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1 © Penerbitan Pelangi Sdn. Bhd.
Paper 2
1. y
A4
–2
2
20 4(3, 1)
(2, 1)
x
A T (2, 1) T (3, −1)
The coordinates of the image = (3, −1)
2. y
R
x
y = 2
0 2 4 6
(5, 3)
(5, 1)
–2
2
4
6
R T (5, 3) P (5, 1)
The coordinates of the image = (5, 1)
3. y
J
x = 1
x0 2 4
(4, 3)
(3, 5)
–2–4
2
4
L
J P (4, 3) Q (3, 5)
The coordinates of the image = (3, 5)
4.
0
2
–2
–4
4
2 4 6–4 –2
y
x
H
5.
0
6
4
2
8
2 4–2
y
x = 1
x
R
6.
0 2
y
x4 6 8
F2
4
6
8
CHAPTER
23 Transformations IIICHAPTER
Page 2
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Mathematics SPM Chapter 23
© Penerbitan Pelangi Sdn. Bhd.
7.
0 2
y
x
2
4
4
–2P
–4
–2–4R
8. y
x
P
R
Q
0 2 4–2–4
5
(0, 1)
–2
–4 –5
2
4
6
(a) Translation = – 55
(b) Centre of enlargement = (0, 1)
Paper 2
1. (a)
2
2 4 6 8 10 12 14 160
4
6
8
10
12
y
x
A
B
A��
A�
A�
90°y = 8
(i) A T (9, 8)
(ii) A P (6, 7) T (9, 6)
(iii) A R (11, 12) P (11, 4)
(b) (i)
2
2 4 6 8 10 12 14 160
4
6
8
10
12
y
x
E
G
F
(4, –1)
y = x
(a) U is a reflection in the line y = x.
(b) V is an enlargement about point (4, –1) with scale factor 2.
(ii) Area of G = k2 × area of E = 22 × 24 = 96 m2
2. (a) y
x
x = 3–2 2
2
4
6
8
(0, 2) (6, 2)
(4, 7)(–2, 7)
4 60
(i) (6, 2) T (4, 7)
(ii) (6, 2) R (0, 2) T (−2, 7)
(b) (i) (I) V is a rotation of 180° about point D.
(II) W is an enlargement about point E with scale factor 3.
(ii) Area of EHJL = k2 × area of EFGD = 32 × 34.5 = 310.5 unit2
Area of the shaded region = 310.5 − 34.5 = 276 unit2
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Mathematics SPM Chapter 23
© Penerbitan Pelangi Sdn. Bhd.
3. (a) y
x
x = 3
G
0–2 2
2
4
6
490°
6 8
Gʹ
Gʹʹ
(i) G(5, 1) R (1, 1) T (–2, 3)
(ii) G(5, 1) V (0, 4) R (6, 4)
(b) (i) U is a reflection in the line x = –1.
(ii) W is an enlargement about point K with scale factor 3.
(c) Area of KLMN = k2 × area of ABCD = 32 × 12 = 108 m2
Area of the shaded region = 108 – 12 = 96 m2
4. (a)
2
2
0
4
6
–2
–2
4 6
y
xEA
A�A�
90°
(i) A(5, 1) T (2, 3) T (–1, 5)
(ii) A(5, 1) R (1, 3) T (–2, 5)
(b) (i) y
x
2
2–2
–2
4 60
4
6G
H
JBA
E
F
D
C
(–1, –2)
(a) W is a reflection in the line AB.
(b) U is an enlargement about centre (–1, –2) with scale factor 3.
(ii) Area of FGHJA = k2 × area of ABCDE = 32 × 52 = 468 m2
Area of the shaded region = 468 – 52 = 416 m2
5.
