Chapter 1 Directed Numbers Chapter 8 · PDF fileChapter 1 Directed Numbers Chapter 2 Basic Algebra Chapter 3 Basic Geometry Chapter 4 Linear Equations in One Unknown Chapter 5 Percentages

2010 Chung Tai Educational Press. All rights reserved.

Chapter 1 Directed Numbers

Chapter 2 Basic Algebra

Chapter 3 Basic Geometry

Chapter 4 Linear Equations in One Unknown

Chapter 5 Percentages

Chapter 6 Statistics in Daily Life

Chapter 7 Algebraic Expressions and Polynomials

Chapter 8 Symmetry and Transformation

Chapter 9 Introduction to Coordinates

Chapter 10 Statistical Graphs

Chapter 11 Linear Equations in Two Unknowns

Chapter 12 Ratio and Rate

Chapter 13 Angles in Rectilinear Figures

Chapter 1 Estimation and Approximation

Chapter 2 Measurement and Errors

Chapter 3 Identities and Factorization

Chapter 4 Simultaneous Equations

Chapter 5 Congruent and Similar Triangles

Chapter 6 Square Roots and Pythagoras’ Theorem

Chapter 7 More about Statistical Graphs

Chapter 8 Inequalities

Chapter 9 Algebraic Fractions and Formulae

Chapter 10 Areas and Volumes

Chapter 11 Introduction to Trigonometric Ratios

Chapter 12 Polygons

Chapter 13 Measures of Central Tendency

Chapter 1 Laws of Integral Indices

Chapter 2 More about Factorization

Chapter 3 Study of 3-dimensional Figures

Chapter 4 Mensuration

Chapter 5 Theorems Related to Triangles

Chapter 6 Introduction to Probability

Chapter 7 More about Percentages

Chapter 8 Properties of Quadrilaterals

Chapter 9 Coordinate Geometry

Chapter 10 Applications of Trigonometry

Chapter 11 Use and Misuse of Statistics

6.1


1. Without using a calculator, find the square roots of the following numbers.

(a) 289 (b) 900

(c) 0.000 4 (d) 529

64

2. Is each of the following a surd?

Yes No

(a) 32 □ □

(b) 484 □ □

(c) 6.3 □ □

(d) 50

18 □ □

3. Without using a calculator, find two consecutive numbers between which each of the following lies.

(a) 30

30

30

6.2


i.e.

30

30 lies between and .

(b) 180

180

180

i.e.

180

180 lies between and .

4. Find the values of the following numbers. (Give your answers correct to 3 significant figures if necessary.)

(a)

80

(b)

024 1

(c)

62.3

(d)

15

4

(e)

310 22

(f)

415 22

6.3


1. Find the value of the unknown in each of the following figures. (Give your answers correct to 3 significant figures if necessary.)

(a) Q P

R

p

24

18

) () ( 222

p (Pyth. theorem)

p

(b) A

15

B Ca

15

6.4


(c)

c

C

5

A

18

B

(d)

r17

32

P

Q

R

6.5


2. In the figure, S is the mid-point of QR and PS QR. Find the value of p.

3. In the figure, find the area of rectangle ABCD.

Q

P

R

S

p

15

18

D C

A B

17 cm15 cm

6.6


4. In the figure, find the area of pentagon ABCDE. (Give your answer correct to 3 significant figures.)

C D

B

A

12 cm

4 cm

7 cm

E

6.7


5. The figure shows rhombus ABCD. The lengths of the diagonals AC and BD are 10 cm and 8 cm respectively. The diagonals are perpendicular to each other and bisect each other at F. Find the perimeter of rhombus ABCD. (Give your answer correct to 3 significant figures.)

DB

C

F

A

6.8


6. In the figure, D is a point on AC such that DCAD .

C

A

B

19 m6 m

D

(a) Find the length of AB. (Give your answer correct to 3 significant figures.)

(b) Find the length of BC. (Give your answer correct to 3 significant figures.)

6.9


1. Determine whether each of the following triangles is a right-angled triangle. If yes, indicate the right angle.

(a)

C

B

A

20

36

39

22

BCAB

2

AC

22 BCAB □ 2AC

ABC

(b)

Q

34

RP 30

16

22

RQPR

2

PQ

22 RQPR □ 2PQ

PQR

6.10


2. (a) Prove that ABC in the figure is a right-angled triangle.

