Chapter 7 Directed Numbers 7A p.2 Chapter 8 Using Algebra ...

1

Chapter 7 Directed Numbers

7A p.2

7B p.18

Chapter 8 Using Algebra to Solve Problems (I)

8A p.34

8B p.47

8C p.61

8D p.73

Chapter 9 Using Algebra to Solve Problems (II)

9A p.82

9B p.95

9C p.113

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For mathematics problems consultation, please email to the following address:

[email protected]

For Maths Corner Exercise, please obtain from the cabinet outside Room 309

2

F1B: Chapter 7A

Date Task Progress

Lesson Worksheet

○ Complete and Checked ○ Problems encountered ○ Skipped

(Full Solution)

Book Example 1

○ Complete ○ Problems encountered ○ Skipped

(Video Teaching)

Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)

Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)

Consolidation Exercise


(Full Solution)

Maths Corner Exercise 1A Level 1


Teacher’s Signature

___________ ( )




___________ ( )




___________ ( )

3

Maths Corner Exercise 1A Multiple Choice

○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s Signature

___________ ( )

E-Class Multiple Choice Self-Test

○ Complete and Checked ○ Problems encountered ○ Skipped Mark:

_________

4

Book 1B Lesson Worksheet 7A (Refer to §7.1)

7.1 Review on Areas of Simple Polygons

(a) Splitting method

Step 1111: Divide the polygon into simple figures.

Step 2222: Add up their areas.

e.g.

Example 1 Instant Drill 1

Find the area of the pentagon ABCDE in the figure.

Sol Straight line CE divides the pentagon into a

square and a triangle.

Area of square ABCE = 4 × 4 m2

= 16 m2

Area of △CDE =2

1× 4 × 3 m2

= 6 m2

Area of pentagon ABCDE = (16 + 6) m2

= 22 m2

Find the area of the pentagon PQRST in the figure.

Sol Straight line PS divides the pentagon into a

______________ and a ______________.

Area of △______

=

Area of rectangle ________

=

Area of pentagon PQRST

=

1. Find the area of the pentagon ABCDE in the 2. Find the area of the pentagon ABCDE in the

P

Q R

S

T

4 cm

3 cm

___ cm ___ cm

(5 – ___) cm

= ___ cm

___ cm

A

B C

D E

4 m 3 m

4 m

The required area:

+

A

B C

E

4 m

4 m

D

3 m

C

E

4 m

= +

area of the

polygon

area of the

rectangle area of the

triangle

A

B C

D E

4 m 3 m

4 m

P

Q R

S

T

4 cm

5 cm

3 cm

5

figure.

Draw a straight line FC such that the

pentagon is divided into a rectangle and a

______________.

figure.

Extend ______ such that the pentagon is

divided into a ______________ and a

______________.

3. Find the area of the polygon ABCDEFG in the

figure.

4. Find the area of the polygon ABCDEF in the

figure.

○○○○→→→→ Ex 7A 1–6

(b) Filling method

A B 2 mm

8 mm

D C

E F

3 mm

2 mm

4 mm

A B

C

E

9 cm

D

7 cm

5 cm

5 cm

A

5 cm B C

D E 4 cm

3 cm

4 cm

A

B

C D

E F

G

3 m

3 m 2 m

2 m

2 m

2 m

6

Step 1111: Use simple figures to fill the polygon to form a larger simple figure.

Step 2222: Subtract the areas of the filled part from the area of the larger figure.

e.g.


Find the area of the shaded region below.

Sol Area of rectangle AEFG

= 8 × 5 cm2

= 40 cm2

Area of △BCD =2

1× 2 × 2 cm2

= 2 cm2

Area of the shaded region = (40 − 2) cm2

= 38 cm2

Find the area of the shaded region below.

Sol Area of △ECB

=2

1× ( ) × ( ) cm2

=

Area of △AED =

Area of the shaded region =

5. Find the area of the shaded region below.


A B 6 cm

12 cm D

C

E F

2 cm

G

4 cm 5 cm

A B

D C

E F

8 cm

G 2 cm

4 cm

H

2 cm

2 cm 2 cm

A B 2 cm

8 cm

D

C

E

F

2 cm

G

5 cm

The required area:

–

A

8 cm

E

F G

5 cm

B 2 cm D

C

2 cm

= –

area of the

polygon

area of the

rectangle area of the

triangle

A

B

5 cm D C 4 cm

3 cm

6 cm

E

The required area: area of △______ – area of △______

7



○○○○→→→→ Ex 7A 7–12

○○○○→→→→ Ex 7A 13, 14


Find the unknown in the figure.

Find the unknown in the figure.

Sol Area of the shaded region

= 12 × 6 cm2 � Using (i)

= 72 cm2

Area of the shaded region

= y × 8 cm2 � Using (ii)

i.e. y × 8 = 72

y = 9

Sol Area of the shaded region

=2

1× 14 × ( ) cm2

=

Area of the shaded region

=2

1× 6 × ( ) cm2

=

○○○○→→→→ Ex 7A 15, 16

y cm 8 cm 6 cm

12 cm

Consider the area of the parallelogram in two ways:

(i) (ii)

6 cm y cm

3 cm

14 cm

y cm 8 cm

12 cm

6 cm

A B

6 cm

C

E

F 2 cm

G 4 cm

6 cm D

A

B

5 m

C

E

2 m

8 m D

4 m Area of the shaded region = area of trapezium EBCD –

area of △_____ –

area of △_____

(base)

(height)

(base) (height)

8

9. The area of polygon ABCDEF in the figure is

20 m2. Find the unknown.

Area of rectangle ________ =

Area of parallelogram ________ =

Area of polygon ABCDEF = 20 m2

( ) + ( ) = 20

=

10. Find the unknown in the figure.

○○○○→→→→ Ex 7A 17

�� ‘Explain Your Answer’ Question

11. Figures I and II are shown below. Which of them has a larger area? Explain your answer.

Area of Figure I =

Area of Figure II =

∵ ________ ( < / > ) ________ ∴ Figure has a larger area.

A

C E

2 cm

D

F

B

3 cm

2 cm

2 cm

Figure I Figure II

3 cm

Find the areas of Figures I and II respectively.

G

I

K 3 cm

J

L

H

1 cm

2 cm

5 cm

A B

p m

C

E

2 m

D

F

5 m

Area = 52 cm2

A B

C

E

10 cm

D F

k cm

6 cm

6 cm

Area of the pentagon = area of rectangle ________ –

area of ___________ This polygon is formed by rectangle __________ and parallelogram __________.

Remember to write down the reason.

9

�� Level Up Questions

12. The figure shows the floor plan of an indoor

playground in a restaurant.

(a) Find the area of the indoor playground.

(b) If we need to cover the floor of the

playground with carpet and the cost of

each m2 of carpet is $300, find the total

cost required.

(a) Area of rectangle ADEF

=

(b) Total cost =

13. In the figure, BCF, DCG and HGFE are straight lines. Find the area of the shaded region.

Area of the shaded region: + –

A

C

E

5 cm

D

F

B

4 cm 2 cm G H

6 cm

2 cm

5 cm

A

C

1 m

D

F

B

4 m 5 m

4 m

G

H

9 m

E

Area of the indoor playground = area of rectangle ADEF – area of _________________ –

area of _________________

10

New Century Mathematics (2nd Edition) 1B

7 Area and Volume (I)

Level 1

Find the areas of the following polygons. [Nos. 1–6]

1. 2. 3.

4. 5. 6.

Find the areas of the shaded regions in the following figures. [Nos. 7–14]

7. 8. 9.

Consolidation Exercise 7A ��

11

10. 11. 12.

13. 14.

Find the unknowns in the following figures. [Nos. 15–18]

15. (a) (b)

16. (a) (b)

17. (a) (b)

12

18. (a) (b)

19. A logo in the shape of number ‘7’ is printed on the T-shirt as

shown. Find the area of the logo.

20. The figure shows the floor plan of a lawn.

(a) Find the area of the lawn.

(b) Suppose we need to hire workers to mow the lawn. The cost

of mowing each m2 of the lawn is $15. Find the cost required.

21. The figure shows a garden ABCDE.

(a) Find the area of the garden ABCDE.

(b) Part of the garden (region ABCE) is used to plant roses. Find

the area of the region ABCE.

