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1 © Penerbitan Pelangi Sdn. Bhd. Paper 2 1. y A 4 –2 2 2 0 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 x 0 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 x 4 6 8 F 2 4 6 8 CHAPTER 23 Transformations III

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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

2

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

3

Mathematics SPM Chapter 23

© Penerbitan Pelangi Sdn. Bhd.

3. (a) y

x

x = 3

G

0–2 2

2

4

6

490°

6 8

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

4

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)

5

Mathematics SPM Chapter 23

© Penerbitan Pelangi Sdn. Bhd.

(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).

6

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

7

Mathematics SPM Chapter 23

© Penerbitan Pelangi Sdn. Bhd.

(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)

8

Mathematics SPM Chapter 23

© Penerbitan Pelangi Sdn. Bhd.

(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)

9

Mathematics SPM Chapter 23

© Penerbitan Pelangi Sdn. Bhd.

(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 .

10

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.