Transformations

Engage!

Enga

ge!

Types of Transformations *dilations* *rotation* enlarge / reduce turn *translation* *reflection* slide flip

Isometry Definition: In a plane, an isometry is a transformation that maps every segment to

congruent segment. Also called a congruence transformation. Examples: Decide whether the transformation is an isometry. If it is, name the transformation. a) b) c) d)

dilation Isometry - reflection

Isometry - reflection

dilation

Transformation (mapping)

Definition: Each pre - image point has exactly one image point, and the converse is also true.

The new figure is called the ________ and the original is called the ___________.

image pre-image

Examples Identify each of the transformations shown as a reflection, rotation, or translation. In each case the image is shown in bold. a) b) c) d)

rotation rotation

translation reflection

9.1 Translations

Translation Translation: The result of a movement in _____ . It is also referred to as a ______. It is can also be the end result of two __________ ______________ called a composite of reflections. Properties of Translations: 1. preserves ___________ 2. preserves ____________

one direction

reflections

congruence orientation

Slide successive

Translations: A translation is a transformation that can be described in coordinate notation this way: (x,y) ( , ) Every point shifts h units __________ and k units _____________.

x + h y + k

left or right up or down

Find the coordinates of the given figure. Then, find the coordinates of the image after each translation: Given: A(__,__) B(__, __) C(__, __) and D(__, __) 1. (x,y) ( x + 3, y - 5)

A’ ( , ) B’ ( , ) C’ ( , ) D’ ( , )

2. (x,y) ( x - 3, y) A’ ( , ) B’ ( , ) C’ ( , ) D’ ( , )

-3 3 0 3 1 1 -4 1

0 -2 3 -2 4 -4 -1 -4

-6 3 -3 3 -2 1 -7 1

Coordinate notation (x,y) → ( x + 5, y + 3)

A. <9, -2> B. <-8, 0>

A B

C

B’ A’

C’

A’(5, 2), B’(7, 3), and C’(6, -1)

What is the length of ? 'CC

Examples Describe each translation using coordinate notation: a) Every point moves to the left 5 units and

down 3 units. b) Every point moves to the right 6 units

and up 2 units.

(x, y) → (x - 5, y - 3)

(x, y) → (x + 6, y + 2)

Repeat, but do in vector notation

<-5, -3>

<6, 2>

GROUPWORK - Transformations

• Ws practice 1, 3, 5, 8

9.3: Reflections

Reflections One type of transformation is called ____________. Line m is called the _________ ___ ___________. Since P is on the line of reflection, its reflected image is itself. X is the ____________ and X’ is the

________.

reflection line of reflection

pre-image image

Properties of Reflections

1. Preserves ______________ 2. Changes ________________________ 3. Line of reflection:

congruence Orientation (the way it faces)

The PERPENDICULAR BISECTOR of every segment that connects a point and its image.

Examples Sketch the image of the polygon after a reflection over the given line. a) b)

Examples Find the number of lines of reflectional

symmetry for each figure. In each case, sketch each line on the figure.

a) b)

c) d)

Examples Find the reflection image of each point in the: a) x-axis. b) y- axis. A(3, 5) → _____ C(–4, 2) → ____ B(–2, 1) → _____ D(3, 1) → _____

A’ (3, -5) C’(4,2)

B’(-2,-1) D’(-3,1)

A

A’

B

B’

C C’ D D’

Reflecting Over the Coordinate Axes:

• Reflecting over x-axis:_______________

• Reflecting over y-axis: _______________

• Reflecting over line y=x: ______________

• Reflecting over line y=-x: ______________

(x, y) → (x, - y)

(x, y) → (-x, y)

(x, y) → (y, x)

(x, y) → (-y, -x)

Examples a) Given M(0,0), I(4,0), L(2, 3), and K(2,-3) reflect over y = x: b) Given T(-3,0), I(-2,2), A(3, 0), and N(-1,4) reflect over y = -x:

M’ (0,0) I’(0,4) L’(3,2) K’(-3,2)

T’ (0,3) I’ (-2,2) A’ (0,-3) N’ (-4,1)

Locating best tie in:

Existing line

GROUPWORK - Reflections

• Ws practice 11, 14, 17

Rotations

9.4 – Rotations • Definition: A rotation about a point O through an

angle of x° is a transformation that maps each point P of the plane with a point P′ such that m∠POP′ = x° and OP = OP′. Point O is called the _________________ and x° is called the ______________.

Center of rotation Angle of Rotation

O x°

P P’

Example 1) If PQRST is a regular pentagon and O is the center of the polygon, find

a) the 72° clockwise rotation about O of point Q. b) the 144° counterclockwise rotation about O of point R. c) the 360° clockwise rotation about O of point P.