Level 2

Find the areas of the following polygons. [Nos. 22–25]

22. 23.

13

24. 25.


26. 27.

28. 29.


30. 31.

14

32.

In the figure, the area of the polygon ABCDE is 104 cm2. Find the value of y.

33. In the figure, ABCG is a trapezium. BCDE is a

parallelogram. BE and CG intersect at F. It is given

that the area of the polygon ABCDEFG is 99 cm2.

(a) Express the areas of trapezium ABCG,

parallelogram BCDE and △BCF in terms of y.

(b) Find the value of y.

34. In the figure, a road sign is formed by three parts: right-angled

triangle ABC, square DEIJ and parallelogram FGHI.

(a) Find the area of the polygon ABCDEIJ.

(b) If the area of the road sign is 13 300 cm2, find the length

of EF.

15

35.

The map shows a concert hall ABC which is in a triangular shape. The hall is located by a straight

road DEF, where DE = 40 m and EF = 60 m. The length of the paths AD, BE, and CF are 50 m, 30 m

and 80 m respectively. Find the area of the concert hall ABC.

36.

The figure shows two icons of letters ‘F’ and ‘Y’. Which icon has a smaller area? Explain your answer.

37.

The figure shows the floor plans of Brian’s courtyard and garage. He wants to cover the floors of these

two places with cement. It is known that the cost of each m2 of cement is $320. If Brian’s budget is

$28 000, does he have enough money to buy the cement required? Explain your answer.

16

38. (a) Find the area of the shaded region in the figure.

(b) Grace makes four identical shapes as shown in (a) to

form a logo on the right. She wants to cover the logo

with gold foil priced at $0.6 per mm2. She claims that

the total cost is less than $140. Do you agree? Explain

your answer.

(Note: For the remaining part with area less than 1 mm2,

its cost is also $0.6.)

17

Consolidation Exercise 7A (Answer)

1. 36 cm2 2. 21 cm2

3. 53 cm2 4. 66 mm2

5. 30 cm2 6. 55 m2

7. 33 cm2 8. 15 cm2

9. 150 mm2 10. 38 m2

11. 27 cm2 12. 98 cm2

13. 21 cm2 14. 34.5 cm2

15. (a) 6 (b) 12

16. (a) 7.2 (b) 4.5

17. (a) 10 (b) 4

18. (a) 2 (b) 3

19. 130 cm2

20. (a) 78 m2 (b) $1 170

21. (a) 220 m2 (b) 100 m2

22. 87 cm2 23. 51 mm2

24. 99.5 cm2 25. 69 m2

26. 39 m2 27. 48 m2

28. 48 cm2 29. 35 cm2

30. 60 cm2 31. 56 cm2

32. 9

33. (a) trapezium ABCG: (8 + 4y) cm2,

parallelogram BCDE: 12y cm2,

△BCF: 3y cm2

(b) 7

34. (a) 6 800 cm2 (b) 70 cm

35. 1 600 m2

36. icon of letter ‘Y’

37. yes

38.(a) 61.45 mm2 (b) no

18

F1B: Chapter 7B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 6


(Video Teaching)

Book Example 7


(Video Teaching)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)

Book Example 10


(Video Teaching)

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)

19



(Full Solution)

Maths Corner Exercise 7B Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 7B Multiple Choice


___________ ( )



_________

20

Book 1B Lesson Worksheet 7B (Refer to §7.2)

7.2A Prisms

(a) A solid with a uniform cross-section in

the shape of a polygon is called a prism.

(b) A prism is named according to the number of sides or the shape of

its bases.

e.g.

Triangular prism Rectangular prism Square prism

1. Write down the shape of the base of each of the following prisms.

(a) (b)

2. Write down the height of each of the following prisms.

(a) (b)

7.2B Total Surface Areas of Prisms

For prisms of any shape,

total surface area = total area of two bases + total area of all lateral faces

where total area of all lateral faces = perimeter of base × height

e.g. Total surface area of the prism on the right

= base + base + all lateral faces


11 m

5 m 2 m 3 cm

7 cm

4 cm

base

height lateral

faces

uniform cross-section

base

� Rectangular prisms and square prisms are also called cuboids.

21

In the figure, the base of the prism is a triangle.

(a) Find the area of base.

(b) Find the total area of all lateral faces.

(c) Find the total surface area of the prism.

In the figure, the base of the prism is a trapezium.

(a) Find the area of base.

(b) Find the total area of all lateral faces.

(c) Find the total surface area of the prism.

Sol (a) Area of base

=2

1× 5 × 12 cm2

= 30 cm2

(b) Total area of all lateral faces

= (5 + 12 + 13) × 10 cm2

= 300 cm2

(c) Total surface area of the prism

= (2 × 30 + 300) cm2

= 360 cm2

Sol (a) Area of base

=2

1( + ) × ( ) cm2

=

(b) Total area of all lateral faces

= ( ) × ( ) cm2

=

(c) Total surface area of the prism

= ( ) cm2

=

3. In the figure, the base of the prism is a square.

Find the total surface area of the prism.

Total area of the two bases

= 2 × ( × ) cm2

=

Total area of all lateral faces

=

∴ Total surface area of the prism

= ( + ) cm2

=

4. Find the total surface area of the prism.

○○○○→→→→ Ex 7B 1–8 (a)


5 cm

5 cm 8 cm

3 cm 7 cm

7 cm

4 cm

4 cm

9 cm

13 cm

12 cm

10 cm

5 cm

The required area:

base + base + all lateral

faces

4 cm

6 cm 5 cm

3 cm

5 cm

� perimeter

of base × height

22

In the figure, the total surface area of the prism is

232 cm2. Find the value of p.

In the figure, the total surface area of the prism is

60 cm2. Find the value of y.

Sol Total area of the two bases

= 2 × (10 × 8) cm2

= 160 cm2


= 2(10 + 8) × p cm2

= 36p cm2

∴ Total surface area = (160 + 36p) cm2

i.e. 160 + 36p = 232

36p = 72

p = 2

Sol Total area of the two bases

=


=

∴ Total surface area =

i.e. =

5. In the figure, the total surface area of the prism

is 384 m2. Find the value of a.

6. In the figure, the total surface area of the prism

is 560 cm2. Find the value of d.

○○○○→→→→ Ex 7B 11, 14

6 m a m

13 m

8 cm

10 cm

p cm 3 cm

4 cm

5 cm 5 cm

y cm

15 cm

11 cm 8 cm

d cm

� perimeter of

base × height

23

7.2C Volumes of Prisms

Volume of prism = area of base × height

e.g. Volume

= 30 × 7 cm3

= 210 cm3


Find the volume of the prism below.

Find the volume of the prism below.

Sol Volume of the prism

=

×+ 4)73(

2

1× 5 cm3

= 100 cm3

Sol Volume of the prism

=

× ( ) cm3

=

7. Find the volume of the prism below.

8. Find the volume of the prism below.

○○○○→→→→ Ex 7B 1–8 (b)

3 cm

4 cm

7 cm

5 cm

A

B

C

D

E

F

6 cm

8 cm

15 cm

6 m

7 m

8 m

4 m

7 m 8 m

7 cm

area of base

= 30 cm2

Identify the base and the height of the prism first.

24


It is given that the volume of the prism below is

180 cm3. Find the value of y.

Sol Volume of the prism = 180 cm3

20 × y = 180

y = 9

It is given that the volume of the prism below is 42

cm3. Find the value of a.

Sol Volume of the prism = ( ) cm3

( ) × ( ) = ( )

=

9. It is given that the volume of the prism below

is 24 cm3. Find the value of p.

Volume of the prism = ( ) cm3

( ) × ( ) × ( )= ( )

=

10. It is given that the volume of the prism below

is 60 m3. Find the value of k.

○○○○→→→→ Ex 7B 9, 10, 12, 13

11. As shown in the figure, water of 2 000 cm3 is poured into an empty container in the shape of a prism

with base area 250 cm2. If no water overflows, find the depth of the water.

Let ____ cm be the depth of the water.

○○○○→→→→ Ex 7B 19

�� ‘Explain Your Answer’ Questions

area of base

= 20 cm2

y cm

7 cm

area of base = a cm2 x cm

k m

6 m

5 m

2 m 3 cm

2 cm

p cm

Consider the water in the container as a prism. The depth

of the water is regarded as the __________ of the prism.

