chapter 12 enhance transformations iii

8/2/2019 Chapter 12 Enhance Transformations III

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CHAPTER 12 TRANSFORMATIONS III

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

TRANSLATIONAll the points on a given plane move along a straight line by the same distance atthe same direction.

The shape, size and orientation remain the same.

It is written as AA with a translation of

k

h.

REFLECTIONAll 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 directionabout 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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ABCDE FGHIJ 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 giventransformations.

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




(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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3. T = Translation

3

4



4. T = Translation

2

1

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)

(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.

12.3 Description of a transformation when the object and image are

given


(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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Full description must be given by stating

(i) the type of transformation,

(ii) details of the transformation required.

Type oftransformations

Details required Example

TranslationABor

k

h

Translation

32

or

Translation UV

Reflection Axis of reflection A reflection in the line x = 2

A reflection in the line y = 3

A reflection in the line y = x

Rotation Angle of rotation

(90o

or 180o

)Direction of rotation

(clockwise or anticlockwise)

Centre of rotation

A rotation of 90o in the clockwise

direction about the point (1,3).A rotation of 90o in the anticlockwise

direction about the origin.

A rotation of 180o in the clockwisedirection about the point K.

Enlargement Scale Factor (, 2 or 3)Centre of enlargement

An enlargement with scale factor 3 atthe centre (4,1).

An enlargement at the point (4,1)

through a scale factor of 2.

In the following diagrams, II is the image ofI under a transformation V. Describe in full,

transformation V.

Example Exercise

1.

V = Translation

31

2

V =

3 4


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

II

I

II

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


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V = Reflection at the line x = 2. V =

5

V =

6

V =

7

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

8

V =

9 10


I II III

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

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

I

O

A

B

A

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


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V = V =

11

V = Enlargement with scale factor 2

at (0, 0).

12

V =

13

V =

14

V =

12.4 Description of the combined transformations involved when

the object and image are given


I

II

II

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

I

II

I

II

O

2

3

y

x1

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

-1

-2

-3

II

I

O

2

3

y

x1

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

-1

-2

-3

II

I


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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 2

at (-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

915

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 =2

ImageofArea

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 = 3Area of A =

8

Area of A = 4.2, k = 3Area of shaded region

= 4.232 4.2

= 33.6

9

Area of A = 21.6, k = 1.5Area of shaded region

=

10

Area of A = 47 cm2, k =

Area of shaded region

=

11

Area of A = 21 cm2, k =

Area of shaded region

=

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


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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 combinationof two isometric

transformations of the same type only.

Example Exercise

1.

V = Translation

1

2,W = Translation

34

WV = Translation

42

.

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


15/26

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

23 . 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 atransformation 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( ,


17/26

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 unit2, find the area represented by

triangle DEF.

Transformations III

Object Point E D B

Image Point F A L

17

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


18/26

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 atransformation 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 ofJKLM under a transformation S.

(i) Describe in full transformation S .


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


20/26

(ii) If the area of JKLMis 17 unit2, calculate the area ofNPQR.

[5 marks]

Question 5 (2000J)

(a) Transformation T represents a translation

3

6and transformation P represents a

reflection at the line y = -2.

State the coordinates of the image of point (4, 3) under the following transformation:(i) T,

(ii) P,

(iii) TP. [4 marks]

(b) In Diagram 10, triangleHJKis the image of triangleEFG under a transformation V

and triangleLMNis the image of triangleHJKunder a transformation W.

Describe in full

(i) transformation V ,

(ii) transformation W, and

(iii) a single transformation which is equivalent to WV.


DIAGRAM 10

2 4-3 -2 0 6 81 3 5 7 x-1

2

1

3

-1

-2

-3

-4

-5

y

4

5

JK

H

F

G E

M

N L


21/26

[8 marks]



22/26

Question 6 (2000)

The graph in Diagram 8 shows the quadrilateralsDEFG,DKJHandLMNH.

(a) Transformation S represents a translation

3

2. Transformation T represents a

reflection at lineDEK.

State the coordinates of the image of point (1, 2) under the followingtransformation:

(i) S,

(ii) T,

(iii) ST.

