transformation iii

8/3/2019 Transformation III

1/19

CHAPTER 12 TRANSFORMATIONS III

12.1 Revision of the 4 types of transformations which you have learnt.

TRANSLATION

All the points on a given plane move along a straight line by the same distanceat the same direction.

The shape, size and orientation remain the same.

It is written as AA with a translation of

k

h.

REFLECTION

All the points of an object are reflected in a line called the axis of reflection orline of reflection.

It is written as AA with a reflection in the line ..

ROTATIONAll points on the object are rotated through a fixed angle at the same direction

about a fixed point.

The direction of rotation is either clockwise or anticlockwise.The fixed point about which the rotation takes place is called the centre of

rotation.

A Rotation is determined by (a) the centre of rotation(b) the angle of rotation

(c) the direction of rotation

ENLARGEMENT

All points on the object move from a fixed point (the centre of enlargement)according to a fixed ratio. (the scale factor)

Example

Transformations III 1


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ABCDEFGHIJ with a translation

43

.

ABCDE KLMNO with a reflection at the line x = 4.

ABCDE PQRST with a rotation of 90o clockwise about the point (-1, 5).

ABCDE AUVWX with an enlargement at point A(1, 2) with ascale factor of 2.

12.2Combination of two types of transformations

Symbol of combination of 2 transformations

P(A) represents the image of point A under transformation P.

PQ represents transformation Q followed by transformation P.

QP represents transformation P followed by transformation Q.

P2 represents two consecutive transformations of P.

Skills assessed To determine the image of a given point or shape under the combination of 2

transformations.

To find a single transformation which is equivalent to a combination of 2

given transformations.

Describe fully two consecutive transformations which map an object to itsimage.

Calculate the area of the image (or object) under an enlargement.

12.2a To Find the Image of a Point under the Combination of Translation and

Reflection

1. T = Translation

13

P = Reflection at y = 1.Find the image of A under (a) PT (b) TP.

2. T = Translation

16

P = Reflection at y = 1.Find the image of A under (a) PT (b) TP.

3. T = Translation

3

4

4. T = Translation

2

1


(a) A(2,4)A(5,3) A(5,-1) (a) A(1,3)A( , ) A( , )

(b) A(1,3)A( , ) A( , )(b) A(2,4)A(2,-2) A(5,-3)


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P = Reflection at y = 3.

Find the image of A under (a) PT (b) TP.

P = Reflection at x = 2.


5. T = Translation

23

P = Reflection at x = -1.


6. T = Translation

25

P = Reflection at y = x.



(a) A(4,-1)A( , ) A( , )

(b) A(4,-1)A( , ) A( , )

(a) A(-2,4)A( , ) A( , )

(b) A(-2,4)A( , ) A( , )

(a) A(-2,3)A( , ) A( , )

(b) A(-2,3)A( , ) A( , )

(a) A(2, 2)A( , ) A( , )

(b) A(2, 2)A( , ) A( , )


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12.2b To Find the Image of a Point under the Combination of Translation and

Rotation

1. T = Translation

13

R = Rotation 90o clockwise about point P

Find the image of A under (a) RT (b) TR.

2. T = Translation

2

4

R = Rotation 90o clockwise about point P.


3. T = Translation

3

2

R= Rotation 90o anticlockwise about point P


4. T = Translation

2

1

R= Rotation 90o anticlockwise about point P



(a) A(2,3)A(5,2) A(2,-3)

(b) A(2,3)A(3,0) A(6,-1)

b

(a) A(2,1)A( , ) A( , )

(b) A(2,1)A( , ) A( , )

(a) A(3,-1)A( , ) A( , )

(b) A(3,-1)A( , ) A( , )

(a) A(6,-1)A( , ) A( ,

(b) A(6,-1)A( , ) A( ,


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12.2c To Find the Image of a Point under the Combination of Reflection and

Rotation

1. V = Reflection at y = 1.

R = Rotation 90o clockwise about P.

Find the image of A under (a) RV (b)VR.

2. U = Reflection at y = 2.

R = Rotation 90o clockwise about P.

Find the image of A under (a) RU (b) UR.

3. P = Reflection at x = -1.

R = Rotation 90o clockwise about H.Find the image of A under (a) RP (b)

PR.

4. P = Reflection at x = 4.

R = Rotation 90o anticlockwise about O.Find the image of A under (a) RP (b) PR.


