transformations iii 2011(old)

8/3/2019 Transformations III 2011(Old)

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A

CHAPTER 3 :

TRANSFORMATIONS III

revision TRANSLATION

A translation is a transformation that takes place when all pointsin a plane are moved in the same direction through the same distance

Object A moves in the samedirection for the same distancein the same plane

A

A


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DESCRIBING TRANSLATION

xy

A translation means a movement ofx units

parallel to the x-axis and a movement ofy unitsparallel to the y-axis

The translation shown in the diagram 1

is4

2

A

A

DIAGRAM 1

x

y


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Translation

y

x

x represents the movement to the right or left which

is parallel to x-axis

y represents the movement upwards or downwards

which is parallel to the y-axis


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Translation

4

3

represents a movement of a distance3 units to the right and 4 units upwards

Translation

4

3

represents a movement of a distance

3 units to the left and 4 units downwards


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y

x0 1 2 3 4 5

5

4

3

2

1

A

A

31

Write the coordinates ofthe image of A(1,3) under

a translation

(1 , 3)

(4 , 4)

3

1 Coordinate of image,A= (4 , 4)


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y

x0 1 2 3 4 5

5

4

3

2

1

A

A

mn

m

If the object A (x , y) ismoved under

a translation

then its image will beA(x + m , y + n)n

Coordinate of image,A= (1 + 3 , 3 + 1)= (4 , 4)

(1 , 3)

(4 , 4)

3

1


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Determining the coordinates of the image

under a translation

POINT TRANSLATION IMAGE

(-2,5)

(0,-3)

(6,-4)

(-8,-7)

(-9,1)

5

2

2

3

6

0

1

7

1

5

)10,0()55,22( !

? A )1,3(23),3(0 !

)2,6()64,06( !

? A )8,1()1()7(,78 !

? A )0,14()1(1),5(9 !


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A

A

4

3

The pointAis the image ofpoint A under a translation

4

3


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A

A

4

-5


4

-5


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A

A

-5

5


-5

5


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AA


-5

-5

0


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A

-5

-2

The A is the image of A under a translation

-5

-2

A


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y

x0 1 2 3 4 5

5

4

3

2

1A B

C

A B

C

3

1

3

1

TriangleABC

is the image oftriangle ABC under

a translation


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Important Tips

Find the translation that maps the point (2 , 4) onto the point (0 , 1)

Translation =0

1

2

4-

-2

-3=

Image Object

To find a translation under these conditions,subtract the coordinates of the object fromthe coordinates of its image.


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Revision Translation

Properties ofTranslation

The object and its image are identical in shape, size and orientation

All points in the object change their positions to those of the image

Every point is moved in the same direction for the same distance

All lines in the image, if any, are parallel and equal in length to those

of the object


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CHAPTER 3 :

TRANSFORMATIONS III

revisionREFLECTION

Reflection is a form of transformation in which all points in a plane arelaterally inverted in a line called the line of reflection or the axis of

reflection

All points in object A are reflectedin the axis of reflection

A A

Axis of reflection


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CHAPTER 3 :

TRANSFORMATIONS III

revisionREFLECTION

The coordinates of an image can be determined if the coordinatesof the object are provided

A

x

y

1 2 30

1

2

3

-1-2-3

With reference to the diagram, find

the coordinates of the image of

point A(3,2) under a reflection in

y-axis,


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CHAPTER 3 :

TRANSFORMATIONS III

solution

A

x

y

1 2 30

1

2

3

-1-2-3

The perpendicular distance of the

object and its image from the axisof reflection is equal A

The coordinates of the image of

point A is (-3,2)


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

0

Find the coordinates of the image of point

Q (3,5) under a reflection in the line x = 6

Q(3,5) Q(9,5)

x=6


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

0


Q (9,5) under a reflection in the line x = 6

Q(3,5) Q(9,5)

x=6


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Find the coordinates of the image of point Q (3,5)

under a reflection in the line x = 6

Distance of the object from the axis of reflection

= 6- 3

= 3

Image of point Q(3,5) under a reflection in

the line x = 6

= (6 +3,5)

= (9,5)


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Find the coordinates of the image of point Q(9,5)

under a reflection in the line x = 6


= 9 - 6

= 3


the line x = 6

= (6 - 3,5)

= (3,5)


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

0


Q (7,9) under a reflection in the line y = 6

Q(7,9)