Area of base = 250 cm2

25

12. Two tanks in the shape of prisms are shown

in the figure. Tank A is full of water, and

tank B is empty.

(a) Find the volume of water in tank A.

(b) If the water in tank A is poured into

tank B, will the water overflow?

Explain you answer.

(a) Volume of water in tank A =

(b) Volume of tank B

=

∵ __________ ( < / > ) __________

∴ The water (will / will not) overflow.

13. The figure shows a watch box in the shape of a prism.

Its base is a regular pentagon.

(a) Find the total surface area of the watch box.

(b) The surface of the watch box is printed in colour and

the printing cost of each cm2 is $0.02. The manager

claims that the total cost of printing one watch box is

not more than $10. Do you agree? Explain you answer.


14. Hillary won a competition. She got a trophy in the shape of a

10 cm 10 cm

30 cm

Tank A Tank B

16 cm

12 cm 22 cm

14 cm

Compare the volumes of tank B and the water.

10 cm

30 cm

5 cm

5 cm

5 cm

5 cm

Area of base

= 84 cm2

7 cm

10 cm


26

prism as shown on the right. Find

(a) the area of base,

(b) (i) volume,

(ii) total surface area

of the trophy.

15. The figure shows a souvenir in the shape of a prism.

Its base is a triangle. It is given that the volume of

the souvenir is 480 cm3.

(a) Find the value of y.

(b) Find the total surface area of the souvenir.

12 cm

y cm

13 cm 13 cm

8 cm

Use splitting method or filling method to find the area of base first.

10 cm

10 cm

5 cm 5 cm

5 cm

5 cm

27


7 Area and Volume (I)

Level 1

For each prism in Nos. 1–8, find

(a) the total surface area,

(b) the volume.

1. 2. 3.

4. 5. 6.

7. 8.

Find the value of the unknown in each of the following prisms. [Nos. 9–14]

9.

x cm

area = 25 cm2

volume = 200 cm3 10.

Consolidation Exercise 7B ��

28

11. 12.

13. 14.

15. The figure shows a plastic box (with a lid) in the shape of a

rectangular prism.

(a) Find the total surface area of the box.

(b) If each m2 of plastic sheet costs $150, what is the total cost of

making 6 such boxes?

16. The figure shows a block in the shape of a prism. Its base is a

regular pentagon.

(a) Find the total surface area of the block.

(b) The surface of the block is painted in red. If the cost of

painting each cm2 of the block is $0.05, what is the total

cost of painting 800 such blocks?

17. The figure shows a chocolate box in the shape of a prism,

whose base is a right-angled trapezium. Find the volume of

the chocolate box.

18. A factory produces 500 wooden wedges in the shape of prisms.

The figure shows the size of each wedge. Find the total volume

of wood required to produce these 500 wedges.

5 cm

14 cm

3.5 cm

29

19.

As shown in the figure, water of 1 575 cm3 is pumped away from a container in the shape of a prism

with base area 225 cm2. Find the decrease in the water level.

20. The figure shows an empty container in the shape of a rectangular

prism. It is 20 cm long and 10 cm wide. Manson pours 1 700 cm3 of

milk into the container. If no milk overflows, find the depth of milk

in the container.

Level 2

For each prism in Nos. 21–26, find

(a) the total surface area,

(b) the volume.

21. 22.

23. 24.

30

25. 26.

27. The figure shows an E-shaped wooden block.

(a) Find the volume of the wooden block.

(b) If each cubic centimetre of the wood weighs 0.4 g,

find the weight of the wooden block.

28. The figure shows a cubic container which is made of glass of

1 cm thick. It has 4 lateral faces and one base.

(a) Find the volume of the space inside the container.

(b) If 1 cm3 of glass weighs 2.5 g, find the weight of the container.

29. In the figure, the volume of a soap in the shape of a rectangular

prism is 180 cm3.

(a) Find the value of t.

(b) Find the total surface area of the soap.

30. The figure shows a prism whose base is a parallelogram. The

total surface area of the prism is 577 mm2.

(a) Find the value of x.

(b) Find the volume of the prism.

32 cm

31

31. The figure shows a model in the shape of a prism. The base

of the prism (the shaded part) is a hexagon formed by a

rectangle and a trapezium. If the volume of the model is

159 m3, find

(a) the value of x,

(b) the total surface area of the model.

32.

The figure shows a piece of rectangular cardboard of length 80 cm and width 40 cm. Sharon cuts a

square of side 5 cm away from each of the four corners of the cardboard. Then, she folds the

cardboard along the dotted lines to make a box. Find the capacity of the box obtained.

33.

As shown in the figure, a rectangular tank is full of water originally. After draining water from the

tank twice as shown, what is the final volume of water in the tank?

34. The figure shows a metal block in the shape of a prism. Some

identical blocks are put into a rectangular water tank of length

0.4 m, width 0.3 m and height 0.2 m. The original depth of water

in the tank is 17 cm and all blocks are immersed in water. What is

the maximum number of blocks to be put into the tank such that

no water overflows? Explain your answer.

32

35. In the figure, a reservoir in the shape of a prism is 300 m long and 100 m wide. The depth of water in

the reservoir is 0.5 m at the shallow end and 2.5 m at the deep end. The difference in height between

the edge of the reservoir and the water surface is 8 m.

300 m

8 m

2.5 m

100 m

0.5 m

(a) Find the volume of water in the reservoir.

(b) If some water is added to the reservoir so that the volume becomes four times of that

in (a), what is the difference in height between the edge and the water surface?

36.

The figure shows a rectangular prism and a triangular prism.

(a) Find the volume of the rectangular prism.

(b) It is given that the two prisms are of the same volume.

(i) Find the value of x.

(ii) Which prism has a greater total surface area? Explain your answer.

37. In the figure, the metal block is in the shape of a prism whose base is a trapezium. It is melted to make

30 identical metal sticks in the shape of hexagonal prisms.

(a) Find the volume of the metal block.

(b) If the height of each metal stick is 12 cm, find its base area.

33

Consolidation Exercise 7B (Answer)

1. (a) 294 cm2 (b) 343 cm3

2. (a) 104 mm2 (b) 60 mm3

3. (a) 408 cm2 (b) 416 cm3

4. (a) 108 cm2 (b) 48 cm3

5. (a) 984 m2 (b) 1 440 m3

6. (a) 244 cm2 (b) 220 cm3

7. (a) 972 mm2 (b) 1 440 mm3

8. (a) 1 776 cm2 (b) 3 240 cm3

9. 8 10. 6

11. 30 12. 12

13. 8 14. 9

15. (a) 27 m2 (b) $24 300

16. (a) 190.7 cm2 (b) $7 628

17. 800 cm3 18. 61 250 cm3

19. 7 cm 20. 8.5 cm

21. (a) 446 mm2 (b) 630 mm3

22. (a) 284 cm2 (b) 264 cm3

23. (a) 476 cm2 (b) 420 cm3

24. (a) 704 m2 (b) 570 m3

25. (a) 768 cm2 (b) 1 152 cm3

26. (a) 468 m2 (b) 540 m3

27. (a) 504 cm3 (b) 201.6 g

28. (a) 27 900 cm3 (b) 12 170 g

29. (a) 6 (b) 216 cm2

30. (a) 9.5 (b) 798 mm3

31. (a) 5 (b) 172.6 m2

32. 10 500 cm3 33. 54 000 cm3

34. 51

35. (a) 45 000 m3 (b) 3.5 m

36. (a) 1 080 cm3

(b) (i) 16 (ii) rectangular prism

37. (a) 1 440 cm3 (b) 4 cm2

34

F1B: Chapter 8A

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 1


(Video Teaching)

Book Example 2


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )



___________ ( )



_________

35

Book 1B Lesson Worksheet 8A (Refer to §8.1B)

8.1B Rectangular Coordinate Plane

Fig. 1 shows a rectangular coordinate plane. The x-axis

and the y-axis intersect at a point called the origin.

The position of a point can be represented by an ordered

pair (a , b), i.e. the coordinates of that point.

e.g. The coordinates of the origin are (0 , 0),

the coordinates of point P are (2 , 3), and

the coordinates of point Q are (–2 , –1).