(a) A(2,3)A(2,-1) A(1,4)

(b) A(2,3)A( , ) A( , )

(a) A(3,3)A( , ) A( , )

(b) A(3,3)A( , ) A( , )

(a) A(1,2)A( , ) A( , )

(b) A(1,2)A( , ) A( , )

(a) A(3,1)A( , ) A( , )(b) A(3,1)A( , ) A( , )


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12.4 Description of the combined transformations involved when

the object and image are given

In the following diagrams, II is the image ofI under a transformation V followed by

another transformation W.

Describe in full, transformations V and W.

Note that there are many possible combinations.

Example Exercise

1.

V = Translation

1

2

W = Enlargement with scale factor 2at (-3, 1).

2

V =

W =

3

V = Reflection at the line x = 2.

W = Enlargement with scale factor

at (3, 0).

4

V =

W =


O

2

3

y

x1

31 2-1 4 5-2-3-4-1

-2

-3

I

II

O

2

3

y

x1

31 2-1 4 5-2-3-4

-1

-2

-3

I

II

O

2

3

y

x1

31 2-1 4 5-2-3-4

-1

-2

-3

III

O

2

3

y

x1

31 2-1 4 5-2-3-4-1

-2

-3

I

I

II


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5

V =

W =

6

V =

W =

7

V = Rotation of 90

o

clockwise at (2,0)W = Enlargement with scale factor 2

at (1,-3)

8

V =W =

9

V =

W =

10

V =

W =


O

2

3

y

x

1

31 2-1 4 5-2-3-4

-1

-2

-3

O

2

3

y

x1

31 2-1 4 5-2-3-4

-1

-2

-3I

II

O

2

3

y

x1

31 2-1 4 5-2-3-4

-1

-2

-3

II

O

2

3

y

x1

31 2-1 4 5-2-3-4

-1

-2

-3

I

O

A

BA

B O

2

3

y

x1

31 2-1 4 5-2-3-4

-1

-2

-3

O

2

3

y

x1

31 2-1 4 5-2-3-4

-1

-2

-3

II

I

II

I

IIII

I

I

I

II


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12.5 To calculate the area of the image (or object) under an

enlargement with scale factor k

Use the formula

Or

12.5ANo. Area of Object Scale Factor, k Area of Image

1 5 cm2 2 225 = 20 cm2

2 12 cm2 3

3 36 cm2

2

1

4 17 unit2 1.5

5 2.5 unit2 4

6 22

162

64cm=

2 64 cm2

7 4 560 cm2

8

2

1 24 cm2

9 2 72 cm2

10 3 157.5 unit2

11 1.5 108

12 42 cm2 2.5

13 5 2800 cm2

14 315 2

15

15 cm2

3

9

15

1352

=

==

k

k

135 cm2

16 24 cm2 384 cm2

17 108 cm2 27 cm2


Area of Image = k2 Area of Object

Area of Object = 2ImageofArea

k


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12.5B In the diagrams below, A is the image of A under an enlargement with scale factor k.

1

Area of A = 18.5, k = 2

Area of A= 18.522

= 74

2

Area of A = 14, k = 1.5

Area of A=

3

Area of A = 48, k =

Area of A=

4

Area of A = 108, k = 2

Area of A = 10822

= 27

5

Area of A = 153, k = 1.5

Area of A =

6

Area of A = 48, k =

Area of A =

7

Area of A = 927, k = 3

Area of A =

8

Area of A = 4.2, k = 3

Area of shaded region

= 4.232 4.2= 33.6

9

Area of A = 21.6, k = 1.5


=

10

Area of A = 47 cm2, k =


=

11

Area of A = 21 cm2, k =


=

12

Area of A = 33.6 cm2, k =

Area of image A

=


AA

AA

AA

AA

AA

AA

AA

A

A

A

A

A A

A

A

A

A


10/19

12.6 Description of a single transformation which is equivalent to two

combined transformations

Using the object and image given in the following diagrams, describe in full, a single

transformation which is equivalent to the combination of the two given transformations.

Note that you are required to know the cases for combination of two isometric

transformations of the same type only.

Example Exercise

1.

V = Translation

1

2,W = Translation

34

WV = Translation

4

2

.

2

V = Translation

, W = Translation

WV =

3

V = Reflection at the line x = 2.W = Reflection at the line x = -1.

WV = Translation

0

6

4

V = Reflection at ______________ .W = Reflection at ______________

WV =


O

2

3

y

x1

31 2-1 4 5-2-3-4-1

-2

-3

I

O

2

3

y

x1

31 2-1 4 5-2-3-4

-1

-2

-3

O

2

3

y

x1

31 2-1 4 5-2-3-4

-1

-2

-3

O

2

3

y

x1

31 2-1 4 5-2-3-4-1

-2

-3

I

I

II

III

II

III

IIIIII

I IIIII


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5

V = Reflection at the line y = -2.