Q(7,3)

y=6


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Find the coordinates of the image of point Q(7,9)

under a reflection in the line y = 6


= 9 - 6

= 3


the line y = 6

= (7,6 - 3)

= (7,3)


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1

2

3

4

5

6

0 x

y

-1

-2

-3

-4

-5

-

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

Find the coordinates of the image

of point Q (2,3) under a reflection

in the x-axis Q(2,3)

Q(2,-3)


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1

2

3

4

5

6

0 x

y

-1

-2

-3

-4

-5

-

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


of point Q (-4,3) under a

reflection in the x-axisQ(-4,3)

Q(-4,-3)


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1

2

3

4

5

6

0 x

y

-1

-2

-3

-4

-5

-

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



in the y-axis

Q(3,5)Q(-3,5)


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1

2

3

4

5

6

0 x

y

-1

-2

-3

-4

-5

-

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



in the y-axis

Q(3,-2)Q(-3,-2)


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

0


Q (3,8) under a reflection in the line y = x

Q(3,8)

Q(8,3)

y=x


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

0



Q(3,8)

Q(8,3)

y=x


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

0



Q(3,8)

Q(8,3)

y=x


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

0



Q(10,5)

Q(5,10) y=x


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1

2

3

4

5

6

0 x

y

-1

-2

-3

-4

-5

-

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


of point Q (-3,4) under a

reflection in the line y=x

Q(-3,4)

Q(4,-3)

y=x


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x

y

-6 -4 -2 0 2 4 6

2

4

6

8

10

-8


Q (-5,3) under a reflection in the line y = -x

Q(-3,5)

Q(-5,3)

y=-x

y


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1

2

3

4

5

6

0 x

y

-1

-2

-3

-4

-5

-

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



in the line y=-x

Q(2,3)

Q(-3,-2)

y=-x


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under a reflection

POINT REFLECTION

IN THE

IMAGE

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

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

(x,y) line y=x (y,x)

(x,y) line y=-x (-y,-x)


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under a reflection

POINT REFLECTION

IN THE

IMAGE

(-2,5) x-axis

(4,-3) y-axis

(6,-9) line y=x

(-8,-7) line y=-x

(-2,-5)

(-4,-3)

(-9,6)

(7,8)


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y

x

5

4

3

2

1

A B

C

B

C

FigureABCis theimage of figure ABC under areflection in the line x = 2

x = 2

0 1 2 3 4 5

DESCRIBING REFLECTIONEXAMPLE

When an object and itsimage under a certainreflection are given,the reflection can bedescribed

A


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x

y

0

4

3

2

1

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

A B

DESCRIBING REFLECTION

The point B is the image of point Aunder a reflection in the y-axis


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x

y

0

4

3

2

1

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

DESCRIBING REFLECTION

The B is the image of Aunder a reflection in the y-axis

BA


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y

x


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CHAPTER 3 :

TRANSFORMATIONS III

revisionROTATION

Rotation is a form of transformation in which all points in a planeare rotated through an angle in a certain direction about a point

The pointis called the centre of rotation while the angle is referred toas the angle of rotation

0

AA

In the diagram, point A is rotatedclockwise through an angle of 900

about the point 0

Centre of rotation

Angle of rotation

Direction of rotation


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The transformation R represents a 900 clockwise

rotation about the center(3,6). State the coordinates

of the image of the point Funder the transformation R.

x

y

2 4 6 8 10 12 14

2

4

6

8

10

0

F

F


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

0

F




F


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C

900 clockwise rotation

about the center C.


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C

900 clockwise rotation

about the center C.

C


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

0

F




F


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

0

F




F


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

0

F

The transformation S represents a 900 anticlockwise


of the image of the point Funder the transformation S.

F


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

0

F





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x

y

2 4 6 8 10 12 14

2

4

6

8

10

0

F




F(6, 2)


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The transformation R represents a 900 anticlockwise


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y

x0

4

2

2 4 6

-2

-4

-2-4

K

The transformation R represents a 900 anticlockwise


of the image of the point Kunder the transformation R.

K(4,-2)

0


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y

x0

4

2

2 4 6

-2

-4

-2-4

K

The transformationR represents a 1800 rotation about the center

(1,1). State the coordinates of the image of the point Kunder the

transformation R.

The transformation R represents a 1800 rotation about


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y

x0

4

2

2 4 6

-2

-4

-2-4

K

The transformation R represents a 180 rotation about

the center(1,1). State the coordinates of the image of

the point Kunder the transformation R.