[4 marks]

(b) QuadrilateralDKJHis the image of quadrilateralDEFG under a transformation Vand quadrilateralLMNHis the image of quadrilateralDKJH under a transformation W.

Describe in full(i) transformation V,(ii) transformation W.

[6 marks]

(c) If the area of quadrilateral DEFG is 12.7 square unit , calculate the area of

quadrilateralDKJH.

[2 marks]


DIAGRAM 8

4

2

6

8

0 2-4 -2 4 6x

y

-8 -6

M

N H

L

J

K

E

F

G D


23/26

Question 7 (2001J)

(a) Transformation P represents a reflection at the line that passes through (0, 0) and

(6, 6). Transformation T represents a translation

12

.

State the coordinates of the image of point (3, 2) under the following

transformation:

(i) T,(ii) PT,

(iii) TP.[5 marks]

(b) In Diagram 8, triangleDECis the image of triangleABC under a transformation

V and triangleDFG is the image of triangleDECunder a transformation W.

(i) Describe in full transformation V.

(ii) Given that transformation W is an enlargement. State

the centre and scale factor of the enlargement.

(iii) Calculate the area of triangleABCif the area of

quadrilateral CEFG is 24 square units.

[7 marks]


DIAGRAM 8

A

B

C

D

E

F

G


24/26

Question 8 (2001)

(a) Transformation M represents a translation

1

4. Transformation P represents a

reflection at the line y = 2.

State the coordinates of the image of point (-3, 0) under the following

transformation:

(i) P,(ii) MM,

(iii) PM.[5 marks]

(b) Diagram 9 shows quadrilateral ODEF, quadrilateral OFGH and quadrilateral

OLKJwhich are drawn on square grids.

(i) Given that transformation Q is a reflection at line OFL and transformation

Ris a reflection at line OHJ. If quadrilateral ODEFexperiences

transformation RQ, describein fulla single transformation which is

equivalent to RQ.

(ii) Given that quadrilateral OLKJis the image of quadrilateral OFGHunder a

transformation W.

(a) Describe in full transformation W.(b) Calculate the area of the shaded region if the area of quadrilateral

OFGHis 12.7 square units.

[7 marks]


DIAGRAM 9

K J

L

E

D O

HG

F


25/26

Question 9 (2002J)

The graph in Diagram 9 shows the trianglesKLM,KRQ andPNM.

(a) TrianglePNM is the image of triangleKLMunder a transformation G whereas

triangleKRQ is the image of triangleKLMunder a transformation H.

Describe in full

(i) transformation G,(ii) transformation H.

(b) Given that transformation D is a reflection at the line y = 1 and transformation E is a

reflection at the x-axis.

(i) State the coordinates of(a) the image of pointMunder the transformation D,

(b) the image of pointR under the transformation DE.

(ii) If transformation W is a single transformation which is equivalent to

transformation DE, describe in full transformation W.

(c) Given that the area of triangleKLMis 6.3 square unit, calculate the area of the image

of triangleKLMunder an enlargement with a scale factor of 5.

[7 marks]


DIAGRAM 9

x

N

2 4-4 -2 0

2

4

6

y

8

-2

-4

-6

R

Q

P

N

M

K

L


26/26

Question 10(2002)

(a) Transformation Rrepresents a rotation of 90o in the anti-clockwise direction atpoint (1, 4). Transformation P represents a reflection at the line y = 2.

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

(i) R,

(ii) PR.

[3 marks]

(b) The graph in diagram 8 shows quadrilateralsA, B, C andD.

(i) QuadrilateralB is the image of quadrilateralA under a transformation V, whereasquadrilateral C is the image of quadrilateralB under a transformation W.

Describein full

(a) transformation V,

(b) a single transformation which is equivalent to transformation WV.

(ii) QuadrilateralD is the image of quadrilateralA under a certain enlargement.

(a) State the scale factor of the enlargement.

(b) Find the coordinates of the centre of the enlargement.

(c) If the area of quadrilateralA is 7.5 square units, calculate the area of quadrilateralD.

[9 marks]

DIAGRAM 8

126 8 14 16x

4

y

4

2

6

8

0 2 10

D

AB

C

chapter 12 enhance transformations iii

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