1. Write down the x-coordinate of each of the following points.

(a) A(3 , 6) (b) B(–2 , –7)

2. Write down the y-coordinate of each of the following points.

(a) C(–9 , 1) (b) D(4 , –5)

○○○○→→→→ Ex 8A 1, 2


Write down the coordinates of point A in the figure

below.

Sol The coordinates of A are (–2 , 3).

Write down the coordinates of points B and C in the

figure below.

Sol The coordinates of B are ( , ).

The coordinates of C are ( , ).

3. Write down the coordinates of points E and F

in the figure below.

4. Write down the coordinates of points G and H

in the figure below.

○○○○→→→→ Ex 8A 3, 4, 6, 7

A

C

B

E

F

H

G

Fig. 1

P(2 , 3)

Q(–2 , –1)

y-axis

x-axis

origin

x-coordinate of P

y-coordinate of P

Note the scales of the axes.

Write x-coordinate first, then y-coordinate.

36

The x-axis and the y-axis divide a rectangular coordinate

plane into 4 quadrants. The signs of coordinates of the

points in each quadrant are indicated in Fig. 2.

e.g. Both the x-coordinate and the y-coordinate of any

point in quadrant I are positive.

In Fig. 1 on the previous page, point P lies in

quadrant I while point Q lies in quadrant III.

Note: Points lying on the x-axis and the y-axis do not belong to any quadrant.


Write down the coordinates of point A in the figure,

and the quadrant where A lies in.

Sol The coordinates of A are (–1 , 2), and A lies in

quadrant II.

Write down the coordinates of point B in the figure,

and the quadrant where B lies in.

Sol The coordinates of B are ( , ),

and B lies in quadrant .

5. Write down the coordinates of point C in the

figure, and the quadrant where C lies in.

6. Write down the coordinates of point D in the

figure, and the quadrant where D lies in.

○○○○→→→→ Ex 8A 5, 8, 9


A

B

C

D

Fig. 2

quadrant II

(– , +)

quadrant I

(+ , +)

quadrant IV

(+ , –)

quadrant III

(– , –)

37

Plot points E(3 , 4) and F(–6 , –2) in a rectangular

coordinate plane.

Sol

Plot points G(5 , 1), H(4 , –2) and I(–3 , 2) in a

rectangular coordinate plane.

Sol

7. Plot points J(2 , 0) and K(– 4 , –3) in the given


8. Plot points M(0 , –2), N(0 , 2.5) and P(2 , –1)

in the given rectangular coordinate plane.

○○○○→→→→ Ex 8A 10


Write down the coordinates of the point of

intersection of the two given straight lines in the

figure below.

Sol The coordinates of the point of intersection are

(–1 , 1).

Write down the coordinates of the point of

intersection of the two given straight lines in the

figure below.

Sol The coordinates of the point of intersection are

( , ).

9. Write down the coordinates of the point of 10. Write down the coordinates of the point of

E

F

The y-coordinate of J is ____.

It lies on the (x / y) axis. Both the x-coordinates of M and N are ____.

They lie on the (x / y) axis.

38

intersection of the two given straight lines in

the figure below.

intersection of the two given straight lines in

the figure below.

○○○○→→→→ Ex 8A 13

11. (a) Two points A and C are shown in the rectangular

coordinate plane on the right. Plot points B(–5 , 1)

and D(3 , 1), and then draw lines AB, AC and BD.

Which number is formed?

(b) Find the coordinates of the point of intersection of

lines AC and BD.

(a) The number formed is ‘ ’.

(b) ○○○○→→→→ Ex 8A 11, 12, 14


12. (a) Plot the following points in a rectangular coordinate plane:

A(– 4 , 4), B(4 , 4), C(2 , 2) and D(– 4 , – 4).

(b) Kathy claims that a quadrilateral can be formed by drawing lines AB, BC, CD and DA. Do you

agree? Explain your answer. (a), (b)

From the graph, after drawing lines AB, BC, CD and DA, a quadrilateral (can /

cannot) be formed. ∴ The claim is (agreed / disagreed).


C

A

39


A(–1 , 5), B(3 , –3), C(2 , 4) and D(–5 , –3).

(b) Draw a line passing through A and B. Let P be the point of intersection of AB and the y-axis. Find

the coordinates of P.

(c) Draw a line passing through C and D. Let Q be the point of intersection of CD and the x-axis. Find

the coordinates of Q.

(a)

(b) From the graph, the coordinates of P are ( , ).

(c)

14. (a) Plot points A(– 4 , 5), B(6 , –5), C(5 , 0) and D(0 , –5) in a rectangular coordinate plane.

(b) Let M be the point of intersection of AB and CD.

(i) Write down the coordinates of M.

(ii) Which quadrant does M lie in?

(a), (b)

40


8 Introduction to Coordinates

Level 1

1. Write down the x-coordinate of each of the following points.

(a) M(3 , −2) (b) N(−8 , −10)

2. Write down the y-coordinate of each of the following points.

(a) P(5 , 4) (b) Q(6 , −3)

3. Write down the coordinates of points A and B in the given


4. Write down the coordinates of point P in each of the following rectangular coordinate planes.

(a) (b)

(c) (d)


41

5. Write down the coordinates of point Q in each of the following rectangular coordinate planes.

(a) (b)

(c) (d)

6. The figure on the right shows a rectangular coordinate plane.

(a) Write down the coordinates of points K, L, M and N in the


(b) Write down the quadrants where K, L, M and N lie in.

7. In the rectangular coordinate plane on the right, which points are

(a) with x-coordinates equal to −2?

(b) with y-coordinates equal to −2?

8. The figure shows the map of an estate.

(a) Write down the coordinates of the blocks

A, B, C, D, E, F and G.

(b) Which block(s) has/have the least x-coordinate?

(c) Which block(s) has/have the largest y-coordinate?

9. In the rectangular coordinate plane on the right, how

many points

42

(a) lie on the x-axis?

(b) lie on the y-axis?

(c) lie in quadrant II?

(d) lie in quadrant III?

10. In the rectangular coordinate plane on the right, write down the

coordinates of the point(s) which lie(s) in

(a) quadrant I,

(b) quadrant IV.

11. In each of the following, plot the specified point in the given rectangular coordinate plane.

(a) A(2 , −1) (b) B(0 , −4)

(c) C(−2 , −1.5) (d) D(5 , −3)

43

12. In each of the following, plot the specified point in the given rectangular coordinate plane.

(a) E(4 , 2) (b) F(−3 , 0)

(c) G(−5 , 3) (d) H(−7 , 5)


P(−1 , 4), Q(−1 , −2), R(3 , 4), S(−1 , 1) and T(3 , −2).

(b) Draw lines PQ, RS and ST. Which English letter is formed?


A(6 , −2), B(6 , 4), C(3 , 1) and D(−3 , 1).

(b) Draw lines AB, BC and CD. Which line is perpendicular to

(i) the x-axis?

(ii) the y-axis?

15. In each of the following, write down the coordinates of the point of intersection of the two given

straight lines.

(a) (b)

44

16. In each of the following, write down the coordinates of the point of intersection of the two given

straight lines, and also the quadrant where the point lies in.

(a) (b)

Level 2

17. Write down the coordinates of points A, B, C, D and E in the

rectangular coordinate plane on the right.

18. An icon is drawn on the rectangular coordinate plane on the

right. Write down the coordinates of points A, B, C, D, E, F, G

and H in the rectangular coordinate plane.

19. Write down the quadrant where each of the following points lies in:

P(5 , −3), Q(−5 , 0), R(−3 , 4), S(−6 , −6), T(2 , 7) and U(0 , 9).

20. Consider the following points in a rectangular coordinate plane:

A(10 , 1), B(−2 , 5), C(3 , 0), D(−6 , −8), E(0 , −1),

F(2 , −3), G(−1 , 9), H(0 , 4), I(−5 , 3), J(−7 , 0)

(a) Write down the point(s) which lie(s) in quadrant II.

(b) Write down the point(s) which do(es) not lie in any quadrant.

45

21. (a) Plot the following six points in a rectangular coordinate plane:

A(2 , 0), B(−4 , 3), C(−2 , 2), D(0 , 1), E(4 , −3) and F(6 , −2).