K(-2,-1)


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A B

C

A

B C

(1) Construct the perpendicular bisectorof line AA

(2) Construct the perpendicular bisector

of line CC

(3) The point of intersection between thetwo perpendicular bisectors isthe centre of rotation

Centre of rotation

In the diagram below, ABC is the image of ABC

under a rotation. Determine the centre of rotation


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A B

C

A

BC

If the object and its image under a rotation are given, the centre, angle and direction

of the rotation can be determined.

A B

C

A

B C


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AB

C

A

BC



A B

C

A

B C


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A B

C

A

BC



Centre of rotation

A B

C

A

B C


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A B

C

A

BC

Centre of rotation




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A A

Determining The

Centre of Rotation

In the Diagram 4, point A is the imageof point A under the rotation through

900 anti clockwise. Determine the

centre of rotation.

DIAGRAM 4

centre of rotation


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Q(3,6)

Q(6,3)

In the Diagram, point Q is the image

of point Q under the rotation through

900 anticlockwise. Determine the centre of rotation.


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Q(3,6)

Q(6,3)

Centre

In the Diagram, point Q is the image

of point Q under the rotation through

900 anticlockwise. Determine the centre of rotation.


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Determining The Centre of

Rotation

In the Diagram 5,point Q is the image

of point Q under the

rotation through

90

0

anti clockwise.Determine the

centre of rotation.

DIAGRAM 5

QQ

centre of rotation


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Determining The Centre of Rotation

In the Diagram 7,point U is the image

of point U under the

rotation through

900 clockwise.

Determine the

centre of rotation.

DIAGRAM 7

UU

centre of rotation


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Determining The Centre of

Rotation

In the Diagram 8, pointW is the image

of point W under the

rotation through

900

clockwise.Determine the centre

of rotation.

DIAGRAM 8

WW

centre of rotation

Triangle ACG is the image of triangle PQR under


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

O

C

A

GP

RQ


a rotation. Determine the centre of rotation.



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x

y

2 4 6 8 10 12 14

2

4

6

8

10

O

C

A

GP

RQ

PC



(7, 0)


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

O

C

A

GP

RQ

PC



(7, 0)


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

O

C

A

GP

RQ

PC



(7, 0)



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x

y

2 4 6 8 10 12 14

2

4

6

8

10

O

C

A

GP

RQ

PC



(7, 0)


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y

xO

4

2

2 4 6

-2

-4

-2-4

A

B

C

D

H

E F

G

Trapezium EFGH is the image of trapezium ABCD

under a rotation. Determine the centre of rotation.

A

E

(1,1)


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y

xO

4

2

2 4 6

-2

-4

-2-4

A

B

C

D

H

E F

G



A

E

(1,1)

T i EFGH i th i f t i ABCD


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y

xO

4

2

2 4 6

-2

-4

-2-4

A

B

C

D

H

E F

G



A

E

(1,1)

T i EFGH i th i g f t i ABCD


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y

xO

4

2

2 4 6

-2

-4

-2-4

A

B

C

D

H

E F

G



A

E

(1,1)

Trapezium ABCD is the image of trapezium PQRS


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A B

CD

P Q

R

S

2

4

6

8

0 2 4 6



y

x


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T i ABCD i th i g f t i PQRS


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A B

CD

P Q

R

S

2

4

6

8

0 2 4 6



P

D

(1, 3)

y

x



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A B

CD

P Q

R

S

2

4

6

8

0 2 4 6

p g p Q


P

D

(1, 3)

y

x



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A B

CD

P Q

R

S

2

4

6

8

0 2 4 6



P

D

(1, 3)

y

x

CHAPTER 3 :


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CHAPTER 3 :

TRANSFORMATIONS III

revision ENLARGEMENT

Enlargement is a form of transformation involving invariant pointcalled the centre of enlargement

O

AA

B

C

B

C

Under an enlargement,all points on a plane,except the centre ofenlargement, will move

from the centre ofenlargement according

to a certain ratio knownas the scale factor

For instance,

CHAPTER 3 :


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CHAPTER 3 :

TRANSFORMATIONS III


Under enlargement with centre O and scale factor k , ABC ismapped onto ABC.

