Starter• Convert the following:• 4000 m = __________km• 20 mm = __________cm• 100 cm = __________ m• 45 cm = __________ m• 5 km = __________ m

mm cm m km

÷ 10 ÷ 100 ÷ 1000

× 10 × 100 × 1000

421

0.45

5000

GeometryTransformations

B

CA

A` C`

B`

Why is it important that an airplane is symmetrical?

Are the freight containers mirror images of each other?

How are the blades of the engine symmetrical?

ReflectionRotation

Which aircraft does not have a symmetrical seating plan?

Is it possible for a symmetrical aircraft to have an odd number of seats in a row?

Which aircraft has a 2-3-2 seating plan in economy class?

Symmetry

Axes of Symmetry• A line of symmetry divides a shape into two

parts, where each part is a mirror image of the other half.

• Example:

2 axes of symmetry

Line Symmetry - How many axes of Symmetry can you find?

Note 1: Order of Rotational Symmetry• The order of rotational symmetry is how many

times the object can be rotated to ‘map’ itself. (through an angle of 360° or less)

Rotational order of Symmetry

Which one of these cards has a Rotational Order of Symmetry = 2?

Note 1: Total order of symmetry

Total order of Symmetry

=Number of Axes of Symmetry

+Order of Rotational Symmetry

(Line Symmetry)

Shape Axes of Symmetry

Order of Rotational Symmetry

Total Order of Symmetry

4

0

12

1 1 2

2 2

4 8

6 6

Task !

Choose 3 objects in the room and describe their axes of symmetry, order of rotational symmetry and total order of symmetry.

Can you find an object with a total order of symmetry greater than 4 ?

Note 1: Total order of symmetry

• Total order of symmetry = the number of axes of symmetry + order of rotational symmetry.

The number of axes of symmetry is the number of mirror lines that can be drawn on an object.

The order of rotational symmetry is how many times the object can be rotated to ‘map’ itself.

(through an angle of 360° or less)

IWB Ex 27.01 pg 745-747 Ex 27.02 pg 750

Note 2: Reflection

• A point and its image are always the same distance from the mirror line

• If a point is on the mirror line, it stays there in the reflection. This is called an invariant point.

ReflectionTo draw an image:• Measure the perpendicular distance from each point to the

mirror line.• Measure the same perpendicular distance in the opposite

direction from the mirror line to find the image point. (often it is easier to count squares).

e.g. Draw the image of PQR in the mirror line LM.

IWB Ex 26.01 pg 704-705

Analyze the

Notice the letter B, H and E are unchanged if we take their horizontal mirror image? Can you think of any other letters in the alphabet that are unchanged in their reflection?

What is the longest word you can spell that is unchanged when placed on a mirror?

Can you draw an accurate reflection of your own name?

IWB Ex 26.02 pg 707-708

To draw a mirror line between a point and it’s reflection:

1. Construct the perpendicular bisector between the point and it’s image.

e.g. Find the mirror line by which B` has been reflected from B.

Practice Drawing a Reflections and mirror

lines!• Count squares or measure

with a ruler

• Handouts – Reflection, Mirror lines

• Homework - Finish these handouts.

Draw the Mirror lines for these shapes using a compass

What are these equivalent angles

of Rotation?

• 270° Anti clockwise is _______ clockwise• 180 ° Anti clockwise is ______ clockwise• 340 ° Anti clockwise is _______ clockwise

Rotations are always specified in the anti clockwise direction

Drawing Rotations

A

B C

D B’

D’ C’

¼ turn clockwise =

90º clockwise

Rotate about point A

To draw images of rotation:• Measure the distance from the centre of rotation to

a point.• Place the protractor on the shape with the cross-

hairs on the centre of rotation and the 0o towards the point.

• Mark the wanted angle, ensuring to mark it in the anti-clockwise direction.

• Measure the same distance from the centre of rotation in the new direction.

• Repeat for as many points as necessary.

Examples• Rotate flag FG, 180

about O

• Draw the image A`B`C`D` of rectangle ABCD if it is rotated 90o about point A.

A

Rotation• In rotation every point rotates through a

certain angle about a fixed point called the centre of rotation.

• Rotation is always done in an anti-clockwise direction.

• A point and it’s image are always the same distance from the centre of rotation.

• The centre of rotation is the only invariant point.

• Rotation game

By what angle is this flag rotated about point C ?

180º

Remember: Rotation is always measured in the anti clockwise direction!

C

By what angle is this flag rotated about point C ?

90º

C

By what angle is this flag rotated about point C ?

