Lesson 7.1 Rigid Motion in a Plane

Lesson 7.1

Rigid Motion in a Plane

Today, we will learn to…> identify the 3 basic transformations> use transformations in real-life

situations

Transformations

The original figure is called the ____________ and the new figure is called

the ____________.

preimage

image

Transformations

Preimage: A , B , C , D

Image: A’ , B’ , C’ , D’

Rotation

Translation

Isometriespreserve length, angle measures,

parallel lines, & distances between points

Theorems 7.1, 7.2, & 7.4

Reflections, translations, and rotations are

isometries.

1. Name and describe the transformation.reflection over

ABC

the y-axis

A’B’C’

2. Name the coordinates of the vertices of the preimage and image.

(-4,0)

(-4,4)

(4,0)

(4,4)

(0,4)

3. Name and describe the transformation.reflection over

ABCD

x = -1

HGFE

4. Is the transformation an isometry? Explain.

NO

YES

YES

NO

5. The mapping is a reflection. Which side should have a length of 7?

Explain.WX = 7

6. Name the transformation. Find x and y.

Reflection

x = y = 40 4

Reflection

7. Name the transformation. Find x and y.

x = y = 12 4

8. Name the transformation. Find a, b, c, and d.

a =b =c =d =

7353158

reflection

9. Name the transformation. Find p, q, and r.

p =q =r =

193

7.5

rotation

10. Name the transformation and complete this statementGHI ____LKP

reflection

11. Name the transformation that maps the unshaded turtle onto the shaded turtle

reflection

translation

rotation

Lesson 7.2Reflections

Today, we will learn to…> identify and use reflections > identify relationships between

reflections and line symmetry

Reflection

2 images required

1. Is this a reflection?What is the line of reflection?

YES x = -2

2. Is this a reflection?

NO


YES

y = 1


YES

y = x


YES

y = - x

When can I use this in “Real Life?”

Finding a minimum distance

Telephone Cable - Pole PlacementTV cable (Converter Placement)

Walking Distances

Helps you work smarter not harder


6. A new telephone pole needs to be placed near the road at point C so that the length of telephone cable (AC + CB) is a minimum distance. Two houses are at positions A and B. Where should you locate the telephone pole?

A B

C

A’


1) reflect A2) connect A’ and B3) mark C

GSP

http://7.2.gsp/

http://7.2.gsp/

Line of Symmetry

1 image reflects onto itself

7. How many lines of symmetry does the figure have?

1 23

4

5

678

8. How many lines of symmetry does the figure have?

2

m A = can be used to calculate the angle between the mirrors

in a kaleidoscope

n = the number of lines of symmetry

180˚ n

1 23

4

5

6

78

180˚

8= 22.5˚

http://kaleidoscopeheaven.org

180˚

9= 20˚

10. Find the angle needed for the mirrors in this kaleidoscope.

180˚

4= 45˚

Project?1) Identify a reflection in a flag

2) Identify a line of symmetry

Reflection

Line of Symmetry

Reflection

Line of Symmetry

Section 7.2 Practice Sheet !!!

Lesson 7.3Rotations

students need tracing paper

Today, we will learn to…> identify and use rotations

Rotation

Angle of Rotation?

Center of Rotation?

Direction of Rotation?

Clockwise rotation

of 60°

Center of Rotation?

Angle of Rotation?

60˚60˚

Counter-Clockwise rotation

of 40°40°

40°

Theorem 7.3A reflection followed by a

reflection is a rotation.

If x˚ is the angle formed by the lines of reflection, then the angle of rotation is 2x°.

A

B’

A’

A’’

B’’

B 2x˚x˚

1. What is the degree of the rotation?

140˚

70˚

AA’

A’’

2. What is the degree of the rotation?

110˚

A A’

A’’

?125˚55˚

3. Use tracing paper to rotate ABCD 90º counterclockwise about the origin.

B (4, 1)Figure ABCD

Figure A'B'C'D'

A (2, –2)

A’ (2, 2) B ‘ (–1, 4)

C (5, 1)

C ‘ (–1,5)

D (5, –1)

D ‘(1, 5)

Rotational Symmetry

A figure has rotational symmetry if it can be mapped

onto itself by a rotation of 180˚ or less.

I had another dream….

6. Describe the rotations that map the figure onto itself.

= 45˚360˚

81

2

3

45

6

7

8

___ rotational symmetry 45˚

Describe the rotations that map the figure onto itself.

360 2

= 180˚ 1

2 ____ rotational symmetry 180˚

Describe the rotations that map the figure onto itself.

___ rotational symmetryno

Describe the rotational symmetry.

12

3

4

5

6

3606 = 60˚

60˚ rotational symmetry

Which segment represents a 90˚clockwise rotation of AB about P?

CD

Which segment represents a 90˚counterclockwise rotation of HI about Q?

LF

Project?1) Identify a rotation in a flag

2) Identify rotational symmetry in a flag

Rotation

60° Rotational symmetry

A

B

C

D

E

J

P KM

H F

G

L

Section 7.3 Practice!!!