(b) Find the area of ABC.

C

17.5

BA

14

10.5

6.11


3. (a) Find the area of PQR in the figure.

(b) Express the area of PQR in terms of h. Hence find the value of h.

R

25

QP 24

7h

6.12


4. In the figure, D is a point on AB.

(a) Prove that 90ADC .

(b) Find the perimeter of ABC.

C

24

D BA 32 10

26

6.13


1. At 12:00 noon, aeroplane A is 800 km due west of the airport, while aeroplane B is 1 000 km due north of the airport. If aeroplane A and aeroplane B are on the same horizontal line, what is the distance between these two aeroplanes? (Give your answer correct to the nearest km.)

1 000 km

800 kmA

BN

Airport

Let x km be the distance between these two aeroplanes.

6.14


2. Given that the perimeter of a square wooden board is 60 cm, find the length of the diagonal of the board. (Give your answer correct to 3 significant figures.)

3. Find the perimeter of OAB in the figure.

y

x

A (0, 20)

OB (21, 0)

Perimeter 60 cm

Diagonal

6.15


4. A car and a bus travel at the speeds of 80 km/h due east and 70 km/h due south respectively from point A at the same time. What is the distance between the car and the bus after two hours? (Give your answer correct to the nearest km.)

80 km/h

70 km/h

A

N

6.16


5. A pole of 2.5 m long leans against a vertical wall and its upper end is 1.8 m above the ground. If the upper end of the pole slides down by 0.2 m, how far will the lower end of the pole move from its original position? (Give your answer correct to 3 significant figures.)

2.5 m

0.2 m

1.8 m

6.17


Simplify the following. (1 4)

1. (a)

) () (3

) () (33

2

25

(b)

) () (2

) () (22

2

26

(c)

) (7

) (77

2

27

(d)

58

2. (a)

2

) (244

) (

) (

(b)

3

) (363

) (

) (

(c)

135

(d)

252

6.18


3. (a)

) (

) (

169

7

(b)

) (

) (

) (

) (

4

27

(c)

81

14

(d)

225

32

4. (a)

322

(b)

) (

) (

12

216

(c)

) (

) (

50

126

(d)

) (

) (

288

42

6.19


Rationalize the denominators of the following. (1 2)

1. (a)

) (

) (

6

1

6

1

(b)

) (

) (

5

2

5

2

(c)

11

9

(d)

10

15

2. (a)

) (

) (

30

5

30

5

(b)

) (

) (

21

72

21

72

(c)

) (

) (

) (

) (

) (

) (

5

81

5

81

(d)

3

44

6.20


Simplify the following. (3 4)

3. (a)

5) (

5255

(b)

14314

(c)

1391316

(d)

5

6262

4. (a)

188

(b)

5

24554

(c)

12754

(d)

73285

6.21


1. Determine whether each of the following numbers is a perfect square. If yes, find the square roots of the number.

(a) 24 (b) 36

(c) 110 (d) 450

(e) 784 (f) 1 089

2. Determine whether each of the following is a surd.

(a) 45 (b) 120

(c) 200

98 (d)

224

55

(e) 69.1 (f) 8.4


(a) 81 (b) 196

(c) 484 (d) 144

1

(e) 4

169 (f)

100

121

4. Find the values of the following numbers. (Give your answers correct to 3 significant figures.)

(a) 21 (b) 120

(c) 950 (d) 7.0

(e) 6

55 (f)

5

227

6.22


5. Without using a calculator, find two consecutive numbers between which each of the following lies.

(a) 22 (b) 35

(c) 60 (d) 140

(e) 380 (f) 480


(a) 400

361 (b)

729

256

(c) 324

625 (d) 2.89

(e) 4.84 (f) 0.002 5

7. Find the values of the following expressions. (Give your answers correct to 3 significant figures.)

(a) 22 139 (b) 22 1721

(c) 22 1029 (d) 22 1832

(e) 22 25408 (f) 132435 22

8. Find the values of the unknowns in the following. (Give your answers correct to 3 significant figures.)

(a) B

7

9C A

a

(b)

R

38

Q

P

b

(c)

B

6

c

CA

11

6.23


9. Find the values of the unknowns in the following. (Give your answers correct to 3 significant figures.)

(a) 10

C

B

A

14

p

(b) Q

511

RP q

(c)

R

1420

PQ r

10. In the figure, BDC is a straight line. Find the value of c. (Give your answer correct to 3 significant figures.)

11. In the figure, find the length of each side of square ABCD. (Give your answer correct to 3 significant figures.)

12. In the figure, the area of ABC is 2cm 364 . Find the length

of AB. (Give your answer correct to 3 significant figures.)