(b) Draw a line passing through A and B. Which of the above points does not lie on the line drawn?

22. (a) Plot points A(−9 , −2), B(−8 , −6), C(−3 , −5) and D(−2 , −2) in a rectangular coordinate plane.

(b) Let E be the point of intersection of AC and BD.

(i) Write down the coordinates of E.

(ii) Which quadrant does E lie in?

23. (a) Draw a quadrilateral with vertices A(−6 , −3), B(−4 , 3), C(9 , 2) and D(6 , −7) in a rectangular

coordinate plane. Is the origin O outside the quadrilateral?

(b) Find the coordinates of the point of intersection of AB and the x-axis.

(c) Find the coordinates of the point of intersection of AD and the y-axis.

(d) Let M be the point of intersection of the two diagonals of quadrilateral ABCD.

(i) Which axis does M lie on, x-axis or y-axis?

(ii) Find the coordinates of M.

24. (a) Plot the following four points in a rectangular coordinate plane:

E(1 , 3), F(−0.5 , 2), G(−2 , −1) and H(−3.5 , 1).

(b) (i) Let M(m , 3) be a point in the rectangular coordinate plane such that EM is a horizontal line

and EM intersects GH at M. Find the value of m.

(ii) Let N(−0.5 , n) be another point such that FN is a vertical line and FN intersects GH

at N. Gary claims that m is less than n. Do you agree? Explain your answer.

25. (a) Which quadrant does P(1 , −5) lie in?

(b) It is given that P, Q and R are three points in different quadrants. Find the coordinates of Q and R

such that

(i) ∠PQR = 90°,

(ii) ∠QPR = 90°.

46


1. (a) 3

(b) −8

2. (a) 4

(b) −3

3. A(−3 , 3), B(2 , −1)

4. (a) (−1 , −2)

(b) (1 , 0)

(c) (−2 , 5)

(d) (1.5 , 2.5)

5. (a) (0 , 3)

(b) (−3 , −6)

(c) (5 , −5)

(d) (9 , 10)

6. (a) K(−4 , 2), L(0 , −3), M(3 , −4), N(4 , 1)

(b) K: quadrant II, L: none,

M: quadrant IV, N: quadrant I

7. (a) Q, R

(b) R, S

8. (a) A(−8 , 16), B(−16 , 4), C(−8 , −8),

D(8 , −4), E(16 , −8), F(24 , 12),

G(12 , 16)

(b) block B

(c) block A and block G

9. (a) 1

(b) 3

(c) 4

(d) 0

10. (a) C(5 , 15)

(b) D(5 , −5), E(15 , −5)

13. (b) K

14. (b) (i) AB

(ii) CD

15. (a) (1 , 2)

(b) (−2.5 , 3)

16. (a) (2 , −2), quadrant IV

(b) (−3 , −4), quadrant III

17. A(−1.5 , −2), B(1 , −1.7), C(0.7 , 0.9),

D(−1.2 , 0.6), E(−1.8 , −0.7)

18. A(−5 , −1.8), B(−5.6 , −5.2), C(−4.4 , −7.8),

D(−3.2 , −5.2), E(0.8 , −5.2), F(2 , −7.8),

G(3.2 , −5.2), H(2.6 , −1.8)

19. P: quadrant IV, Q: none, R: quadrant II,

S: quadrant III, T: quadrant I, U: none

20. (a) B, G, I

(b) C, E, H, J

21. (b) E

22. (b) (i) (−5 , −4)

(ii) quadrant III

23. (a) no

(b) (−5 , 0)

(c) (0 , −5)

(d) (i) y-axis

(ii) (0 , −1)

24. (b) (i) −5

(ii) yes

25. (a) quadrant IV

(b) (i) Q(−1 , −5), R(−1 , 1)

(or other reasonable answers)

(ii) Q(−1 , −5), R(1 , 1)


47

F1B: Chapter 8B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 3


(Video Teaching)

Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)

Book Example 6


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )



___________ ( )


○ Complete and Checked ○ Problems encountered

Mark:

48

○ Skipped _________

49


8.2 Lengths of Line Segments in the Rectangular Coordinate System

(a) For points lying on a horizontal line,

they have the same y-coordinate.

(b) If A(a , k) and B(b , k) are two points

lying on a horizontal line and a > b,

then AB = a – b.


In the figure, AB is a horizontal line. Find the

length of AB.

Sol AB

= [4 – (–3)] units �4 – (–3) = 4 + 3

= 7 units

In the figure, CD and EF are horizontal lines.

(a) Find the length of CD.

(b) Find the length of EF.

Sol (a) CD = [( ) – ( )] units

=

(b) EF =

(a) For points lying on a vertical line,

they have the same x-coordinate.

(b) If P(h , p) and Q(h , q) are two points

lying on a vertical line and p > q,

then PQ = p – q.

F(–2 , –1) E(–8 , –1) O

C(1 , 4)

D(6 , 4) A(–3 , 5) B(4 , 5)

O

A(a , y)

O

B(b , y) C(c , y)

A(a , k)

O

B(b , k)

O Q(x , q)

P(x , p)

R(x , r)

Q(h , q)

O

P(h , p)

50


In the figure, GH is a vertical line. Find the length

of GH.

Sol GH

= [(–1) – (–5)] units

= 4 units

In the figure, IJ and KL are vertical lines.

(a) Find the length of IJ.

(b) Find the length of KL.

Sol (a) IJ = [( ) – ( )] units

=

(b) KL =

1. In the figure, AB is a vertical line while PQ is

a horizontal line.

(a) Find the lengths of AB and PQ.

(b) Hence, determine which line segment is

longer, AB or PQ.

(a) AB =

PQ =

(b) ∵ __________ > __________

∴ The line segment (AB / PQ) is

longer.

2. In the figure, DC is a horizontal line while RS

is a vertical line.

(a) Find the lengths of DC and RS.

(b) Hence, determine which line segment is

longer, DC or RS.

○○○○→→→→ Ex 8B 1, 2, 4

R(– 4 , –1)

C(3 , –3)

S(– 4 , –8)

O

D (–5 , –3)

P(1 , 4) Q(7 , 4)

A(–1 , 5)

O

B(–1 , –2)

I(–3 , 4)

L(1 , –2)

J(–3 , 2)

K(1 , 3)

O

G(2 , –1)

H(2 , –5)

O

51

3. (a) Determine whether the line passing

through T(2 , 10) and U(2 , 8) is a

horizontal line or a vertical line.

(b) Calculate the distance between T and U.

(a) ∵ T and U have the same __-

coordinate.

∴ TU is a (horizontal / vertical) line.

(b) Distance between T and U

= [( ) – ( )] units

=

4. (a) Determine whether the line passing

through R(– 4 , 3) and S(11 , 3) is a

horizontal line or a vertical line.

(b) Calculate the distance between R and S.

○○○○→→→→ Ex 8B 3


The figure shows a rectangle ABCD, whose sides

are either horizontal or vertical.

(a) Find the coordinates of B.

(b) Find the perimeter of rectangle ABCD.

The figure shows a rectangle EFGH, whose sides

are either horizontal or vertical.

(a) Find the coordinates of E.

(b) Find the perimeter of rectangle EFGH.

Sol (a) Let (p , q) be the coordinates of B.

∵ BC is a vertical line.

∴ x-coordinate x-coordinate

of B of C

p = 7

∵ AB is a horizontal line.

∴ y-coordinate y-coordinate

of B of A

q = 5

∴ The coordinates of B are (7 , 5).

(b) AB = (7 − 1) units = 6 units

BC = (5 − 2) units = 3 units

Perimeter of rectangle ABCD

= (AB + BC) × 2

= (6 + 3) × 2 units

= 18 units

Sol (a) Let (a , b) be the coordinates of E.

∵ _____ is a vertical line.

∴ x-coordinate x-coordinate

of ___ of ___

=

(b) EF = [( ) − ( )] units

=

Perimeter of rectangle EFGH

= [( ) + ( )] × ( )

=

5. The figure shows a rectangle ABCD, whose 6. In the figure, ABCD is a trapezium. AB and

E

G

F(6 , 4)

H(2 , –3)

O

B

D

A(1 , 5)

C(7 , 2)

O

Are their x-coordinates or y-coordinates the same?

