7. The preimage of 11 is 7

Geometry Unit 11 – Notes Transformational Geometry

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

Example: If function , find the Image of 8 and the Preimage of 11. : 2 3h x x→ −Solution:

( )

What would the outcome beif 8?

: 8 2 8 3 ,13x

h=

→ − =

The image of 8 is 13.7. The preimage of 11 is 7.x =

What value of would give an outcome of 11?

: 2 3 11 2 14

x x

x

hx

→ − ==

Isometry – a transformation that preserves length. Mapping – an operation that matches each element of a set with another element, its image, in the same set. Transformation – the operation that maps, or moves, a preimage onto an image. Three basic transformations are reflections, rotations, and translations. The three main Transformations are:

Reflection Flip!

Rotation Turn!

Translation

Slide!

Unit 11 Transformational Geometry Page 1 of 24

http://www.mathsisfun.com/geometry/reflection.html

http://www.mathsisfun.com/geometry/rotation.html

http://www.mathsisfun.com/geometry/translation.html

Syllabus Objective: 11.1 - The student will distinguish among the basic mapping functions: reflections, translations, and rotations. Reflection – type of transformation that uses a line that acts like a mirror, called a line of reflection, with a preimage reflected over the line to form a new image. {Flip} A reflection is a FLIP over a line.

• Every point is the same distance from the central line! • The reflection has the same size as the original image.

Line of reflection – the mirror line. Theorem: A reflection is an isometry.

Example: Given rectangle ABCD, write the coordinates of the each point by reflection in:

a) The x-axis

( ) ( ) ( ) ( )1,1 , 5,1 , 5,6 , 1,6

As the rectangle is reflected over the -axis:A B C D

x( ) ( ) ( ) ( )' 1, 1 , ' 5, 1 , ' 5, 6 , ' 1, 6 .B CA D− − − −

b) The y-axis

( ) ( ) ( ) ( )1,1 , 5,1 , 5,6 , 1,6

As the rectangle is reflected over the i :A B C D

y( ) ( ) ( ) ( )' 1,1 , ' 5,1 , ' 5,6 , ' 1,6 .A B C D− − − −

-ax s

c) The line y = x.

( ) ( ) ( ) ( )1,1 , 5,1 , 5,6 , 1,6

As the rectangle is reflected over the :A B C D

7

6

5

4

3

2

1

2 4

D C

BA

6

y x= ( ) ( ) ( ) ( )' 1,1 , ' 1,5 , ' 6,5 , ' 6,1 .A B C D


Notice the patterns: reflection over the x-axis → change the sign of the y- coordinate, (x, y) → (x, -y) reflection over the y-axis → change the sign of the x-coordinate, (x, y) → (-x, y) reflection over y = x → switch the order of the x and y-coordinates. (x, y) → (y, x) And if all else fails, just fold your sheet of paper along the mirror line and hold it up to the light!

Line of symmetry – a line that a figure in the plane has if the figure can be mapped onto itself by a reflection in the line.

Examples: Determine and draw all lines of symmetry in the following figures.

Example: Reflect over the y-axis: Solution:


Example: Notice that some letters possess vertical line symmetry, some possess horizontal line symmetry, and some possess BOTH vertical and horizontal line symmetry.

A point reflection exists when a figure is built around a single point called the center of the

figure. It is a direct isometry.

( ) ( ), ,P x y x y→ − − . Rotation – a type of transformation in which a figure is turned about a fixed point, called a center of rotation. {Turn} Center of rotation – the fixed point. Angle of rotation – the angle formed when rays are drawn from the center of rotation to a point and its image. Counterclockwise rotation is considered positive and clockwise is considered negative.

"Rotation" means turning around a center.

• The distance from the center to any point on the shape stays the same. • Every point makes a circle around the center.

Theorem: A rotation is an isometry.


Examples: O is the center of regular pentagon ABCDE. State the images of A, B, C, D, and E under each rotation.

a) RO,144 b) RO,-72

Rotate counterclockwise144 about point .OABC

E

ED

°→→→→→ .

CD

AB .

ABCD

Rotate clockwise 72 about point .OABCD

E

E

°→→→→→

Examples:

A

B 72° E O

D C

a) Each of these figures has rotation symmetry. Estimate the center of rotation and the angle of rotation?

a) For each shape, the center of rotation is the center of the figure. The angles

of rotation, from left to right, are 120°, 180°, 120°, and 90°. b) Do the regular polygons have rotation symmetry? For each polygon, what are

the center and angle of rotation?

b) For each shape, the center of rotation is the center of the figure. The angles

of rotation, from left to right, are 120°, 180°, 120°, and 90°. c) Name the vertices of the image of KLMΔ after a rotation of 90°. K(4, 2), L(1,

3), and M(2, 1). c) K’(-2, 4), L’(-3, 1), and M’(-1, 2).