16

B

c

D C

A

5

DA

CB

10 cm

C

A

B

26 cm

6.24


13. In the figure, cm 34 AB and 15:8: BCAC . Find the length of BC.

C

34 cm

B

A

14. In the figure, find the perimeter of trapezium ABCD.

C

28 cmEB

DA 12 cm

12 cm

15. The figure shows parallelogram ABCD. E is a point on AB such that 4:1: EBAE . If cm 20 CD and cm 12 AD , find the area of parallelogram ABCD. (Give your answer correct to 3 significant

figures.)

D C

A E B

12 cm

20 cm

16. In the figure, cm 18 PQ , cm 12 QR and cm 10 PS .

P

18 cm

S

Q R

10 cm

12 cm

(a) Find the length of QS.

(b) Find the length of RS.

(Give your answers correct to 3 significant figures.)

6.25


17. In the figure, find the perimeter of quadrilateral ABCD. (Give your answer correct to 3 significant figures.)

C

D

BA

5 cm

7 cm

13 cm

18. In the figure, 47WX , 40XY and 32YZ .

X

YZ

W

q

p

47

32

40

(a) Find the value of p.

(b) Find the value of q.

(c) Find the area of quadrilateral WXYZ.

(Give your answers correct to 3 significant figures.)

19. In the figure, find the perimeter of pentagon PQRST. (Give your answer correct to 3 significant figures.)

P S

RQ

25 cm

7 cm

30 cm

T

6.26


20. Determine whether each of the following triangles is a right-angled triangle. If yes, indicate the right angle.

(a)

B

24

32

40

C

A (b) 15

189

B

C

A (c) 7.5

18

19.5

A

C

B

21. In the figure, the areas of squares ABCD, DPQR and ARST are 2cm 65 ,

2cm 90 and 2cm 155

respectively. Determine whether ADR is a right angle.

Area 90 cm2

SR

TA

B

C

P

D

Q

Area 65 cm2

Area 155 cm2

22. (a) Prove that PQR in the figure is a right angle.

(b) Find the area of PQR.

3

Q R

P

5

34

6.27


23. In the figure, D is a point on BC and DCBD .

(a) Prove that ABC is a right-angled triangle.

(b) Find the value of h. (Give your answer correct to 3 significant figures.)

24. (a) Prove that ABC in the figure is a right-angled triangle.

(b) If PRPQ 43 , prove that PQRABC ~ .

25. In the figure, DCBADB .

(a) Prove that DBCABD ~ .

(b) Find the perimeter of quadrilateral ABCD. (Give your answer correct to 3 significant figures.)

26. In the figure, AC and BD intersect at E.

(a) Prove that BDAC .

(b) Find the lengths of BC and DC.

(c) Is BCD a right-angled triangle? Explain briefly.

B36D C

A

15 h39

C

2

B Q R

P

A

2.5

1.5

D

5

A

B

C

7

74

B C

A

E

D

15

30

16

9

12

6.28


27. The lengths of the hour hand and minute hand of a watch are 8 mm and 15 mm respectively. What is the distance between the tips of these two hands when it is 9 o’clock?

12

6

8 mm

15 mm9 3

28. A ladder of 5 m long leans against a vertical wall. If the upper end of the ladder is 3.5 m above the ground, what is the distance between the lower end of the ladder and the wall? (Give your answer correct to 3 significant figures.)

5 m3.5 m

29. In each of the following figures, find the coordinates of B and the length of OA.

(a)

x

y

A (12, 16)

O B

(b)

x

y

A (12, 9)

O

B

6.29


30. One end of a rope of 6.4 m long is fixed at point D on the ground, while the other end is at point B on a lamp post. The distances from point B to top A and to bottom C of the lamp post are the same. If m 6.3 DC , find the height of the lamp post. (Give your answer correct to 1 decimal place.)