270º

C

Define these terms• Mirror line• Centre of rotation• Invariant

What is invariant in• Reflection

• rotation

The line equidistant from an object and its imageThe point an object is rotated aboutDoesn’t change

The mirror lineThe size of angles and sidesThe area of the shapeCentre of rotationSize of angles and sidesThe area of the shape

Translations

Each point moves the same distance in the same direction

There are no invariant points in a translation(every point moves)

Vectors

• Vectors describe movement

( )xy

← movement in the x direction (left and right)

← movement in the y direction (up and down)

Each vertex of shape EFGH moves along the vector

( )-3

-6

To become the translated shape E’F’G’H’

• Translate the shape ABCDEF by the vector to give the image A`B`C`D`E`F`.

( )- 4

- 2

Enlargement

• In enlargement, all lengths and distances from a point called the centre of enlargement are multiplied by a scale factor (k).

To draw an enlargement

1. Measure the distance from the centre of enlargement to a point.

2. Multiply the point by the scale factor and mark the point’s image point.

3. Continue for as many points as necessary.

Enlarge the ABC by a scale factor of 2 using the point O as the centre of enlargement.

To find the centre of enlargement1. Join each of the points to it’s image point. 2. The point where all lines intersect is the centre of

enlargement.

B

CA

A` C`

B`

B

CA

A` C`

B`

Calculating the scale factorTo calculate the scale factor (k) we use the

formula :

lengthobject

lengthedenlk

arg

Scale factor (k) =

=

=

P QPQ` `

1624

23

Negative Scale Factors

When the scale factor is negative, the image is on the opposite side of the centre of enlargement from the object.

To draw images of negative scale factors:1. Measure the distance from the centre of rotation to

a point.2. Multiply the distance by the scale factor.3. Measure the distance on the opposite side of the

centre of rotation from the point.4. Repeat for as many points as necessary.

Enlarge XYZ by a scale factor of –2 about O.

Z

YX

O Y

Z

X

Do Now:1. Match up the terms with the correct definition2. Write them into your vocab list

Image

Mirror line

Perpendicular

Bisector

Reflection

Cuts a line into two equal parts (cuts it in half) – also called the mediator

The transformed object

A transformation which maps objects across a mirror line

A line which intersects a line at right angles

The line in which an object is reflected

Reflect the shape in the red mirror line, translate the image by the vector

Enlarge the image scale factor 3, centre PRotate 45o , centre A’’’

3

4

6

A

P

Do Now: What transformations are in each example?

1. 2. 3.

4. 5.

ReflectionEnlargement

Rotation

Translation Reflection

Do Now

1.) Reflection is a transformation which maps an object across a __________.2.) In rotation, the only invariant point is called the __________________.3.) A _______ describes the movement up and down, and across, in a translation.4.) All of the _________ points in reflection lie on the mirror line.5.) The area of the object, the size of the angles and the length of the sides are invariant in both rotation and ___________

mirror line

centre of rotationvector

invariant

reflection

invariant, centre of rotation, reflection, vector, mirror line

Koru Design

Using the templates provided, or your own, create a pattern of at least 5 transformations, which consists of at least:

• One reflection• One translation• One rotation

How to Write Instructions

• Reflect ABCD through the mirror line (M)

• Translate the image A’B’C’D’ 4 cm to the right

• Rotate the image A’’B’’C’’D’’, 90o counter clockwise about the point P.M

4 cm

P

Now its your turn!Write the appropriate instruction for each

transformation, in the order that it appears.

M

5 cm

6 cm

A

A’

A’’

A’’’ A’’’’

P

Writing Instructions

x-10 -5 5 10

y

-10

-5

5

10

1.) Label your object

2.) Reflect image about mirror line M

3.) Translate the imagea’b’c’d’ by the vector

( ) → ( )

4.) Rotate the image a’’b’’c’’d’’ about point P 90º

M

a

b c

d a’

b’c’

d’1

x-7

d’’

c’’ b’’

a’’

P

3

4d’’’c’’’

a’’’b’’’

4y

2

Writing Instructions

x-10 -5 5 10

y

-10

-5

5

10

1.) Construct equilateral triangle abc(Label your object)

2.) Rotate object abc 90º about point P to give a’b’c’.

3.) Reflect image a’b’c’ through mirror line M to give image a’’b’’c’’

4.) Translate the object a’’b’’c’’ by the vector

( ) to give image a’’’b’’’c’’’ M

a

b

c

a’

b’

c’

1

48

c’’

b’’

a’’

P

2 3

4

c’’’

a’’’

b’’’

Koru Design

Using the templates provided, or your own, create a pattern of at least 5 transformations, which consists of at least:

• One reflection• One translation• One rotation* Write a set of instructions so another student

could reproduce your pattern.

Choose a Task

Design a LogoUsing at least 2 different construction techniques and at least 2 different types of transformations with Instructions

(You have 20 minutes to complete it!)

or“The Backyard”Follow the set of instructions to complete a plan of a backyard.

Exchange your Work

Logo CreatorsFollow your partners instructions and recreate their logo

Backyard DesignersMark your partners work against the marking sheet