O

AA

B

C

B

C

The scale factor, k

= OA = OB = OC

OA OB OC


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Scale Factor

Scale Factor= Distance of a point on the image from the centre of

enlargementDistance of the corresponding point on the object from the

centreof enlargement

Scale Factor

= Length of one side of the imageLength of the corresponding side of the object


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PROPERTIES OF ENLARGEMENT

The object and its image are similar

Every side of the object is parallel to the corresponding side of its image

If A is the image for A and P is the centre of enlargement, then

the scale factor k = PA

PA

If the scale factor > 1, then the image is bigger than the object

If the scale factor < 1, then the image is smaller than the object

If the scale factor < 1, then the image is smaller than the object

If the scale factor = 1, then all points on the plane remain in their respective

positions

PROPERTIES OF ENLARGEMENT

S l F t k 1


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Scale Factor, k > 1

The image is bigger than the object

S l F t k 1


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Scale Factor, k < 1

The image is smaller than the object


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0 < k < 1

k = - 1k < - 1

Scale Factor, k


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BO

DESCRIBING ENLARGEMENT

B

Square B is the image of square Aunder an enlargement with centreat point O and a scale factor of 2

A



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y

x0 1 2 3 4 5

A B

C

A B

C

TriangleABCis the

image of triangle ABCunder an enlargementwith centre (1,1) and ascale factor 2

k > 0, k > 15

4

3

2

1



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y

x

0 1 2 3 4 5

A B

C

A B

C

TriangleABCis theimage of triangle ABCunder an enlargementwith centre (1,1)

and a scale factor

k > 0, k< 1


21

5

4

3

2

1



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An enlargementwith centre O anda scale factor of 3

OABC is the image of

OA1B1C1



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An enlargement

with centre P anda scale factor of 2

PQRST is the imageOf PKLMN



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An enlargement

with centre P anda scale factor of 1

2

PKLMN is the imageOf PQRST



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y

x

AB

C

A B

C

k < 0, k = -1

TriangleABCis theimage of triangle ABCunder an enlargement

with centre (3,3) and ascale factor -1

0 1 2 3 4 5

5

4

3

2

1



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y

x

AB

C

A B

C

k < 0, k < -1

TriangleABCis theimage of triangleABC under anenlargement

with centre (4,2)and a scale factor -2

0 1 2 3 4 5

1

2

3

4

-1

-2

yDetermining The Centre of Enlargement


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y

x

A B

C

A B

C

TriangleABCis theimage of triangle ABC

Join the points of theobject to thecorresponding pointsof the image

Extend these linesuntil they intersectone another at thecentre of

enlargement, O

O

0 1 2 3 4 5

5

4

3

2

1

AREA OF THE IMAGE


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P Q Q

S

S

R

R

= K2 x Area of The Object

By Counting

Area of PQRS = 24 units2

In the diagram above, PQRS is the image ofPQRS under an enlargement with centre P andscale factor 2

Area of PQRS = 6 units2

AREA OF THE IMAGE

= K2 x Area of The Object

= 22 x 6 units2

= 24 units2

AREA OF THE IMAGE


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Scale Factor (k) = 3

Area of the object

= 24.5 cm2

Area of the image

=22

5.2205.243 cm!x

OA1B1C1 is the object

AREA OF THE IMAGE


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Area of the object

=410.4 cm2

Area of the image

=26.454

3

1( cm!10.4x)2

OA1B1C1 is the image

AREA OF THE COLORED REGION


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Area of the object

= 24.5 cm2

Area of the red colored

region

2

2

196

5.24)5.243(

cm

x

!

!

OA1B1C1 is the object

AREA OF THE OBJECT PQRST is the image


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Scale Factor (k) = 2Area of the greencolored region

= 124.5 cm

2

Area of the object (y)

2

2

5.41

5.1243

5.124)2(

cmy

y

yy

!

!

!x

Of PKLMN

AREA OF THE OBJECT


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Scale Factor (k) = 2Area of the image= 220 cm2

Area of the object

2

2

55

2

220

cm!

!