3.6 m

6.4 m

D C

B

A

31. In the figure, two vertical poles are 5 m apart and the height of the shorter pole is 3 m. If the distance between the tops of the poles is 7 m, what is the height of the longer pole? (Give your answer correct to 3 significant figures.)

7 m

3 m

5 m

32. In the figure, find the perimeter of ABC.

x

y

A (9, 6)

O

B (3, 6)

C (3, 3)

6.30


33. In the figure, the lighthouse is 20 m high. The top A of the lighthouse is connected to points C and D on the ground by two wires, where CBD is a straight line. If AC AD 30 m, find the distance between C and D. (Give your answer correct to 3 significant figures.)

30 m30 m20 m

A

C B D

34. The figure shows a rectangular top ABCD of a desk. If the area of ABCD is 2m 98.0 and BCAB 2 ,

find the length of diagonal AC. (Give your answer correct to 3 significant figures.)

CD

BA

35. The length and width of a field are 130 m and 74 m respectively.

74 m

130 m

A D

CB

Field

The following are two routes to walk from point A to point C.

Route 1: Walking along the sides of the field (i.e. A B C).

Route 2: Walking diagonally across the field (i.e. A C).

What is the difference in length between the above two routes? (Give your answer correct to 3 significant figures.)

36. Given that the area of a square wooden board is 2cm 125 , find the length of the diagonal of the board.

(Give your answer correct to 3 significant figures.)

Area 125 cm2

Diagonal

6.31


37. Oscar walked due west from point A at a speed of 60 m / min. After 3 minutes, he reached point B. A tree is 425 m and 385 m away from points A and B respectively.

N

425 m385 m

B A

(a) Prove that point B is due north of the tree.

(b) If Oscar continued to walk due west from point B at the same speed for 5 minutes, he reached point C finally. Find the distance between point C and the tree. (Give your answer correct to 3 significant figures.)

38. In the figure, find the length of PR. (Give your answer correct to 3 significant figures.)

P (2, 4) Q (4, 4)

S (4, 2) R (2, 2)

Ox

y

39. Find the area of the polygon in the figure. (Give your answer correct to 3 significant figures.)

1 cm

2 cm

3 cm

4 cm

6.32


40. Simplify the following.

(a) 72 (b) 45

(c) 317 (d) 510

(e) 411 (f) 96


(a) 28 (b) 99

(c) 104 (d) 140

(e) 264 (f) 756


(a) 64

3 (b)

100

27

(c) 49

80 (d)

144

32

(e) 50

216 (f)

784

135


(a) 546 (b) 4510

(c) 4260 (d) 5

125

(e) 15

180 (f)

33

243

6.33



(a) 752015 (b) 18

12540

(c) 54

7021 (d)

10512

315

45. Rationalize the denominators of the following.

(a) 10

3 (b)

13

6

(c) 6

27 (d)

21

70

(e) 42

18 (f)

8

36


(a) 2

15 (b)

120

6

(c) 45

147 (d)

5

1082

(e) 54

33 (f)

18

154

6.34



(a) 7

64 (b)

15

32

(c) 48

11 (d)

128

25

(e) 72

135 (f)

30

168


(a) 31334 (b) 7278

(c) 2

10310 (d) 5512

(e) 116119 (f) 3

152

3

1514


(a) 272123 (b) 60154

(c) 639285 (d) 50128

(e) 203458 (f) 5431502

6.35


35

Worksheet 6A (page 6.1) 1. (a) 17 or 17 (b) 30 or 30

(c) 0.02 or 0.02 (d) 238

or

238

2. (a) Yes (b) No (c) Yes (d) No

3. (a) Between 5 and 6 (b) Between 13 and 14

4. (a) 8.94 (b) 32 (c) 1.90 (d) 0.516 (e) 10.4 (f) 14.5

Worksheet 6B (page 6.3) 1. (a) 30 (b) 21.2 (c) 17.3 (d) 27.1

2. 12

3. 120 cm2

4. 82.1 cm2

5. 25.6 cm

6. (a) 18.0 m (b) 21.7 m

Worksheet 6C (page 6.9) 1. (a) No (b) Yes, PRQ

2. (b) 73.5

3. (a) 84 (b) Area 72.6 ;2

25 h

h

4. (b) 108

Worksheet 6D (page 6.13) 1. 1 281 km

2. 21.2 cm

3. 70 units

4. 213 km

5. 0.186 m

Worksheet 6E (page 6.17) 1. (a) 39 (b) 8 (c) 7343 (d) 625

2. (a) 112 (b) 73 (c) 153 (d) 76

3. (a) 13

7 (b)