Notice the patterns: rotation of 90° → change the sign of the y-coordinate then switch the x and y-coordinates, (x, y) → (-y, x) rotation of 180° → change the signs of the x and y- coordinates, (x, y) → (-x, -y) rotation of 270° → change the sign of the x-coordinate then switch the x and y-coordinates. (x, y) → (y, -x) Translation – a type of transformation that maps every two points P and Q in the plane to points P’ and Q’, so that the following two properties are true. 1. PP’ = QQ’. 2. ' 'PP QQ or 'PP and 'QQ are collinear. {Slide} When you are sliding down a water slide, you are experiencing a translation. Your body is moving

a given distance (the length of the slide) in a given direction. You do not change your size, shape or the direction in which you are facing.

"Translation" simply means Moving ...

... without rotating, resizing or anything else, just moving.

Every point of the shape must move: • the same distance,

• in the same direction.

Theorem: A translation is an isometry.

Examples: Explain the meaning of: a) (x, y) → (x – 7, y + 3) Slide the original point(s) by moving it left 7 units and up 3 units. {mapping notation} b) T(-7, 3) or T-7, 3(x, y) Slide the original point(s) by moving it left 7 units and up 3 units. {symbol notation}


Examples: a) Which of the following lettered figures are translations of the shape of the

purple arrow? Chose ALL that apply.

Solution: a, c, and e only.

b) Which of the following translations best describes the diagram at the left?

i. 3 units right and 2 units down. ii. 3 units left and 2 units up. iii. 3 units left and 2 units down.

Solution: (i) 3 right and 2 down.

Glide reflections – a transformation in which every point P is mapped onto a point P” by the following two steps. 1. A translation maps P to P’. 2. A reflection in a line k parallel to the direction of the translation maps P’ to P”.

When a translation (a slide or glide) and a reflection are performed one after the other, a transformation called a glide reflection is produced. In a glide reflection, the line of reflection

is parallel to the direction of the translation. It does not matter whether you glide first and then reflect, or reflect first and then glide. This transformation is commutative.

Since translations and reflections are both isometries, a glide reflection is also an isometry.


Examples: a) Does this paw print illustration depict a glide reflection?

Solution: Yes.

b) Examine the graph below. Is triangle A"B"C" a glide reflection of triangle ABC?

Solution: Yes. c) A triangle has vertices A(3,2), B(4,1) and C(4,3). What are the coordinates of

point B under a glide reflection, 0,1 0xT r = ? Solution: After the reflection, B’(-4, 1). After the translation, B”(-4, 2).

d) Given triangle ABC: A(1,4), B(3,7), C(5,1). Graph and label the following composition: 5, 2 x axisT r− − − .


Syllabus Objective: 11.3 - The student will explore relationships among transformations.

Line Reflection Point Reflection Translations Rotations Dilations

Opposite isometry Direct isometry Direct isometry Direct isometry

NOT an isometry.

Figures are similar.

Properties preserved:

distance angle measure

parallelism collinearity

midpoint




midpoint




midpoint




midpoint


angle measure parallelism collinearity

midpoint Lengths not

same. Reverse

Orientation (letter order

changed)

Same Orientation (letter order the

same)


same)


same)

Same Orientation

(letter order the same)

Notation:

( ) (, ,x axis

r x y x−

= − )y

( ) ( ), ,y axis

r x y x y−

= −

( ) ( ), ,y x

r x y y x=

=

( ) (, ,y x

r x y y=−

= − − )x

Notation: ( ) ( ), ,

originr x y x y= − −

Notation: ( ) ( ),

, ,h k

T x y x h y k= + + Notation: ( ) ( )90

, ,R x y y x°

= −

( ) (180, ,R x y x y

°= − − )

( ) (270, ,R x y y x

°= − )

Notation: ( ) ( ), ,

kD x y kx ky=

Glide Reflection Opposite isometry

Properties preserved: distance

angle measure parallelism collinearity

midpoint Reverse Orientation

(letter order changed)


Syllabus Objective: 11.4 - The student will design examples of each type of symmetry. Reflection Symmetry (sometimes called Line Symmetry or Mirror Symmetry) is easy to recognize, because one half is the reflection of the other half. With Rotational Symmetry, the shape or image can be rotated and it still looks the same. The image is rotated (around a central point) so that it appears 2 or more times. The number of times it appears is called its order.