=

=

=

52

sides are either horizontal or vertical.

(a) Find the coordinates of D.

(b) Find the perimeter of rectangle ABCD.

○○○○→→→→ Ex 8B 8, 9

DC are horizontal lines while AD is a vertical

line.

(a) Find the coordinates of A.

(b) If BC = 13 units, find the perimeter of

trapezium ABCD.

○○○○→→→→ Ex 8B 10

7. Given that the distance between points L and

M in the figure is 8 units, find the value of a.

∵ LM = _____ units

∴ ( ) − ( )= ( )

a =

8. Given that the distance between points E and

F in the figure is 10 units, find the value of d.

○○○○→→→→ Ex 8B 11, 12 �� ‘Explain Your Answer’ Questions

9. A(10 , 5), B(10 , 3) and C(–1 , 3) are three points in a rectangular coordinate plane. Is △ABC a right-

L(2 , 3) M(a , 3)

O

8 units

B(8 , 7)

D(4 , 2) C(20 , 2)

O

A

B

D

A(–8 , –3)

C(–2 , –7)

O

F(11 , d)

E(11 , 8)

O

10 units

Are there any horizontal lines or vertical lines among AB, BC and AC?

Is LM a horizontal line? Why?

53

angled triangle? Explain your answer.

∵ A and B have the same ____-coordinate.

∴ AB is a (horizontal / vertical) line.

10. The figure shows a quadrilateral KLMN, whose sides are

either horizontal or vertical.

(a) Find the coordinates of L and N.

(b) Is KLMN a square? Explain your answer.

(a) Let ( , ) and ( , ) be the

coordinates of L and N respectively.

∵ ____ is a vertical line.

∴ ____-coordinate of ____= ____-coordinate of ____

=

(b)


11. In the figure, A(6 , a) and B(6 , –3) are two points lying on a

Compare the lengths of the sides of KLMN.

L

N

O

K(3 , 4)

M(–5 , –3)

A(6 , a)

O C

54

vertical line. C is the point of intersection of AB and the x-axis.

(a) Find the coordinates of C.

(b) If AC = 2CB, find the value of a.

(a) ∵ C is a point lying on the ____-axis.

∴ ____-coordinate of C = ____

∵ C is a point lying on the (horizontal / vertical) line AB.

∴ ____-coordinate of C = ____-coordinate of ____

=

(b) CB = [( ) − ( )] units =

AC = 2 × _____

AC = ( ) × ( ) units

( ) – ( ) = ( )

=

12. In the figure, all the straight lines are either horizontal

or vertical.

(a) Find the coordinates of A, B, C, D and E.

(b) In the figure, an ant moves from A to E via B, C

and D in order. Find the total length of the ant’s

path.

(a) From the figure, the coordinates of A, B, C, D and E are

( , ), ( , ), ( , ), ( , ) and ( , ) respectively.

(b) AB =

BC =

Total length of the ant’s path

= AB + ( ) + ( ) + ( )

=

Find the length of each line segment, and then add

them up.

O

E

D C

B A

55



Level 1

1. Find the length of each line segment in the figure.

2. In the figure, KL is a horizontal line while MN is a vertical line.

(a) Find the lengths of KL and MN.

(b) Hence, determine which line segment is longer,

KL or MN.

3. For each of the following pair of points,

(i) determine whether the line passing through them is a horizontal line or vertical line,

(ii) calculate the distance between them.

(a) A(−5 , 3), B(6 , 3) (b) C(−4 , 2), D(−8 , 2)

(c) E(−1 , −5), F(−1 , 4) (d) G(−7 , −6), H(−7 , −13)

4. In the figure, a car travels from P to S via Q and R in order. Find

the total distance travelled by the car.

5. The figure shows △ ABC, where AB is vertical and BC is

horizontal.


(b) Find the lengths of AB and BC.

(c) Is △ABC an isosceles triangle? Explain your answer.

6. Find the perimeter of the rectangle EFGH in the figure.


56

7. The figure shows a square ABCD. Find its perimeter.

8. In each of the following figures, PQRS is a rectangle whose sides are either horizontal or vertical.

(i) Find the coordinates of Q and S.

(ii) Find the perimeter of rectangle PQRS.

(a) (b)

9. In each of the following figures, TUVW is a rectangle whose sides are either horizontal or vertical.

(i) Find the coordinates of U and W.

(ii) Find the perimeter of rectangle TUVW.

(a) (b)

10. In each of the following figures, ABCD is a rectangle whose sides are either horizontal or vertical.

(i) Find the coordinates of B and D.

(ii) Is ABCD a square? Explain your answer.

57

(iii) Find the perimeter of rectangle ABCD.

(a) (b)

11. In the figure, PQRS is a trapezium.

(a) Find the coordinates of R.

(b) If QP = 13 units, find the perimeter of the trapezium PQRS.

12. In the figure, AB is a horizontal line and CB is a vertical line.


(b) If AO = OC = 5 units, find the perimeter of the

quadrilateral OABC.

13. Given that the distance between points M and N in the figure is

9 units, find the value of n.

14. Given that the distance between points P and Q in the figure is

12 units, find the value of q.

Level 2

15. For each of the following pair of points,

(i) determine whether the line passing through them is a horizontal line or vertical line,

(ii) calculate the distance between them.

58

(a) K(−5.4 , 2.7), L(9.6 , 2.7)

(b) M(−0.8 , −3.2), N(−0.8 , −7.9)

(c)

3

26,

3

14P ,

−

3

12,

3

14Q

(d)

−−

8

1,

2

13R ,

−−

8

1,

5

36S

16. In the figure, B is the point of intersection of AC and the x-axis.


(b) Find the lengths of AB and BC.

(c) If DE = 2BC, determine which line segment, DE or AB, is longer.

17. E(−5 , 3), F(−5 , −2), G(2 , −2) and H(2 , 3) are four points in a rectangular coordinate plane. Given

that EFGH is a rectangle, what is its perimeter?

18. The figure shows a polygon ABCDEF, whose sides are

either horizontal or vertical.

(a) Find the coordinates of A, B, C, D, E and F.

(b) Find the perimeter of the polygon.

19. The figure shows a rectangle KLMN, where LM = 3KL.

(a) Find the lengths of KL and LM.

(b) Find the coordinates of M and N.

59

20. The figure shows a polygon PQRSTUVW, whose sides are

either horizontal or vertical. Find its perimeter.

21. A(−1 , −4) and C(7 , 3) are two vertices of a rectangle ABCD, whose sides are either horizontal or

vertical.

(a) Is ABCD a square? Explain your answer.

(b) Find the perimeter of ABCD.

22. M(2 , 3) and N(2 , −4) are two points in a rectangular coordinate plane.

(a) Determine whether MN is a horizontal line or a vertical line.

(b) Find the length of MN.

(c) P is another point lying on the line passing through M and N such that MP = MN. Find the

coordinates of P.

23. If the distance between points A(−2 , 3t − 2) and B(−2 , 2t + 1) is 5 units, find the two possible values

of t.

24. Refer to the figure.

(a) Given that the distance between points D and E is

15 units, find the value of x.

(b) If F is a point in a rectangular coordinate plane such

that DF ⊥ DE and DE = 3DF, find the two possible

coordinates of F.

25. The figure shows a rectangle ABCD, whose perimeter is 44

units.

(a) Find the value of a. Hence, find the coordinates of A.

(b) Is ABCD a square? Explain your answer.