Lesson 7.4Translations and

Vectors

Today, we will learn to…> identify and use translations

Translation

One reflection after another in two parallel lines creates a translation.

m n

THEOREM 7.5

PP '' is parallel to QQ''

k mQ

P

Q '

P '

Q ''

P ''

PP '' is perpendicular to k and m.______________

_______

k mQ

P

Q '

P '

Q ''

P ''

2d

d

The distance between P and P” is 2d, if d is the distance between

the parallel lines.

Name two segments parallel to YY”

XX”

ZZ”

Find YY”

6 cm

12 cm

XX”=

ZZ”=

12 cm

12 cm

A translation maps XYZ onto which triangle?

X”Y”Z”

Name two lines to XX”

line k

line m

A translation can be described by coordinate notation.

(x, y) (x + a, y + b) describesmovement

left or right

describesmovementup or down

(x, y) (x + 12, y - 20) means to translate the figure…

right 12 spaces & down 20 spaces

– + + –

1. (x, y) (x + 1, y – 9)

Use words to describe the translation.

2. (x, y) (x – 2, y + 7)

right 1 space , down 9 spaces

left 2 spaces, up 7 spaces

(x, y) (x + 5, y – 3)

3. left 5, down 10

Write the coordinate notation described.

4. up 6

(x – 5, y – 10)

(x, y + 6)(x , y)

(x , y)

5. Describe the translation with coordinate notation.

-2

+3

+3

-2

(x,y) (x – 2, y + 3)

6. Describe the translation with coordinate notation.

-7-2-2 -7

(x,y) (x – 7, y – 2)

-2 -7-2 -7

7. A triangle has vertices (-4,3);(0, 4); and (3, 2). Find the

coordinates of its image after the translation (x, y) (x + 4, y – 5)

(-4, 3) (-4 + 4, 3 – 5)

(3, 2) (3 + 4, 2 – 5)

(0, 4) (0 + 4, 4 – 5)

(7, -3)

(4, -1)

(0,-2)

Graphically, it would be…

(x, y) (x + 4, y – 5)

(-4, 3) (0, -2) (0, 4) (4, -1)

(3, 2) (7, -3)

preimage image (x, y) (x + 6, y – 2)

8. Find the image of (-4, 5)

9. Find the preimage of (9, 5)

(2, 3)

(3, 7)

(-4, 5) (-4 + 6, 5 – 2) ( __, __ )

( _ , _ ) ( x + 6, y – 2) ( 9, 5 ) x + 6 = 9 y – 2 = 5

A vector is a quantity that has both direction

and magnitude (size).

A vector can be used to describe a translation.

3 units up

5 units right

initial point

terminal pointBA

5

4

2 B

A

5 3,

The vector component form combines the horizontal and

vertical components.

5 3,

(x, y) (x + 5, y + 3)

Write this in coordinate notation form

4

2

-2

-4

-5 5

D'

C'B'

A'

D

B

A

C

10. What is the component form of the vector used for this translation?

4 2,

11. Name the vector and write its component form.

XY

X

Y5, 3

Write this in coordinate form.

(x,y) (x + 5, y – 3)

12) Describe the translation which maps ABC onto A’B’C’ by writing the translation in coordinate form and in vector component form.

A(3,6); B(1,0); C(4,8); A’(1,2); B’(-1,-4); C’(2,4)

(x, y) (x – 2, y – 4) – 2, – 4

Project?1) Identify a translation in a flag

Translation

Project?Two Objects

RequiredOne Object

OnlyReflection Line of Symmetry

RotationRotational Symmetry

Translation

Lesson 7.5Glide Reflections and

Compositions

students need worksheets and tracing paper

glide reflection

Example #1 Example #2

To be a “glide” reflection, the translation must be parallel

to the line of reflection.

NOTa glide reflection

NOTa glide reflection

These are just examples of a translation followed by a reflection.

Two or more transformations are combined to create a

composition.

A

A (2, 4) A’ ( , ) A’’ ( , )

1. translation: (x,y) (x, y+2) reflection: in the y-axis

2 6 -2 6

A’A”

2. reflection: in y = x translation: (x,y) (x+2, y-3)

A

A (-3, -2) A’ ( , ) A’’ ( , )

-2 -3 0 -6

A’A”

A

A”

B

A’

B’

B’’

A’’ (-1,- 4) and B’’ ( 2,- 1)

3. translation: (x,y) (x-3, y) reflection: in the x-axis

A (2, 4)

and B (5, 1)

4. translation: (x,y) (x, y+2) reflection: in y = -x

A (0, 4)

and

B (3, 2).

A’’ (-6, 0) and B’’ ( -4,-3)

BA B’A’

B”

A”

5. Describe the composition.

Reflection: in x-axisTranslation:(x,y) (x + 6,y + 2)

6. Describe the composition.

Reflection: in y = ½ Rotation: 90˚ clockwise about (1,-3)

PracticePractice Practice

How do we get better?

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Lesson 7.1 Rigid Motion in a Plane