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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 distance at the same direction. The shape, size and orientation remain the same. It is written as A A with a translation of k . REFLECTION All the points of an object are reflected in a line called the axis of reflection or line of reflection. It is written as AA with a reflection in the line .. ROTATION All 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
h

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ABCDE FGHIJ with a translation 4 . 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 a scale factor of 2. 12.2 Combination 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 its image. Calculate the area of the image (or object) under an enlargement. 12.3 Description of a transformation when the object and image are given Full description must be given. Type of Details required Example transformations Translation h 2 or AB Translation or Translation
k 3
U V

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 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 clockwise direction about the point K. An enlargement with scale factor 3 at the centre (4,1). An enlargement at the point (4,1) through a scale factor of 2.
1 2

Rotation

Enlargement

Angle of rotation (90o or 180o) Direction of rotation (clockwise or anticlockwise) Centre of rotation Scale Factor (, 2 or 3) Centre of enlargement

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12.4 Questions Based on Examination Format Question 1 (a) Transformation P represents a reflection at the line y = 2. Transformation T represents a translation
3 . Transformation R represents a rotation of 90o in the 2

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. N
6 y

M
4 2

K
x -6 -4 -2 0 2 4 6 8

F E

L
-2 -4

DIAGRAM 1

H
-6

(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.

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Model Answer For Question 1 (a) (i) (3, 3) 1 mark (ii) (0, 5) 2 marks (iii) (6, -1) 2 marks

y 4 2

y =2
2 4 6 x

-4

-2

0 -2 -4

(b)

6 y 4 2

K
-6 -4 -2

x 0 -2 -4

F E H

G
-6

(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.)
Area of image = k 2 Area of object

(c)

88 =2 2 l l= 88 =22 unit 4
2

DIAGRAM 1 mark 2

1 mark

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Question 2 Object Point Image Point E F D A B L

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. D
y 4 3 2

E
-6 -5 -4 -3 -2 -1

1 0 -1 -2 -3 -4 -5 1 2 3 4

DIAGRAM 2

(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 V followed 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.

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Question 3 (1998) (a) Transformation G represents a reflection at the line x = 1. Transformation H represents a translation
2 . Transformation K represents a rotation of 90o in the 6

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] N

N
DIAGRAM 3

L M S R Q

(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]

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Question 4 (1999) N y
10

R
6

E H

F
DIAGRAM 8

P K
-8 -6 -4

N M J
-2

The graph in Diagram 8 shows the quadrilaterals EFGH, JKLM and NPQR. (a) Transformation T represents a translation
3 . Transformation V represents a 2

reflection at the line y = 4. Transformation W represents a reflection at the y-axis. (i) translation T. (ii) reflection V. (iii) (iv) Find the coordinates of the image of point J under the By recognizing the image of EFGH under the transformation WT. transformation WV, describe in full a single transformation which is equivalent to transformation WV. [7 marks] (b) NPQR is the image of JKLM under a transformation S. (i) (ii) Describe in full transformation S . If the area of JKLM is 17 unit2, calculate the area of NPQR.
1 7

State the coordinates of the image of point E under the State the coordinates of the image of point H under the

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[5 marks] Question 5 (2000J) (a) Transformation T represents a translation 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]
6 and transformation P represents a 3

y 5 4 3 2 1 -3 -2 -1 0 -1 -2 -3 -4 -5

F H J
1 2 3

K
4

G
5 6

E
7 8 x

M
DIAGRAM 10

(b) In Diagram 10, triangle HJK is the image of triangle EFG under a transformation V and triangle LMN is the image of triangle HJK under a transformation W. Describe in full (i) transformation V , (ii) (iii) transformation W, and a single transformation which is equivalent to WV.

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[8 marks]

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Question 6 (2000)
L 8 y 6 K 4 M J 2 N -8 -6 -4 H -2 0 F G 2 4 D 6 x E DIAGRAM 8

The graph in Diagram 8 shows the quadrilaterals DEFG, DKJH and LMNH. (a) Transformation S represents a translation reflection at line DEK. State the coordinates of the image of point (1, 2) under the following transformation: (i) S, (ii) T, (iii) ST. [4 marks] (b) Quadrilateral DKJH is the image of quadrilateral DEFG under a transformation V and quadrilateral LMNH is the image of quadrilateral DKJH 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 quadrilateral DKJH. [2 marks]
2 . Transformation T represents a 3

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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
2 . 1