60


1. AB = 8 units, CD = 7 units,

EF = 7 units, GH = 4 units

2. (a) KL = 6 units, MN = 5 units

(b) KL

3. (a) (i) horizontal line

(ii) 11 units

(b) (i) horizontal line

(ii) 4 units

(c) (i) vertical line

(ii) 9 units

(d) (i) vertical line

(ii) 7 units

4. 61 units

5. (a) (3 , −5)

(b) AB = 12 units, BC = 12 units

(c) yes

6. 120 units

7. 28 units

8. (a) (i) Q(3 , 0), S(9 , 11)

(ii) 34 units

(b) (i) Q(−2 , 3), S(−17 , 8)

(ii) 40 units

9. (a) (i) U(13 , −4), W(0 , −8)

(ii) 34 units

(b) (i) U(−12 , 2), W(−5 , −9)

(ii) 36 units

10. (a) (i) B(3 , −9), D(12 , −2)

(ii) no

(iii) 32 units

(b) (i) B(6 , −8), D(−4 , 2)

(ii) yes

(iii) 40 units

11. (a) (−5 , −10)

(b) 50 units

12. (a) (4 , −4)

(b) 24 units

13. −1

14. −3

15. (a) (i) horizontal line

(ii) 15 units

(b) (i) vertical line

(ii) 4.7 units

(c) (i) vertical line

(ii) 9 units

(d) (i) horizontal line

(ii) 10

13 units

16. (a) (4 , 0)

(b) AB = 5 units, BC = 2 units

(c) AB

17. 24 units

18. (a) A(−7 , 2), B(−7 , −4), C(1 , −4), D(1 , 0),

E(5 , 0), F(5 , 2)

(b) 36 units

19. (a) KL = 5 units, LM = 15 units

(b) M(6 , −6), N(6 , −1)

20. 42 units

21. (a) no

(b) 30 units

22. (a) vertical line

(b) 7 units

(c) (2 , 10)

23. −2, 8

24. (a) 5

(b) (−13 , 1), (−13 , −9)

25. (a) 3; (−7 , 9)

(b) yes

61

F1B: Chapter 8C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 7


(Video Teaching)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)

Book Example 10


(Video Teaching)

Book Example 11


(Video Teaching)



(Full Solution)

Maths Corner Exercise 8C Level 1



___________ ( )




___________ ( )




___________ ( )

62

○ Skipped

Maths Corner Exercise 8C Multiple Choice


___________ ( )



_________

63

Book 1B Lesson Worksheet 8C (Refer to §8.3)

8.3 Areas of Plane Figures in the Rectangular Coordinate System

To calculate the areas of simple figures in a rectangular coordinate

plane, we can find the lengths of suitable horizontal or vertical line

segments and then use area formulae to obtain the results.


Find the area of rectangle OABC in the figure.

Sol OA = (5 – 0) units = 5 units

OC = (4 – 0) units = 4 units

Area of rectangle OABC

= OA × OC

= 5 × 4 sq. units

= 20 sq. units

Find the area of trapezium ABCD in the figure.

Sol AB =

Area of trapezium ABCD

=

1. Find the area of △ABC in the figure.

2. Find the area of rectangle CDEF in the figure.

○○○○→→→→ Ex 8C 1, 2


C D

F E

A(2 , 4)

C(6 , 1) D(2 , 1)

B(4 , 4)

A

B C

O

A(0 , 5) B(4 , 5)

C(4 , 0)

Can you find the coordinates of A, B and C?

AB and ____ are the two bases of the trapezium, ____ is its

height.

64

In the figure, A(–3 , 0), B(– 4 , 4) and C(5 , 0)

are the vertices of △ABC.

(a) Find the length of AC.

(b) Find the height of △ABC with respect to the

base AC.

(c) Hence, calculate the area of △ABC.

Sol (a) AC = [5 – (–3)] units

= 8 units

(b)

With the notations in the figure,

the coordinates of D are (– 4 , 0).

Height of △ABC with base AC

= BD

= (4 – 0) units

= 4 units

(c) Area of △ABC

=2

1× AC × BD

=2

1× 8 × 4 sq. units

= 16 sq. units

In the figure, E(–1 , 3), F(6 , 3), G(4 , 0) and

H(–3 , 0) are the vertices of parallelogram EFGH.

(a) Find the length of HG.

(b) Find the height of parallelogram EFGH with

respect to the base HG.

(c) Hence, calculate the area of parallelogram

EFGH.

Sol (a) HG =

(b)

(c) Area of parallelogram EFGH

=

3. Find the area of trapezium JKLM in the figure. 4. Find the area of △EFG in the figure.

E F

G H

B

C A D

E F

G H

B

C A

Through B, construct BD perpendicular to CA to meet CA produced at D. Then BD is a height of △ABC.

With the base HG, mark the height.

65

○○○○→→→→ Ex 8C 3–6


Find the area of rectangle PQRS in the figure.

Sol PQ = (6 – 1) units

= 5 units

PS = [5 – (–2)] units

= 7 units

Area of rectangle PQRS

= PQ × PS

= 5 × 7 sq. units

= 35 sq. units

Find the area of △LMN in the figure.

Sol LM = [( ) – ( )] units

=

Area of △LMN

=

5. The figure shows a parallelogram ABCD. 6. The figure shows a trapezium EFGH.

J

K L

M E G

F

N(–1, 1)

M(7 , 4) L(–1 , 4) Q(6 , 5)

S(1 , –2)

P(1 , 5)

R(6 , –2)

Note the scales of the axes.

Mark the height of trapezium JKLM in the figure first.

Are there any horizontal/ vertical line segments in PQRS?

Are there any horizontal/vertical line segments in the figure?

66

(a) Find the value of r.

(b) Find the area of parallelogram ABCD.

(a) r =

(b) BC = [( ) – ( )] units

=

Area of parallelogram ABCD

=

(a) Find the coordinates of K.

(b) Find the area of trapezium EFGH.

○○○○→→→→ Ex 8C 7–10

7. Given that the area of △ABC in the figure is 21 sq. units,

find the value of c.

AB = [( ) – ( )] units

=

BC =

∵ Area of △ABC = ____ sq. units

∴ =

○○○○→→→→ Ex 8C 16, 17


F(2 , 3)

H(–2 , –2)

E(–2 , 1)

G(2 , –3)

K

B(5 , 6)

D(1 , 1)

C(5 , 2)

A(1 , 5)

E(5 , r)

A(–2 , 1)

C(5 , c)

B(5 , 1)

Identify the base and height of parallelogram ABCD.

AE is a (vertical / horizontal) line. __-coordinate of E = __-coordinate of __

67

8. The figure shows a square ABCD and a rectangle

EFGH. Which figure has a smaller area, square

ABCD or rectangle EFGH? Explain your answer.

∵ ________ < ________ ∴ The (square ABCD / rectangle EFGH) has a smaller area.

�� Level Up Question

9. Find the area of polygon ABCDEFG in the figure.

Add horizontal/vertical line(s) in the figure in order to divide

the polygon into simple figures.

A(1 , 5)

D(1 , 2)

B(4 , 5)

E(–6 , –1)

G(–2 , –3)

F(–2 , –1)

C(4 , 2)

H(–6 , –3)

C

D

B A

G F

E


68



Level 1

Find the area of each of the following figures. [Nos. 1–6]

1. 2.

3. 4.

5. 6.


7. 8.

Consolidation Exercise 8C ��

69

9. 10.

11. 12.

13. (a) Draw the quadrilateral PQRS with vertices P(−3 , 4), Q(−3 , −2), R(5 , −2) and S(5 , 4) in a


(b) Find the area of the quadrilateral PQRS.

14. (a) Draw △ABC with vertices A(−1 , 3), B(4 , 3) and C(4 , −4) in a rectangular coordinate plane.

(b) Find the area of △ABC.


15. 16.

17. 18.

19. Given that the area of rectangle PQRS in the figure is 54 sq. units, find

the coordinates of R and S.

70

20. Given that the area of trapezium ABCD in the figure is 35 sq. units, find

the value of c.

Level 2


21. 22.

23. 24.


25. 26.

71

27. 28.

29. 30.

31. (a) Draw a pentagon PQRST with vertices P(−3 , 2), Q(−3 , −4), R(5 , −4), S(6 , −2) and

T(2 , 2) in a rectangular coordinate plane.

(b) Find the area of the pentagon PQRST.

32. (a) Draw a pentagon ABCDE with vertices A(3 , 5), B(−4 , 6), C(−4 , −4), D(3 , −2) and

E(1 , 2) in a rectangular coordinate plane.

(b) Find the area of the pentagon ABCDE.

33. Given that the area of the quadrilateral ABCD in the figure is

22.5 sq. units, find the coordinates of D.

34. Given that the area of the quadrilateral PQRS in the figure is

12 sq. units, find the value of p.