W = Reflection at the line y = 0.

WV = Translation

4

0

6

V = Reflection at the line y = 2.

W = Reflection at the line y = 0.

WV =

7

V = Reflection at the line x = 2.W = Reflection at the line y = -1.

WV = Rotation 180o at the point (2, -1)

8

V = Reflection at the line x = 3.W = Reflection at the line y = 1.

WV =

9

V = Rotation 90o clockwise at (2,0)

W = Rotation 180o clockwise at (-1,1)WV = Rotation 90o anticlockwise at (-2,-2)

10

V = Rotation 90o clockwise at (2,0)

W = Rotation 90o clockwise at (0,2)WV =


O

2

3

y

x

1

31 2-1 4 5-2-3-4

-1

-2

-3

O

2

3

y

x1

31 2-1 4 5-2-3-4

-1

-2

-3II

O

2

3

y

x1

31 2-1 4 5-2-3-4

-1

-2

-3

I

O

O

2

3

y

x1

31 2-1 4 5-2-3-4

-1

-2

-3

O

2

3

y

x1

31 2-1 4 5-2-3-4

-1

-2

-3

I

I

III

I

II

III

O

2

3

y

x1

31 2-1 4 5-2-3-4

-1

-2

-3

O

I

II

III

I

4


12/19

12.7 Questions Based On Examination Format

Question 1

(a) Transformation P represents a reflection at the line y = 2. Transformation T

represents a translation

2

3

. Transformation Rrepresents a rotation of 90o in the

anticlockwise direction about the point (5, 4).

State the coordinates of the image of point (3, 1) under the following transformation:

(i) P,

(ii) TP,

(iii) RT.

(b) In Diagram 1, quadrilateral KLMN is the image of quadrilateral EFGH under a

transformation V followed by another transformation W.Describe in full

(i) transformation V , and

(ii) transformation W.

(c) Given that quadrilateral KLMN represents an area of 88 unit2, find the area

represented by quadrilateral EFGH.


2 4-4 -2 0 6-6 8

x

y

2

4

6

-2

-4

-6

K

L

MN

E

F

GH

DIAGRAM 1


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Model Answer For Question 1

(b)

(i) V = Reflection at the line y = -1. [2 marks] - (get 1 mark if no axis of reflection or

the reflection axis is wrong)

(ii) W = Enlargement with scale factor 2 at centre K.[3 marks] - (get 1 mark if enlargement only,

get 2 marks if enlargement with scale factor 2

or enlargement at centre K.)

(c)2

2

2

224

88

288

unitl

l

objectofAreakimageofArea

==

=

=


(a) (i) (3, 3) 1 mark(ii) (0, 5) 2 marks

(iii) (6, -1) 2 marks

2

-2

4

2 4-4 -2 0 6x

y

-4

y =2

2 4-4 -2 0 6-6 8

x

y

2

4

6

-2

-4

-6

K

L

MN

E

F

GH

(a) A(2,2)A( , ) A(

, )

(b) A(2, 2)A( ,


14/19

Question 2

Table 1(a) Table 1 shows three pairs of corresponding object and image points under the

same translation. State the coordinates of

(i) point A,(ii) point B.

(b) Point D is the image for point J under a reflection. State the image of point K

under the same reflection.

(c) In Diagram 2, triangle JKL is the image of triangle DEF under a transformation Vfollowed by another transformation W

Describe in full

(i) transformation V , and

(ii) transformation W.

(d) Given that triangle JKL represents an area of 169 unit 2, find the area represented by

triangle DEF.

Transformations III

Object Point E D B

Image Point F A L

14

DIAGRAM 2

2 4-4 -2 0 x-6

y

2

4

1

-2

-4

-5

3

-3

-3 -1-1

-5 1 3 5

D

E

FJL

K


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Question 3 (1998)

(a) Transformation G represents a reflection at the line x = 1. Transformation H

represents a translation

62

. Transformation Krepresents a rotation of 90o in the

anticlockwise direction about the point (3, 0).

State the coordinates of the image of point (5, 2) under the following transformation:

(i) G,(ii) HG,

(iii) KH. [5 marks]

(b) In Diagram 3, triangle LMN is the image of triangle RMS under a transformation

V and triangle LMN is also the object which maps to the image triangle LQP under a

transformation W.

Describe in full

(i) transformation V , and(ii) transformation W.