State the coordinates of the image of point (3, 2) under the following transformation: (i) T, (ii) PT, (iii) TP. [5 marks] D E B F A C
DIAGRAM 8

G (b) In Diagram 8, triangle DEC is the image of triangle ABC under a transformation V and triangle DFG is the image of triangle DEC under a transformation W. (i) (ii) (iii) Describe in full transformation V. Given that transformation W is an enlargement. State Calculate the area of triangle ABC if the area of

the centre and scale factor of the enlargement. quadrilateral CEFG is 24 square units. [7 marks] Question 8 (2001)
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(a) Transformation M represents a translation . Transformation P represents a 1

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]

L
DIAGRAM 9

Q G F E D O H

(b) Diagram 9 shows quadrilateral ODEF, quadrilateral OFGH and quadrilateral OLKJ which are drawn on square grids. (i) Given that transformation Q is a reflection at line OFL and transformation R is a reflection at line OHJ . If quadrilateral ODEF experiences transformation RQ, describe in full a single transformation which is equivalent to RQ. (ii) Given that quadrilateral OLKJ is the image of quadrilateral OFGH under a transformation W. (a) Describe in full transformation W. (b) Calculate the area of the shaded region if the area of quadrilateral OFGH is 12.7 square units. [7 marks] Question 9 (2002J)

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N
4

y K L M
6

-4

-2

0 -2

x
DIAGRAM 9

P
-4

-6

The graph in Diagram 9 shows the triangles KLM, KRQ and PNM. (a) Triangle PNM is the image of triangle KLM under a transformation G whereas triangle KRQ is the image of triangle KLM under 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 point M under the transformation D, (b) the image of point R 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 triangle KLM is 6.3 square unit, calculate the area of the image of triangle KLM under an enlargement with a scale factor of 5. [7 marks]

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Question 10(2002) (a) Transformation R represents a rotation of 90o in the anti-clockwise direction at point (1, 4). Transformation P represents a reflection at the line y = 2. State the coordinates of the image of point (3, 1) under the following transformation: (i) R, (ii) PR. [3 marks]
y 8 6 4

A D
DIAGRAM 8

C
x

10

12

14

16

(b) The graph in diagram 8 shows quadrilaterals A, B, C and D. (i) Quadrilateral B is the image of quadrilateral A under a transformation V, whereas quadrilateral C is the image of quadrilateral B under a transformation W. Describe in full (a) transformation V, (b) a single transformation which is equivalent to transformation WV. (ii) Quadrilateral D is the image of quadrilateral A 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 quadrilateral A is 7.5 square units, calculate the area of quadrilateral D. [9 marks]

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ANSWERS
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) o (b) (i) V = Rotation 180 clockwise about point M. (ii) W = Enlargement with scale factor 3 at point L. (c) 168 4(a) (i) (-1, 8) (ii) (2, 3) (iii) (5, 4) o (iv) WV = Rotation 180 clockwise about point (0, 4). (b) (i) Enlargement with scale factor 3 at point (-2, 1). (ii) 153 5(a) (i) (-2, 6) (ii) (4, -7) (iii) (-2, - 4) o (b) (i) V = Rotation 90 anti-clockwise about point (4, 0). (ii) W = Rotation 90o clockwise about point (1, -1). 2 (iii) Translation 4 6(a) (i) (-1, 5) (ii) (3, 4) (iii) (1, 7) (b) (i) V = Enlargement with scale factor 2 at point D. (ii) W = Rotation 90o anti-clockwise about point H. (c) 50.8 7(a) (i) (5, 1) (ii) (1, 5) (iii) (4, 2) (b) (i) V = Rotation 90o clockwise about point C. (ii) Centre D, scale factor = 2 (iii) 8 8 (a) (i) (-3, 4) (ii) (5, 2) (iii) (1, 3) (b) (i) RQ = Rotation 90o clockwise about point O. (ii) (a) W = Enlargement with scale factor 3 at point O. (b) 101.6

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9. (a) (i) Rotation 180o clockwise about point M. (ii) Enlargement with scale factor 3 at point K. (b) (i) (a) (6, 2) (b) (-4, -1) 0 (ii) W = Translation . 2 (c) 157.5 10. (a) (i) (4, 6) (ii) (4, -2) (b) (i) (a) V = Reflection at the line x = 5 (b) WV = Rotation 180o clockwise about point (5, 4) (ii) (a) 3 (b) (6, 7) (c) 67.5

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