2

2.1 Translation 1. Translation is a transformation that moves all the points on a plane through the same

distance and in the same direction.

2. Properties of a translation

a) the shape, size and orientation of the object and the image are the same

b) every point is moved through the same distance and in the same direction

3. A translation is usually expressed in the form , where h represents the horizontal

movement parallel to the x-axis and k represents the vertical movement parallel to

the y- axis

Example 1

i) A is mapped onto B under a

translation or each point in

triangle A is moved 4 units to the right

followed by 2 uints downwards

ii) Under the same translation,

image of H is H(7,5). Exercise :

i) Draw the image for each of the

following object.

ii) State the coordinates of the image for each of

the points under the given translation.1. Translation

Answer : A( )

2. Translation

3. Translation

Answer:

4. Translation

Answer : 5. Translation

EMBED Equation.3 Answer:

2.2 Reflection

Reflection is a transformation which reflects all the points on a plane in a line

called the axis of reflection. Properties of reflection :

i) the shape and size of the object and its image are the same .(congruent)

ii) the orientation of the image is laterally inverted compared to the object.

iii) the position of any point lying on the axis of reflection does not change.

2.2.1 To determine the image under a reflection

Draw and label the image under a reflection for each of the following.

Example :

Exercise:

1.

2

3

4

5

2.2.2 To determine the axis of reflection

Example: Draw the axis of reflection

axis of reflectionExercise

1.

2.

3

2.2.3 State the axis of reflection. Example:

Axis of reflection : x = 2Exersice

1 Axis of reflection :

2.

Axis of reflection :3

Axis of reflection :

2.3 Rotation

A rotation is a transformation which rotates all the points on a plane :

about a fixed point (known as centre of rotation) Centre

through a given angle Angle

in the clockwise or anticlockwise direction. Direction

The properties of rotation :

i) the shape of the object and the image are the same

ii) the size of the object and image are the same

iii) the orientation of object and image remain the same

iv) the centre of rotation is the only point that does not change its position.

Since full rotation is 3600

900 clockwise = 2700 anticlockwise

2700 clockwise = 900 anticlockwise

1800 clockwise = 1800 clockwise ( for a rotation 1800 it is not necessary to mention the

direction of rotation)

Example :

Draw the image of M under a Rotation of 900 clockwise about A.

Exercise Draw the image for each of the object

under the rotation given :

2. Rotation of 1800 anticlockwise about H.

3. Rotation of 900 clockwise about A.

4. Rotation of 900 clockwise about B .

5. Rotation of 900 anticlockwise about C

2.3.2 Determine and mark the centre of rotation

The intersection of two or more lines of perpendicular bisectors is a centre of rotation.

i) Join point D to D and construct the

perpendicular bisector of line DD.ii) Join point C to C and construct the

perpendicular bisector of line CC.iii) Extend the two lines until they

intersect each other.

The point of intersection of these two

perpendicular bisectors is the centre of rotation.

Exercise : Determine and mark the centre of

rotation for each of the following

diagram

1

2

3.

4

To determine centre,angle and direction of rotation

Centre : (0,0)

Angle : 900 , Direction : clockwise

2. Centre : ..

Angle :.

Direction :

Exercise :

1.

Centre : ..

Angle : .

Direction : ...

4

Centre :

Angle :

Direction : ..

5

Centre : ..

Angle : .

Direction : ...

2.4 ENLARGEMENT An enlargement is a transformation which has a fixed point call the centre of enlargement.

All the points on the plane move at a constant ratio from the centre.

The ratio is known as the scale factor.

The image produced is always similar to the object.

To determine the centre of enlargement

and the scale factor

Example: i) Mark the centre of enlargement

ii) State the factor scale of

enlargment

Answer

i) scale factor : 2

Exersice

1.

Answer :

i)

2.

i) Scale factor :

3.

i) Scale factor :

6.

Scale factor :5.

Scale factor :

CHAPTER 2 : TRANSFORMATION I, II

x

H

O 2 4 6

y

EMBED Equation.3

A

H

B

6

4

2

y

O 2 4 6

A

6

4

2

y

x

O 2 4 6

A

6

4

2

B

y

x

Answer:

O 2 4 6 x

C

6

4

2

y

y

2 4 6

C

6

4

2

x

D

6

4

2

2 4 6

y

C

6

4

2

2

4

6

C

2

4

6

C

-4 -2 0 2 4

4

2

-2

x

-4

A

A

A'

P

P'

y =1

y

In the diagram:

A' is the image of A under a reflection

of y axis or x = 0

P is the image of P under reflection

of line y =1

Axis of reflection

-4 -2 0 2 4

4

2

-2

x

-4

A'

A

y

-4 -2 0 2 4

4

2

-2

x

-4

B

y

Axis of reflection FFF

Y

-4 -2 0 2 4

4

2

-2

X

-4

P

Q

R

S

B

B'

AA

C

C'

D

D'

-4 -2 0 2 4

4

2

-2

x

-4

y

0 1 2 3 4 5 6 7

6

4

x

2

y

-4 -2 0 2 4

4

2

-2

x

-4

y

6

4

2

x

0 1 2 3 4 5 6 7

y

C

A

D

P

P'

O

The diagram shows that P is mapped onto P through a rotation about O of an angle in clockwise direction

y

O 2 4 6

A

6

4

2

x

M

M'

90

1. Rotation of 900 anticlockwise about A

A

2

6

4

O 2 4 6

x

H

O 2 4 6

A

6

4

2

x

y

O 2 4 6 x

A

6

4

2

y

O 2 4 6

A

6

4

2

B

x

y

O 2 4 6

C

6

4

2

x

y

y

A B

D C

C D

B A

Perpendicular bisectors

Centre of

rotation

Q

R

P

Q

R

P

A

B

A B bbB

P

Q

R

P

Q

R

S

S

B

C

C

B

A

A

-4 -2 0 2 4

4

2

-2

x

-4

y

A

B

A

B

B

A

A

B

-4 -2 0 2 4

4

2

-2

x

-4

y

-4 -2 0 2 4

4

2

-2

x

-4

y

-4 -2 0 2 4

4

2

-2

x

-4

y

T

T

x

y

-4 -2 0 2 4

4

2

-2

-4

P

P

C

P

Q

R

P

Q

Triangle PQR is the image of

triangle PQR under an enlargement

with centre C and scale factor (k) = 2

Scale factor, k can be calculate by

length of side of mage

length of side of object

EMBED Equation.3

R

k =

k = EMBED Equation.3

= EMBED Equation.3

Area under enlargement

For any enlargement with scale factor k.

Area of image = k2 area of object

C

-4 -2 0 2 4

4

2

-2

x

-4

y

A

A

A

R

R

A

B

C

D

A

B

C

D

B

B

M

M

Transformation I,II10

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_1219426775.unknown

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_1225181971.unknown

_1300244542.unknown

_1221811027.unknown

_1219426894.unknown

_1217201712.unknown

_1217199003.unknown

_1217201250.unknown

_1217197664.unknown

_1150215306.unknown