233

(c) 914

(d) 15

24

4. (a) 8 (b) 23 (c) 56

(d) 43

Worksheet 6F (page 6.19)

1. (a) 66

(b) 552

(c) 11119

(d) 2103

2. (a) 66

(b) 332

(c) 559

(d) 3332

3. (a) 57 (b) 144 (c) 137 (d) 5

68

4. (a) 25 (b) 613 (c) 318 (d) 77

Build-up Exercise 6A (page 6.21) 1. (a) No (b) Yes, 6 or 6 (c) No (d) No (e) Yes, 28 or 28 (f) Yes, 33 or 33

2. (a) Yes (b) Yes (c) No (d) Yes (e) No (f) Yes

3. (a) 9 or 9 (b) 14 or 14

(c) 22 or 22 (d) 121

or

121

(e) 2

13 or

213 (f)

1011

or

1011

4. (a) 4.58 (b) 11.0 (c) 30.8 (d) 0.837 (e) 3.03 (f) 6.74

5. (a) Between 4 and 5 (b) Between 5 and 6 (c) Between 7 and 8 (d) Between 11 and 12 (e) Between 19 and 20 (f) Between 21 and 22

6. (a) 2019

or

2019 (b)

2716

or

2716

(c) 1825

or

1825 (d) 1.7 or 1.7

(e) 2.2 or 2.2 (f) 0.05 or 0.05

7. (a) 15.8 (b) 27.0 (c) 27.2 (d) 26.5 (e) 39.2 (f) 12.5

Build-up Exercise 6B (page 6.22) 8. (a) 11.4 (b) 8.54 (c) 12.5

9. (a) 9.80 (b) 9.80 (c) 14.3

10. 12.5

11. 7.07 cm

12. 38.2 cm

13. 30 cm

6.36


14. 72 cm

15. 226 cm2

16. (a) 15.0 cm (b) 19.2 cm

17. 35.0 cm

18. (a) 51.2 (b) 20.4 (c) 1 120

19. 87.5 cm

Build-up Exercise 6C (page 6.26) 20. (a) Yes, ABC (b) No (c) Yes, BAC

21. Yes

22. (b) 7.5

23. (b) 23.4

25. (b) 38.8

26. (b) 34 ,20 DCBC (c) No

Build-up Exercise 6D (page 6.28) 27. 17 mm

28. 3.57 m

29. (a) B(12, 0), 20OA units (b) B(0, 9), 15OA units

30. 10.6 m

31. 7.90 m

32. 36 units

33. 44.7 m

34. 1.57 m

35. 54.4 m

36. 15.8 cm

37. (b) 488 m

38. 7.21 units

39. 6.41 cm2

Build-up Exercise 6E (page 6.32) 40. (a) 28 (b) 25 (c) 1717

(d) 10100 (e) 121 (f) 6296 1

41. (a) 72 (b) 113 (c) 262

(d) 352 (e) 662 (f) 216

42. (a) 83

(b) 10

33 (c)

754

(d) 32

(e) 536

(f) 28153

43. (a) 18 (b) 215 (c) 706

(d) 5 (e) 32 (f) 3

44. (a) 150 (b) 3

50 (c)

357

(d) 21

Build-up Exercise 6F (page 6.33)

45. (a) 10103

(b) 13

136 (c)

269

(d) 3

2110 (e)

7423

(f) 29

46. (a) 230

(b) 10

5 (c)

15157

(d) 51512

(e) 22

(f) 3302

47. (a) 778

(b) 15

304 (c)

1233

(d) 16

25 (e)

430

(f) 5352

48. (a) 317 (b) 710 (c) 2105

(d) 511 (e) 113 (f) 154

49. (a) 312 (b) 156 (c) 737

(d) 23 (e) 518 (f) 6

Documents

Chapter 1 Directed Numbers Chapter 8 · PDF fileChapter 1 Directed Numbers Chapter 2 Basic Algebra Chapter 3 Basic Geometry Chapter 4 Linear Equations in One Unknown Chapter 5 Percentages