(c) Given that triangle LMN has an area of 21 unit2

, find the area of quadrilateral MQPN.

[7 marks]


DIAGRAM 3

N

L

M

N

S

R Q

P


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Question 4 (1999)

The graph in Diagram 8 shows the quadrilateralsEFGH,JKLMandNPQR.

(a) Transformation T represents a translation

2

3. Transformation V represents a

reflection at the line y = 4. Transformation W represents a reflection at the y-axis.

(i) State the coordinates of the image of pointEunder the

translation T.

(ii) State the coordinates of the image of pointHunder the

reflection V.

(iii) Find the coordinates of the image of pointJunder the

transformation WT.

(iv) By recognizing the image of EFGHunder the

transformation WV, describe in full a single transformation which is

equivalent to transformation WV.[7 marks]

(b) NPQR is the image ofJKLMunder a transformation S.

(i) Describe in full transformation S .(ii) If the area of JKLMis 17 unit2, calculate the area ofNPQR.


DIAGRAM 8

x

N

2 4-4 -2 0

10

-6

8

-8

2

4

6

y

P

Q

R

NL

K J

M

E F

G

H


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[5 marks]

ANSWERS

Chapter 12 Transformations III

12.2a

2 (a) A(7, 2), A(7, 0)

(b) A(1, -1), A(7, -2)

3 (a) A(-2, 5), A(-2, 1)

(b) A(2, 4), A(-2, 7)

4 (a) A(3, 1), A(1, 1)(b) A(0, -1), A(-1, 1)

5 (a) A(1, 2), A(-3, 2)

(b) A(2, 4), A(5, 2)

6 (a) A(3, 1), A(1, 3)(b) A(3, -2), A(8, -4)

12.2b

2 (a) A(6, 3), A(4, -1)

(b) A(3, 0), A(7, 2)

3 (a) A(1, 2), A(3, 0)

(b) A(6, 2), A(4, 5)

4 (a) A(5, 1), A(3, 1)

(b) A(5, 2), A(4, 4)

12.2c

1 (b) A(5, 4), A(5, -2)2 (a) A(3, 1), A(-2, 0)

(b) A(0, 0), A(0, 4)3 (a) A(-3, 2), A(1, 4)

(b) A(1, 0), A(-3, 0)

4 (a) A(5, 1), A(-1, 5)

(b) A(-1, 3), A(9, 3)

12.4

2 V = Translation

0

3

W = Enlargement with scale factor 3 at (0, 3)

4 V = Reflection at the line x = 1

W = Enlargement with scale factor 2 at (-1, 0)

5 V = Reflection at the line y = - 2

W= Enlargement with scale factor 2.5 at (3, -2)

6 V = Reflection at the line y = - 1

W = Enlargement with scale factor 2 at (4, -1)

8 V = Rotation of 90o anti-clockwise about (0, -2).

W = Enlargement with scale factor 2 at (4, 0)

9 V = Rotation of 90o clockwise about (3, 1).

W = Enlargement with scale factor 2 at (2, -1)

10 V = Rotation of 180o clockwise about (2, -2).

W = Enlargement with scale factor 2.5 at (3, -2)

12.5A

2. 108 3. 9 4. 38.25

5. 40 7. 35 8. 96

9. 18 10. 17.5 11. 4812. 262.5 13. 112 14. 1260

16. 4 17. 0.5

12.5B

2. 31.5 3. 12 5. 68

6. 192 7. 103 9. 27

10. k=2, 141 11. k=3, 168 12. k=2, 134.4

12.6

2. V = Translation

0

6

W = Translation

4

3

WV = Translation

43

4. V = Reflection at x = 0

W = Reflection at x = 4

WV = Translation

0

8

6. WV = Translation

40



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8. WV = Rotation 180o clockwise about (3,1)

10. WV = Rotation 180o clockwise about (0,0)

12.7 Questions According to Examination Format

2(a) (i) A(1, 3) (ii) B(2, 0)

(b) (3, -4)

(c) V = Rotation 90o anti-clockwise about point F.W = Enlargement with scale factor 2 at point J.

(d) 42.25

3(a) (i) (-3, 2) (ii) (-1, -4) (iii) (7, 4)

(b) (i) V = Rotation 180o clockwise about point M.

(ii) W = Enlargement with scale factor 3 at point L.

(a) 168

(b)

4(a) (i) (-1, 8) (ii) (2, 3) (iii) (5, 4)

(iv) WV = Rotation 180o clockwise about point (0, 4).(b) (i) Enlargement with scale factor 3 at point (-2, 1).

(ii) 153



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transformation iii

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