0 2
90°90°
4–2
–2
x
PQ'' Q'
P'Q
y
2
4
–4–6
(a) (i) P R P′(2, −1)
(ii) (I) Q T (0, 0) T Q′(−3, 2)
(II) Q R (−3, 0) T Q′′(−6, 2)
(b)
0
2
4
6
8
2 4 6–2x
x = 2
F
H
EP
Q
S
R
M
JL K G
y
–4
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Mathematics SPM Chapter 23
© Penerbitan Pelangi Sdn. Bhd.
(i) (I) V is a reflection in the line x = 2. (II) W is an enlargement about point
(−1, 4) with scale factor 3.
(ii) Area of PQRS = 32 × area of JKLM 116.1 = 9 × area of JKLM
Area of JKLM = 116.1–––––9
= 12.9 m2
Area of the shaded region = 116.1 − 12.9 = 103.2 m2
6. (a) y
x0–2
–2
2
2
4
6
4
(2, 2)
(3, 6)
(0, 5)
(4, –1)6
(i) (4, −1) T (2, 2)
(ii) (4, −1) R (3, 6)
(iii) (4, −1) T (2, 2) T (0, 5)
(b) (i) (I) X is a reflection in the line x = −1. (II) Y is an enlargement about point C with
scale factor 3.
(ii) Area of ABCD = k2 × area of EFCG = 32 × area of JKLM = 9 × 15 = 135 cm2
Area of the shaded region = 135 − 15 = 120 cm2
7. (a) y
2
–2
0
4
2
x–2 4 6
y = 2
(–1, –1)
(1, –2)
(5, 0)
(3, 1)
(6, –1)
(6, 5)
(0, 2)
(i) P is a reflection in the line y = 2. Therefore, h = 2
(ii) (I) (3, 1) T (5, 0)
(II) (3, 1) R (–1, −1) T (1, −2)
(b)
J
M
y
x
– 2
G
F
B
A
L
K E
D
C
– 6 – 4 – 2
2
4
6
420
x = –1
(i) (I) V is a reflection in the line x = –1.
(II) W is an enlargement about point A with
scale factor 12
.
(ii) Area of ABCD = 1 12 2
2 × area of AEFG
22.5 = 14
× area of AEFG
Area of AEFG = 4 × 22.5 = 90 cm2
Area of the shaded region = 90 – 22.5 = 67.5 cm2
Paper 2
1. (a)
0x
A
y
y = 3
–2
2
2
(–1, 1)
(–1, 5)6
4
4
(i) A T (−1, 5)
(ii) A T (−1, 5) R (−1, 1)
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Mathematics SPM Chapter 23
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(b) y
x
M
0–2–4 4 62
–2
–4
4
2
8
6
LN
A C
B
D F
E
V is a reflection in the x-axis. W is a reflection in the y-axis.
(c) (i) The centre of enlargement is (0, 0).
(ii) Scale factor = DE––––AB
= 6—3 = 2
(iii) Area of ∆DEF = k2 × area of ∆ABC = 22 × 10 = 40 cm2
2. (a) y
x0–2 2
2
(–1, 3)
(–1, 1) (2, 1)
4
4
y = 2
(i) (−1, 1) P (−1, 3)
(ii) (−1, 1) P (−1, 3) T (2, 1)
(iii)
0
–2
x
y
y = 2
–2
(–1, 1)2
2(2, –1)
4
4
(2, 5)
(−1, 1) T (2, −1) P (2, 5)
(b) (i) U is a reflection in the line PTR. V is an enlargement about point R with scale
factor 2.
(ii) Area of ∆PQR = k2 × area of ∆TSR = 22 × 5 = 20 cm2
3. (a) y
x
y = 3
0
(1, 4)
(1, 2) (5, 2) (7, 2)
(7, 4)(4, 5)
4 6 82
4
2
6
(i) (1, 2) Q (1, 4)
(ii) (1, 2) Q (1, 4) P (5, 2)
(iii) (1, 2) P (7, 2) Q (7, 4)
(b) (i) y
x0 8(7, 1)
1042 6
2
4
6
8
10
12
F E
A B
C
D
H
G
(I) Scale factor = AF––––HG
= 6—3 = 2
(II) The centre of enlargement is (7, 1).
