Goal 1: Identify the three basic rigid transformations

Transformation The operation that maps, or moves, a preimage onto an image. –Three basic transformations are reflections rotations translations

Preimage The original figure in a transformation of a figure in a plane. Image The new figure that results from the transformation of a figure in a plane.

Reflection A type of transformation that uses a line that acts like a mirror, called the line of reflection, with an image reflected in the line. Line of Reflection

Rotations A type of transformation in which a figure is turned about a fixed point called the center of reflection.

Translation A type of translation that maps every 2 points P and Q in the plane to points P’ and Q’, so that the following two properties are true. 1) PP’ = QQ’ 2) or and are collinear.

Isometry A transformation that preserves lengths. Naming Transformations When naming an image, take the corresponding point of the preimage and add a prime symbol. A’ A

(x + ___ ), ( y - ___ )

Example 1

A B C Example 2 Which of the following transformations appear to be isometries.

Example 3 is mapped onto. The mapping is a translation. Given is an isometry, XY = 8, and m, find the length of and the m.

The coordinates of are J (1, 1), K (-2, 4) and L (-2, -1). The coordinates of are J’ (2, -3), K’ (-1, 0) and L’ (-1, -5). a)Name and describe the transformation b)Is this an isometry? c)If m, find m. Example 4

Think About It What does rigid mean? What does plane mean? What do you think a rigid plane would be? Draw what a preimage and an image of a rigid plane might look like.

Think about it! 1. What does rigid mean? 2. What does plane mean? 3. What do you think a rigid plane would be? 4. Draw what a preimage and an image of a rigid plane might look like below.

1.∆ABC __________ 3. ∆DEF _________5. ∆________ ∆EFD 2. ∆ __________ ∆ACB4. ∆ACB _________ 6. ∆________ ∆CBA Example 3

N Use this to rotate the image

Example 5: Using Algebra Find the value of each variable, given that the transformation is an isometry

Vocabulary Reflection –A type of transformation that uses a line that acts like a mirror, called the line of reflection with an image reflected in the line Line of Reflection –The line that the image reflects over Reflection Theorem –A reflection is an isometry

1.If P is not on m, then m is the perpendicular bisector of. 2. If P is on m, then P = P’. P P’ P

3. If P (2, 3) is reflected in the x-axis, what quadrant will its image be in? 4. If Q (-4, 5) is reflected in the y-axis, what quadrant will its image be in?

Reflections in the coordinate plane have the following properties:

1. If (x, y) is reflected across the x axis, its image is point (x, -y). (x, y) (x, -y) preimage image

2. If (x, y) is reflected across the y axis, its image is point (-x, y). (x, y) ( -x, y) preimage image

A Reflection is an isometry There are four ways to show that a reflection preserves the length of a segment. Consider segment that is reflected in a line m to produce. Sketch each possible case below.

Case 1 P and Q are on the same side of m P Q P’ Q’

Case 2: P and Q are on opposite sides of m m P Q P’ Q’

Case 3: One point lies on m and is not perpendicular to m m P Q’ P’ Q

Case 4: Q lies on m and m P’ Q’ P m Q

Line of Symmetry A figure in a plane has a Line of Symmetry if a figure can be mapped onto itself by a reflection in the line.

Finding Lines of Symmetry: Hexagons can have different lines of symmetry depending on their shape. Draw the possible lines of symmetry for each of the following hexagons. __________ __________ __________ lines of symmetry lines of symmetry lines of symmetry

Graph the reflection W (-3, 3) in the y-axis W

Graph the reflection Z (1, 3) in the line x = 1 Z

Graph the reflection M (2, 3) in y = 4 M

Graph the reflection P (-3, 1) in the x-axis P

Example 2: Determine the number of lines of symmetry in the following quadrilaterals. Explain what a line of symmetry is, in your own words, with sketches. square rectangle kite

Activity 7.2 – Reflections in a Plane – textbook page # 403 Materials: tracing paper (2 for each pair of students) ruler (1 for each pair) protractor (1 for each pair)

Question What is the relationship between the line of reflection and the segment connecting a point and its image?

1.Measure and compare and, and, and and. 2.Measure and compare,, and. 3. How does line m relate to,, and. Investigate

Make a Conjecture 4.How does line m relate to and ? Explain your answer. 5. How does the line of reflection relate to the segment connecting a point and its image?

Vocabulary Rotation a transformation in which a figure is turned about a fixed point Center of Rotation the fixed point that is the center for a rotation Angle of Rotation rays drawn from the center of rotation to a point and its image form the angle of rotation Rotation Theorem A rotation is an isometry

P Clockwise rotation of 85 degrees

Counterclockwise rotation of 60 degrees

Case 1 R, Q and P are noncollinear P Q Q’ R R’

Case 2 R, Q and P are collinear

Case 3 P and R are the same point P RR’ Q Q’

Example 1 Preimage A (2, -2) B (4, 1) C (5, 1) D (5, -1) 1) Sketch the quadrilateral using the points in the first column 2) Then, rotate ABCD 90 ○ counterclockwise about the origin This transformation is (x, y) (-y, x) Image A’ B’ C’ D’

There are 2 reflections here. The final degrees of rotation is 2x. 2x○2x○ A A’ B B’ A’’ B’’ x

In the diagram, - is reflected in line k to produce. - is then reflected in line m to produce The acute angle formed by the intersection of lines k and m measures ____________, Applying the theorem above, the transformation that maps to is a clockwise rotation of 2x or 2(______) = 120.

Rotational Symmetry a figure in a plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180 ○ or less. 0 ○ 45 ○ 90 ○ rotations A B A’ B’ 45° A B A’’ B’’

Do the following figures have rotational symmetry? For those that do, describe the rotations that map the figure onto itself.