P

T

S R

Q O •

Point P

Point T

Point P

Step 1: Find the number of degrees between each point.

o

ofsides72

5360

#360

==

72 72 72

72 72

Example 2) Find the rotation image of each segment or triangle.

a) 90° clockwise rotation about Q of JK. b) 90° counterclockwise rotation about Q of ∆HIP. c) 180° rotation about Q of FM . d) 90° clockwise rotation about P of ∆HOP.

Q H

A B C

D

E F G

O

P

I J K

L

N M

LM

∆FON

BI

∆IHP

Definition: A figure is said to have x° rotational symmetry if the figure can be rotated x° about a point so that the pattern appears identical to its original position.

For example a square has 90° and 180° rotational symmetry.

Example 3) Describe the symmetry of each figure. Include the number of lines of symmetry, and

rotational symmetry of 180° or less.

a. b. c. d.

45o, 90o, 135, 180 NONE

72o, 144o 90o,180o

Ways to do a rotation of 90° Patty paper 1. Plot point on graph 2. Place patty paper on top,

trace point and center of rotation with vertical and horizontal marks.

3. Place pencil on center of rotation and spin counterclockwise (unless instructed otherwise) 90°

4. Locate new point (image) (-1, -3)

Given point (-3, 1)

Rotation Coordinates for 90°, 180°, and 270° COUNTERCLOCKWISE (CCW)

90°: _________________________________________

180°: ______________________________________

270°: _______________________________________

( x ,y ) (-x , -y)

( x ,y ) (-y , x)

( x ,y ) (y , -x)

270 CCW = 90 Clockwise

Example 4) Find the image of : A(3, –4) B(-2,1) and C(-3, -4) under a:

a) 90° CCW rotation b)180° CCW rotation about the origin. about the origin. A’ ( , ) A’ ( , ) B’ ( , ) B’ ( , ) C’ ( , ) C’ ( , )

4 3

-1 -2

4 -3

-3 4

2 -1

3 4

( x ,y ) (-y , x) ( x ,y ) (-x , -y)

Rotation practice

• Do 1, 5, 7

You are probably familiar with the word "dilate" as it relates to the eye. "The pupils of the eye were dilated." As light hits the eye, the pupil enlarges or contracts depending upon the amount of light. This

concept of enlarging and contracting is "dilating".

A dilation is a transformation that produces an image that is the same shape as the original,

but is a different size. larger – enlargement smaller - reduction.

The washers shown in this photo illustrate the concept of dilation.

The washers are the same shape, but they are different in

size.

1.

P' is the image of P under a dilation about O of ratio 2.

OP' = 2OP and

Examples:

Properties 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 or 1.)

Dilations create similar figures.

2.

_____ is the image of _____ under a dilation about O of ratio .

Class Exercises: Given center C and each scale factor K, find the

dilation image of XY.

a. k = ¾

b. k = 3

X

Y C

X

Y C

Determine the scale factor used for each dilation with center C. The dilation image in the figure is shown in RED.

a. b. c.

1/3 5/2

A

A is the center of dilation

The coordinates of B(1,0) and A(2.5,0)

TO / FROM

RED BLUE

= 6 3 = 2

To find the dilation of a figure, multiply the ordered pair by the scale factor.

Example: P (2, -3) dilated by 4 means

P (2 * 4, -3 * 4) P’ (8, -12)

M (2, -3) dilated by ½ means

M (2 * ½, -3 * ½) M’ (1, -3/2)

Remember:

Draw the image of CHS after a dilation with center (0,0) and scale factor of ½. C(0,3) H(-6,-2) S(-4, 3)

C S

H

C’

H’

S’

(0, 1.5)

(-3, -1)

(-2, 1.5)

Draw the image of CHS after a dilation with center (0,0) and scale factor of ½. C(0,3) H(-6,-2) S(-4, 3)

C S

H

C’

H’

S’

(0, 1.5)

(-3, -1)

(-2, 1.5)

Find the resulting image of FROST after each transformation. Be sure that each transformation is dependent on the previous one! F(0,2) R(-2,0) O(-1,-2) S(1, -2) T(2,0)

0 2 -2 4 -1 6 1 6 2 4

-3 0 -5 2 -4 4 -2 4 -1 2

3 0 5 -2 4 -4 2 -4 1 -2

6 0 10 -4 8 -8 4 -8 2 -4

a. reflection over the line y = 2: F’( , ) R’( , ) O’( , ) S’( , ) T’( , ) b. (x,y) (x – 3, y – 2): F’’( , ) R’’( , ) O’’( , ) S’’( , ) T’’( , ) c. rotation of 180º: F’’’( , ) R’’’( , ) O’’’( , ) S’’’( , ) T’’’( , ) d. dilation of 2: F’’’’( , ) R’’’’( , ) O’’’’( , ) S’’’’( , ) T’’’’( , )