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Mathematics SPM Chapter 23
© Penerbitan Pelangi Sdn. Bhd.
(ii) Area of hexagon ABCDEF = k2 × area of the shaded hexagon = 22 × 16 = 64 cm2
Area of the unshaded region = 64 − 16 = 48 cm2
4. (a) y
2
–2
0x
–2 4
(–3, 0)
(–3, 4)
(1, 3)
(1, –1)
–4
y = 12
4
(i) (1, 3) T (−3, 4)
(ii) (1, 3) R (1, –1) T (–3, 0)
(b) y
x
x = –1
W V0 2 4
(2, 2)
6–4 –2
2
4
6
8
10
–2
–4
X
P Q
R
S T
U
(i) J is a reflection in the line x = −1. K is an enlargement about point (2, 2) with
scale factor 4.
(ii) Area of ∆STU = k2 × area of ∆PQR = 42 × 21 = 336 cm2
5. (a) y
x0–4
–4 (1, –4)
(2, 5)
(4, 2)(–4, 1)
(–1, 3)
(6, 5)
–2
–2(–2, –2)
2
2
4
4
6
6
(i) (−2, −2) K (− 4, 1)
(ii) (−2, −2) J (1, −4) H (6, 5)
(iii) (–2, –2) H (4, 2) K (2, 5)
(b)
0
2
4
6
2 4–4–6 –2x
x = 1y
P
Q
(i) R is a reflection in the line x = 1. T is an enlargement about point (0, 6) with
scale factor 3.
(ii) Area of Q = k2 × area of P = 32 × 16 = 144 units2
6.
0
2
(–2, 1)
–2
4
–4
2 4–4 –2x
y = –x
E
G
F
y
A B
C
K
L Q R
P
M
T
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Mathematics SPM Chapter 23
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(a) T
– 4–3
(–2, 1)
(b) (i) V is a clockwise rotation of 90° about the origin.
(ii) W is a reflection in the y-axis.
(iii) Z is a reflection in the line y = −x.
(c)
0
–2
–4
2 4–2x
x = 1y
A'
C'
G
FEB
C
A
U is a reflection in the line x = 1. W is an enlargement about point E with scale
factor 2.
7. (a)
0
(–2, 6)
(–2, 5)
–4 –2 2
2
4
(2, 6)(–4, 6)
x
GC
A
y = 3
y
4
6
(i) (I) C T (− 4, 6)
(II) A Q (−2, 5)
(III) G T (2, 6) P (−2, 6)
(ii)
x
y
y = 3(0, 3)
(2, 6)
0–2–4 42 86
–2
–4
4
6
2
A
B
D
C
E H
FGJ
K
M
L
The single transformation is a rotation of 180° about point (0, 3).
(b) (i) W is an enlargement about point (2, 6) with scale factor 3.
(ii) Area of JKLM = k2 × area of EFGH 153 = 9 × area of EFGH
Area of EFGH = 153––––9 = 17 units2
8. (a)
x
y
y = –x
0 4
4
2
(1, 1)2
6
6
–2
–2
–4
(–3, –2)
(2, 3)
(6, 6)
(4, 5)
–4
(i) (2, 3) Q (−3, −2)
(ii) (2, 3) Q (−3, −2) T (1, 1)
(iii) (2, 3) T (6, 6) P (4, 5)
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Mathematics SPM Chapter 23
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(b) (i) W is an enlargement about point A with scale factor 3.