Example 4: The diagonals of the regular hexagon below form 6 equilateral triangles. Use the diagram to complete the sentences. R S W V Q T a)A clockwise rotation of 60 ○ about P maps R onto ________. b)A counterclockwise rotation of 60 ○ about ________ maps R onto Q. c)A clockwise rotation of 120○ about P maps R onto ________. d)A counterclockwise rotation of 180 ○ about P maps V onto ________.

Homework Answers 1.0°, 45°, 90°, 135°, 180° 2.180° 3.180° 4.none

Section 7.4 Translations and Vectors Goal 1: Identify and use translation in the plane.

Translation A type of translation that maps every 2 points P and Q in the plane to points P’ and Q’, so that the following two properties are true. 1) PP’ = QQ’ 2) ║ or and are collinear.

Translation Theorem A translation is an isometry You can find the image of a translation by __________________ a figure in the ___________. Another way to find the ________________ of a translation is to complete one __________________ after the other in two __________________ lines.

k m d If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is a translation. If P’’ is the image of P, then the following is true: 1. is perpendicular to k and m. 2. PP’ = 2d, is the distance between k and m. ½ d P P’

Example 1: ( x, y) ( x + a, y + b) a and b are the values that will shift the point a units horizontally, and b units vertically. In the coordinate plane to the left, begin with the endpoints P (2, 4) and Q (1, 2), Then use the translation (x, y) (x + 4, y – 2) to make P Q

Example 2: Sketch a triangle with vertices A (-1, -3), B (1, -1), C (-1, 0). Then sketch the image of the triangle after the translation ( x, y) ( x + 4, y - 2).

Example 3: Sketch a parallelogram (this is the preimage) with vertices R (-4, -1), S (-2, 0), T (-1, 3), U (-3, 2). Then sketch the image of the parallelogram after the translation ( x, y) ( x + 4, y – 2)

Vector a quantity that has both direction and magnitude, or size. It is represented by an arrow drawn between two points. Initial Point (P) the starting point of the vector Terminal Point (Q) the ending point of a vector Component Form combines the horizontal (x) and vertical (y) points. P Q 5 units right 3 units up

K J J K = = Example 1

J K K JK J = = =

D C Example 2 A B F E

Example 3: The initial point of a vector is V (-2, 3) and the terminal point is W (-4, -7). Name the vector, and write its component parts. V W ‹, ›

Example 4: The initial point of a vector is E (2, -6) and the terminal point is F (2, -9). Name the vector, and write its component parts. ‹, ›

Example 5: The component form of is. Use to translate the triangle whose vertices are A (3, -1), B (1, 1) and C (3, 5). Name the image

Example 6: The component form of is. Use to translate the triangle whose vertices are R (0, 4), S (3, 1) and T (4, -2). Name the image

The component form of is. Use to translate the quadrilateral whose vertices are G (-3, 5), H (0, 3), J (1, 3) and K (3, -2). Name the vertices G’, H’, J’ and K’. (You do not need to graph this if you have figured out how to calculate the location of the images using the component form). Example 7:

7.5 – Glide Reflections and Compositions Goal 1: Identify glide reflections in a plane.

Glide Reflection As transformation in which every point P is mapped onto point P’ by the following steps: 1.A translation maps P onto P’. 2.A reflection in a line k parallel to the direction of the translation maps P’ onto P’’. m P P’ P’’ Q Q’ Q’’

Composition The result when 2 or more transformations are combined. Composition Theorem The composition of two (or more) isometries is an isometry.

Example 1: Use the information below to sketch the image of after a glide reflection. Translation: (x, y) (x + 10, y) Reflection: in the x - axis C’’ A’’ B’’ B B’ CA C’A’

Q’ P’’P’ P Q Example 2: Sketch the image of after a composition of the given rotation and reflection. P (2, -2), Q (3, -4) Rotation: 90 ° counterclockwise about the origin (0, 0) Reflection: in the y - axis Q’’

Example 4: Repeat Example 3, but switch the order of the composition by performing the rotation first and the reflection second. What do you notice?

Example 3: Sketch the image of after a composition of the given rotation and reflection of C (2, 0), D (3, 3) reflection: in the x – axis rotation: 270 ◦ counterclockwise about the origin.

D’’ D C’’ C C’ D’

Example 4: As in example 3 but with the order reversed rotation: 270 ◦ counterclockwise about the origin. reflection: in the x – axis

D C C’ D’

Example 4: As in example 3 but with the order reversed rotation: 270 ◦ counterclockwise about the origin. reflection: in the x – axis D C C’ D’

Example 5:Describing a Transformation Describe the composition of transformations in the diagram. a) First Quadrilateral Second Quadrilateral: b) Second Quadrilateral Third Quadrilateral:

a) First triangle Second triangle b) Second triangle Third triangle :

7.6 Frieze Patterns

Using Algebra Solve for the variables in the glide reflection of described below. (x, y) (x + 3, y) Reflect across the x - axis J ( -2, -1) J’ (c + 1, -1) J’’ (1, -f) K ( - 4, 2a) K’ (5d – 11, 4) K’’ (- 1, 3g + 5) L ( b – 6, 6) L’ (2, 4e) L’’ ( h + 4, - 6)

Frieze or Border Pattern A pattern that extends to the left and right in such a way that the pattern can be mapped onto itself by a horizontal translation.

More……

How to determine the translation: 1)Decide whether or not the pattern has a 180 rotation. I) YES a) TR b) TRVG c) TRHVG II) NO a) T b) TV None of these has an R, c) TG there is no rotation d) THG All of these have an R in them, they ROTATE

Name of the translation Name the translations in these frieze patterns: Draw your own frieze pattern, then name the translation you used.