PQRST is the imageOf PKLMN

COMBINATION OF TWO TRANSFORMATIONS


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COMBINATION OF TWO TRANSFORMATIONS

Suppose P and Q represent two transformations

PQ means transformation Q is carried out first, followedby transformation P

QP means transformation P is carried out first, followedby transformation Q

y

EXAMPLE3


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y

x

A B

C

A B

C

3

1

TriangleABCis the image of

triangle ABC underthe combinedtransformation QP

AB

C

P is a translation3

1

Q is a reflection in theline x = 3

X = 3

Draw the image oftriangle ABC under

the combinedtransformation QP

5

4

3

2

1

0 1 2 3 4 5

y1

EXAMPLE


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y

x

A B

C

A

BC

1

2

A

BC

TriangleABCis the image oftriangle ABC underthe combinedtransformation PQ

P is a translation 2

Q is a clockwise rotationof 900 about the centre(1,2)

Draw the image of

triangle ABC underthe combinedtransformation PQ

5

4

3

2

1

0 1 2 3 4 5

y

13. (a)Diagram 5 shows the point Kon a Cartesian plane.


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

0

F

DIAGRAM 5

y

13. (a) Diagram 5 shows the point Fon a Cartesian plane.


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

0

F

DIAGRAM 5

5


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Transformation S is a translation .

Transformation T is a reflection in the x = 9.

(i) State the coordinates of the image of point Funder

transformation S.

(ii) State the coordinates of image of point F under

transformation TS. [3 marks]

2

5


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(b) Diagram 6 shows three triangle PQR, ACG and EFG on a

l


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x

y

E

2 4 6 8 10 12 14

2

4

6

8

10

O

F

C

A

GP

RQ

DIAGRAM 6

Cartesian plane.

T i l ACG i th i f t i l PQR d t f ti


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TriangleACG is the image of triangle PQR under transformation

V.

Trapezium EFG is the image of triangleACG under transformationW.

(i) Describe in full transformation :

(a) V

(b)W

[6marks]

(ii) Given that the area of triangle EFG represents a region

of area 72 unit2.Calculate the area, in unit2, of the region

represented by triangle PQR.[3 marks]

13(b) i (a)


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x

y

2 4 6 8 10 12 14

2

4

6

8

10

O

C

A

GP

RQ

PC

Rotation through

900 clockwise

y13(b) i (a)


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x2 4 6 8 10 12 14

2

4

6

8

10

O

C

A

GP

RQ

PC

(7, 0)

V = Rotation through 900

clockwise about the point(7,0)

13(b) i (b) W = An enlargement with centre G and

l f t f 3


122/133

x

y

E

2 4 6 8 10 12 14

2

4

6

8

10

O

F

C

A

GP

RQ

DIAGRAM 6

a scale factor of 3

6

2

Scale factor = 6 = 3

2

(ii) 32x Area of PQR = 72


123/133

x

y

E

2 4 6 8 10 12 14

2

4

6

8

10

O

F

C

A

GP

RQ

DIAGRAM 6

Area of PQR = 8 unit2

272cm

solution


124/133

13. (b) (i) (a) V = Rotation through 900 clockwise about

the point (7,0)

(b) W = An enlargement with centre G and

a scale factor of 3



P3

P3

K1

N1

solution


125/133

13. (b) (i) (a) V = Rotation, 900 clockwise, centre (7,0)

(b) W = Enlargement, centre G, scale factor 3



P3

P3

K1

N1

solution


126/133

13. (b) (i) (a) V = Rotation, 900 clockwise, centre (0,7)

(b) W = Enlargement, centre G, scale factor 3



P2

P2

K1

N1

y

2003No.13


127/133

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

8

6

4

2

G

H

J

H

H

y=3

13(a) (i)

K

(7,0 ) P2

y

( 8 )


128/133

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

8

6

4

2

G

K

H

J

y=3

H

H

13(a) (ii) (7,8 ) P2

y


129/133

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

8

6

4

2

D A

C B

E FE F

13 (b) (i) Reflection in the line AB P2

y13(b) (i) Rotation of 900 anticlockwise

b i (6 5) P3


130/133

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

8

6

4

2

D A

C B

E F

G

K

H

J

about point (6,5)


131/133

y

13 (c) (i) ( 6 2 ) P1


132/133

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

LP

N

D A

C B

13(c) (i) (6,2 )

(ii ) 325.8 = 36.2

32

P1

K1N1

M

8

6

4

2

13. ( a ) (i ) (7,0 ) P2


133/133

( ) ( ) ( , )

(ii ) (7,8 )

( b ) (i) Reflection in the line AB

(ii ) Rotation of900 anticlockwise

about (6,5)

( c ) (i ) (6,2 )

(ii ) 325.8 = 36.2

P2

P2

P2

P3

P1

K1N1

transformations iii 2011(old)

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