(ii) Area of AEFG = k2 × area of ABCD = 32 × 22.5 = 202.5 m2
Area of the shaded region = 202.5 − 22.5 = 180 m2
9. (a)
x
y
y = x
–2
–2
2 40–4
6
4
(–3, 5)
(–1, 1) (3, 1)
(–3, 3)(1, 3)
2
(i) (1, 3) R (3, 1)
(ii) (1, 3) R (3, 1) Q (−3, 3)
(iii) (1, 3) Q (−1, 1) P (−3, 5)
(b)
x
y
0 2 4 6
–2
–4
2
4
R
E
D
F
AB
C
(i) D(2, −3), E(2, −1), F(6, −1) (ii) Area of ∆DEF
––––––––––––Area of ∆ABC
= 22
= 4
10. (a)
x
y
x = 3
0
(2, 6)
(5, 4)
(2, 2) (4, 2)
2 4 6
(2, 8)
–2
2
4
6
8
(i) (2, 2) E (4, 2)
(ii) (2, 2) E (4, 2) F (2, 8)
(iii) (2, 2) F (2, 6) G (5, 4)
(b) (i) X is a rotation of 180° about point T. Y is an enlargement about point P with scale
factor 3.
(ii) Area of ∆PRS = k2 × area of ∆PQT = 32 × 27 = 243 units2
Area of quadrilateral QRST = 243 − 27 = 216 units2
11. (a)
x
y
y = 3
0 2 4 6 8–2
2
(–1, 0)(2, 2)
(–1, 6) (3, 6)(6, 5)
(6, 1)
4
6
(i) (6, 5) P (6, 1)
(ii) (6, 5) P (6, 1) Q (3, 6)
(iii) (6, 5) Q (−1, 6) P (−1, 0)
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Mathematics SPM Chapter 23
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(b) (i) (I) X is a rotation of 180° about point O.
(II) Y is an enlargement about point O with scale factor 3.
(ii) Area of OLMN = k2 × area of OHJK = 32 × 9 = 81 cm2
Area of the shaded region = 81 − 9 = 72 cm2
12. (a)
O
y
x2
2
4
6
4 6(5, –1)
(2, 6)
(–1, 5)
(2, 1)
G
Q
V
S
8
(i) Q Y (2, 6)
(ii) G Z (2, 1)
(iii) S Z (5, −1) Y (−1, 5)
(b)
O x
y
EH
2 4 6 8
2(2, 2)
(3, 1)
4
6 FG
P Q
RS
V U
T
A is a clockwise rotation of 90° about point (2, 2).
B is an enlargement about point (3, 1) with scale factor 2.
(c) Area of RTUV = k2 × area of PQRS = 22 × 12 = 48 m2
13. (a) (i) The coordinates of the centre of rotation are (2, 3).
(ii) The angle of rotation = 180°
(b)
x
X
Y
y
J
20 4 6
2
(6, 4)(0, 4)4
6
translation reflection J (6, 4) (0, 4)
(c)
O x
y
2 4
(4, 8)
(7, 2)(1, 2)
6 8
2
4
6
8
A
B
F
Y
x = 4
(i) G is an enlargement about point (1, 2) with scale factor 3.
H is a reflection in the line x = 4.
(ii) (I) F G (4, 8) H (4, 8)
(II) Y G (7, 2) H (1, 2)
14. (a)
0 2 4 6 8 10 12 14
2
4
6
8
10
12
14
x
y
I
II
III
G
E
F
(5, 11)
(6, 6)180°
(i) The coordinates of the image of point G are (5, 11).
(ii) V is a rotation of 180° about point (6, 6).
W is the translation 25 .
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Mathematics SPM Chapter 23
© Penerbitan Pelangi Sdn. Bhd.
(b) (i) Scale factor = PQ––––ST
= 12–––4 = 3
(ii) RQ––––RT
= 3
RQ = 3 × RT = 3 × 5 = 15 cm TQ = RQ − RT = 15 − 5 = 10 cm
(iii)
12 cm
4 cm
5 cmR
S
P Q
T
1—2
(RT) × h = 7.5
1—2
× 5 × h = 7.5
h = 7.5 × 2–––––––5
= 3 cm H = 3h = 3 × 3 = 9 cm
Hence, the perpendicular distance from P to RQ is 9 cm.