Point Symmetry is when every part has a matching part:

• the same distance from the central point • but in the opposite direction.

Point symmetry exists when a figure is built around a single point called the center of the

figure. For every point in the figure, there is another point found directly opposite it on the other side of the center.

Note: it’s the same as "Rotational Symmetry of Order 2".

A simple test to determine whether a figure has point symmetry is to turn it upside-down and see if it looks the same. A figure that has point symmetry is unchanged in appearance by a 180°

rotation. Examples: Line symmetry:

The white line down the center is the Line of Symmetry.


http://www.mathsisfun.com/geometry/symmetry-rotational.html

a) Which of the following lettered items possesses line symmetry? Consider vertical and/or horizontal symmetry. Chose ALL that apply.

a. the letter F Solution: b, c, d, and e. b. an equilateral triangle

c. the letter X

d. the word BOO

e. a square

f. the word WALLY

g. the letter Q

b) If the left wings of this symmetric butterfly have a total of 32 beige scallops around their edges, how many total beige scallops does the butterfly possess?

Solution: 64

Examples: Rotational symmetry: The US Bronze Star Medal

A Dartboard has Rotational Symmetry of Order 5. has Rotational S ymmetry of Order 10.

a) Does this sign have rotational symmetry? If so, what are the degrees of

symmetry? Solution: Yes, 120° and 240°.


b) Which of the following letters or objects possesses rotational symmetry? Chose ALL that apply.

a. letter F Solution: b, c, d, and e. b. an equilateral triangle

c. letter X

d. a rectangle

e. a square

f. letter W

g. the letter Q

Examples: Point symmetry:

HRotational Symmetry of Order 2H

a) A sign by a swimming hole displays the message shown at the left. The

message is saying that there is no swimming on Mondays. What is special about the way the message is written?

Solution: It has point symmetry. It reads the same when turned upside-down.

b) Which of the following lettered items possesses point symmetry? Choae ALL

that apply. a. the letter D Solution: b, c, d, and g. b. a square c. the letter S d. the word e. the letter B f. the word DAD g. the letter Z


c) Which of these cards has point symmetry? Solution: The 2 of spades.

Syllabus Objective: 11.5 - The student will sketch the results after two or more transformations are applied to a figure. Theorem: If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is a rotation about P. The angle of rotation is 2x°, where x° is the measure of the acute or right angle formed by k and m. {Relections over intersecting Lines Th.}

Example: Image A reflects over line m to B, then image B reflects over line n to C. Image C is a rotation of image A.

Notice the yellow star on quad A, with the first reflection its orientation changes. On the second reflection its orientation changes back to the original. Also, the measure of tangle connecting the star on A, the intersection point of the lines, and the star on C is twice the measure of the angle formed by lines m and n.

he

Example: Illustrated at left, the measure created by corresponding points on the reflected triangles is twice the measure between the intersecting lines.

7

6

5

4

3

2

1

2 4 6 8 10

m∠ABC⋅2 = 100.8°m∠XBX'' = 100.8°

m∠ABC = 50.4°

X"

X'

X

C

B

A


Theorem: 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) "PP is perpendicular to k and m. 2) PP” = 2d, where d is the distance between k and m. {Relections over ║ Lines Th.}

Examples: Lines m and n are parallel and 10 cm apart.

a) Point A is 6 cm from line m and 16 cm from line n. Point A is reflected over line m, and then its image, A’, is reflected over line n to create a second image, point A”. How far is point A from point A”?

10 cm6 cm

nm

A

b) What if A is reflected over n, and then its image is reflected over m? Find the new image and distance.

Solutions: a) By the Reflections over Parallel Lines

Theorem, the distance between A and A” is 20 cm, twice the distance between the lines. A drawing verifies this. 20 cm

4 cm4 cm

6 cm

A"

A'10 cm

6 cm

nm

A

b) By the Reflections over Parallel Lines Theorem, the distance between A and A”is 20 cm. A drawing verifies this.

26 cm

20 cmA"

A'

10 cm6 cm

nm

A

_ Composition of transformations – the result when two or more transformations are combined to produce a single transformation. An example is a glide reflection.


When two or more transformations are combined to form a new transformation, the result is called a composition of transformations. In a composition, the first transformation produces an

image upon which the second transformation is then performed.

The symbol for a composition of transformations is an open circle. The notation is

read as a reflection in the x-axis following a translation of 3,4x axisr T−

(x + 3, y + 4). Be careful!!! The process is done in reverse!!

Examples: a) Write a rule for the composition of a dilation of scale factor 3 following a

translation of 2 units to the right. Solution: Because this has a dilation, it is not an isometry. ( ) ( )3 2,0 , 3 6,3 .D T x y x y= + b) Which composition of transformations will map ' ' '?ABC A B CΔ → Δ ?

A) 90 0.5 2,0D T°RB) 2,0 90 0.5T D°R

C) 90 2 2,0D T°RD) 2,0 90 2T D°R

Solution: B) 2,0 90 0.5T D°R , first a dilation

of one-half, then a rotation of 90°, both of those followed by a translation 2 units to the right.

Theorem: The composition of two (or more) isometries is an isometry. Syllabus Objective: 11.2 - The student will explore scale factor and dilations. Dilation – a type of transformation, with center C and scale factor k, that maps every point P in the plane to a point P’ so that the following two properties are true. 1. If P is not the center point C, then the image point P’ lies on CP . The scale factor k is a

positive number such that CP’ = k (CP), and 1k ≠ . 2. If P is the center point C, then P = P’.


A dilation used to create an image larger than the original is called an enlargement. A dilation used to create an image smaller than the original is called a reduction. A figure and its dilation

are similar figures.

Reduction – a dilation with 0 . 1k< < Enlargement – a dilation with . 1k > A dilation of scalar factor k whose center of dilation is the origin may be written:

Dk (x, y) = (kx, ky). Examples: a) Draw the dilation image of triangle ABC with the center of dilation at the origin

and a scale factor of 2. OBSERVE: Notice how EVERY coordinate of the original triangle has been multiplied by the scale factor (x2).

HINT: Dilations involve multiplication!


b) Draw the dilation image of pentagon ABCDE with the center of dilation at the origin and a scale factor of 1/3.

OBSERVE: Notice how EVERY coordinate of the original pentagon has been multiplied by the scale factor (1/3).

HINT: Multiplying by 1/3 is the same as dividing by 3!

c) Draw the dilation image of rectangle EFGH with the center of dilation at point E and a scale factor of 1/2.

For this example, the center of the dilation is NOT the origin.

The center of dilation is a vertex of the original figure.

OBSERVE: Point E and its image are the same. It is important to observe the distance from the center of the dilation, E, to the other points of the figure. Notice EF = 6 and E'F' = 3.

HINT: Be sure to measure distances for this problem.


Syllabus Objective: 11.6 - The student will explore tessellations of a plane using polygons. Tesselation – a collection of tiles that fill the plane with no gaps or overlaps. The tiles in a regular tessellation are congruent regular polygons. Do you think all shapes will tessellate a surface without overlapping or spaces?

No, there are many that will not. What shapes do you think will tessellate a surface without overlapping or spaces?

Some regular polygons (for example). What shapes do you think will not tessellate a surface without overlapping or spaces?

Non-regular polygons, non-polygons (for example).

Introduce students to the concept of tiling. Polygons can be used to tile a surface. When tiling there can be no gaps or overlaps. To demonstrate what tiling is, square blocks “Geoblocks” can used on the overhead projector. Ask students to predict which shapes will tile and which will not.

Shapes kite oval parallelogram / rhombus triangle pentagon hexagon circle rectangle semicircle

Shapes - Solutions

Kite - Yes Oval - NO parallelogram / rhombus - Yes Triangle - Yes Pentagon - NO Hexagon - Yes Circle - NO Rectangle - Yes Semicircle - NO

Only three regular polygons tessellate in the Euclidean plane: triangles, squares or hexagons.




Some ways to create your own tessellations:


On the left is a true tessellation; on the right is not a tessellation but a pattern. Patterns repeat but do not have clearly defined closed shapes. Tessellations repeat and do have clearly defined closed shapes.

Examples:



This unit is designed to follow the Nevada State Standards for Geometry, CCSD syllabus and benchmark calendar. It loosely correlates to Chapter 7 of McDougal Littell Geometry © 2004, sections 7.1 – 7.5, 8.7, & p. 452. The following questions were taken from the 2nd semester common assessment practice and operational exams for 2008-2009 and would apply to this unit.

Multiple Choice # Practice Exam (08-09) Operational Exam (08-09)

23. Determine the transformation that has mapped Δ ABC to Δ A B C′ ′ ′ .

A. dilation B. reflection C. rotation D. translation

Given the figure below:

What transformation mapped Δ ABC to Δ A B C′ ′A. reflection B. rotation C. translation D. dilation

24. How many lines of symmetry does a square have? A. 0 B. 1 C. 2 D. 4

How many lines of symmetry does a rectangle have? A. 0 B. 1 C. 2 D. 4

B

A'

A

B'

C'

C'

B'

A'

A

B

C

C


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7. The preimage of 11 is 7