72

Consolidation Exercise 8C

1. 4 sq. units

2. 12 sq. units

3. 18 sq. units

4. 24 sq. units

5. 36 sq. units

6. 64 sq. units

7. 36 sq. units

8. 48 sq. units

9. 24 sq. units

10. 18 sq. units

11. 22.5 sq. units

12. 13 sq. units

13. (b) 48 sq. units

14. (b) 17.5 sq. units

15. 108 sq. units

16. 140 sq. units

17. 152 sq. units

18. 64 sq. units

19. R(−4 , −4), S(2 , −4)

20. −1

21. 28 sq. units

22. 29 sq. units

23. 56.5 sq. units

24. 14 sq. units

25. 21 sq. units

26. 15.5 sq. units

27. 21 sq. units

28. 25 sq. units

29. 37.5 sq. units

30. 23 sq. units

31. (b) 45 sq. units

32. (b) 52.5 sq. units

33. (0 , 1)

34. 2

73

F1B: Chapter 8D

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 12


(Video Teaching)

Book Example 13


(Video Teaching)



(Full Solution)

Maths Corner Exercise 8D Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 8D Multiple Choice


___________ ( )



_________

74

Book 1B Lesson Worksheet 8D (Refer to §8.4A)

8.4A Introduction to Polar Coordinates

In the polar coordinate plane on the right, the position

of point A can be represented by an ordered pair (2 , 30°),

i.e. the polar coordinates of A.

In A(2 , 30°), 2 is the radius vector of A,

30° is the polar angle of A.

Similarly, the polar coordinates of B are (3 , 150°),

the polar coordinates of C are (4 , 270°).


The figure below shows a polar coordinate plane.

(a) Write down the polar coordinates of A and B.

(b) Find the lengths of OA and OB.

Sol (a) The polar coordinates of A and B are

(5 , 60°) and (4 , 200°) respectively.

(b) OA = 5 units

OB = 4 units

The figure below shows a polar coordinate plane.

(a) Write down the polar coordinates of P, Q and

R.

(b) Find the lengths of OP, OQ and OR.

Sol (a) The polar coordinates of P, Q and R

are ( , ), ( , )

and ( , ) respectively.

(b) OP = units

OQ = units

OR = units

1. The figure below shows a polar coordinate 2. The figure below shows a polar coordinate

P

Q

R

A

B

The radius vectors and the polar angles of the above two points are shown as follows:

A: B:

60°

5

A

X O

B

200°

4

X O

B

C

A(2 , 30°)

pole polar axis

� A(2 , 30°)

X 30°

2

O

A

B D E

75

plane.

(a) Write down the polar coordinates of A, B

and C.

(b) Which point is closest to O? Explain your

answer.

(a)

(b) ∵ ___ units < ___ units < ___ units

i.e. _____ < _____ < _____

∴ is closest to O.

plane.

(a) Write down the polar coordinates of D, E

and F.

(b) Which point is closest to O? Which point

is farthest away from O? Explain your

answers.

○○○○→→→→ Ex 8D 1, 4


(a) Plot the following points in a polar

coordinate plane:

A(6 , 40°), B(5 , 160°) and C(4 , 330°).

(b) Find ∠AOB.

Sol (a)

(b) ∠AOB

= ∠XOB – ∠XOA

= 160° – 40°

= 120°

(a) Plot the following points in a polar

coordinate plane:

D(3 , 170°), E(5 , 300°) and F(2 , 0°).

(b) Find ∠DOE.

Sol (a)

(b) ∠DOE

= reflex ∠XOE – ∠_____

=

3. (a) Plot L(3 , 50°), M(4 , 180°) and 4. (a) Plot P(4 , 90°), Q(2 , 180°) and

B

A

C

A

X 40°

B

O

160°

E

X

170° D O

300°

76

N(2 , 340°) in a polar coordinate plane.

(b) Which angle is smaller, ∠LOM or

∠MON? Explain your answer.

(a)

(b)

R(5 , 330°) in a polar coordinate plane.

(b) Which angle is larger, ∠ROP or ∠QOR?

Explain your answer.

(a)

(b)

○○○○→→→→ Ex 8D 2, 3


5. (a) Plot A(3 , 30°), B(3 , 120°) and C(4 , 270°)

in the polar coordinate plane on the right.

(b) Is △BOC an acute-angled triangle? Explain

your answer.


6. The figure shows the map of a zoo. The map is drawn

Consider the largest angle in the triangle.

Penguin House

panda

lion

Find acute

∠XOR first.

R X O

330°

77

on a polar coordinate plane, where the Penguin House is

taken as the pole O.

(a) Write down the polar coordinates of the panda and

lion in the figure.

(b) A tiger and a giraffe are located at (3 , 220°) and

(2 , 340°) respectively. Mark their positions in the

figure.

(c) Which animal is farthest away from the Penguin

House? How far is it?

7. (a) Plot A(3 , 140°), B(4 , 230°), C(2 , 310°) and D(5 , 50°) in a polar coordinate plane.

(b) Do B, O and D lie on a straight line? Explain your answer.

(a)

(b)

78



Level 1

1. The figure shows a polar coordinate plane.

(a) Write down the polar coordinates of the points

P, Q, R and S.

(b) (i) Which point is closest to the pole O?

(ii) Which point is farthest away from the pole O?


(a) Plot the following points in the figure:

A(3 , 60°), B(1 , 180°) and C(4 , 240°).

(b) Which line segment does O lie on, AB, AC or BC?


(a) Plot the following points in the figure:

D(5 , 40°), E(1 , 150°) and F(2 , 280°).

(b) Which angle is the smallest, ∠DOE, ∠EOF or ∠FOD?


Consolidation Exercise 8D ��

79

4. The map of a forest is represented by a polar coordinate

plane in the figure, and the lake is at the pole O.

(a) Write down the polar coordinates of each animal in the

forest.

(b) Which animal is closest to the lake? How close is it?

5. The figure shows a polar coordinate plane which takes a

lighthouse as the pole O. Two boats M and N are moving

towards the lighthouse, and their polar coordinates are (6 ,

230°) and (4 , 320°) respectively.

(a) Mark the positions of boats M and N in the figure.

(b) Which boat is closer to the lighthouse? How far is

it closer to the lighthouse than the other one?

Level 2


(a) Write down the polar coordinates of P and Q.

(b) Plot R(2 , 240°) and S(6 , 30°) in the figure.

(c) (i) Among the points P, Q, R and S, which two points

and O lie on the same straight line? Explain your

answer.

(ii) Hence, find the distance between the two points

obtained in (c)(i).


(a) Plot C(4 , 60°) and D(4 , 330°) in the figure.

(b) Consider △OCD.

(i) Is it an isosceles triangle?

(ii) Is it a right-angled triangle?

Explain your answers.

80


(a) (i) Plot the following points in the figure:

A(5 , 150°), B(2 , 240°), C(2 , 330°) and

D(5 , 60°).

(ii) By observation, which kind of quadrilateral

is ABCD?

(b) Find the lengths of AC and BD.

(c) (i) Is AC ⊥ BD? Explain your answer.

(ii) Hence, find the area of quadrilateral ABCD.


(a) Plot A(6 , 150°), B(2 , 150°) and C(5 , 240°) in the

figure.

(b) Find the area of △ABC.

10. The figure shows the map of a district. The map is drawn on a polar coordinate plane, where the

library is taken as the pole O.

(a) Write down the polar coordinates of the following places.

(i) stadium (ii) railway station

(b) What places are represented by the following polar coordinates?

(i) (3 , 150°) (ii) (5 , 330°)

(c) Are there any places with the same distance from the library as

(i) bookstore? (ii) shopping centre?

If yes, what places are they?

81

Consolidation Exercise 8D

1. (a) P(4 , 20°), Q(3 , 80°), R(5 , 180°),

S(1 , 310°)

(b) (i) S

(ii) R

2. (b) AC

3. (b) ∠DOE

4. (a) bear: (4 , 240°), giraffe: (2 , 330°),

monkey: (3 , 120°)

(b) giraffe, 2 units

5. (b) boat N, 2 units

6. (a) P(5 , 0°), Q(4 , 210°)

(c) (i) Q and S

(ii) 10 units

7. (b) (i) yes

(ii) yes

8. (a) (ii) trapezium

(b) AC = 7 units, BD = 7 units

(c) (i) yes

(ii) 24.5 sq. units

9. (b) 10 sq. units

10. (a) (i) (3 , 60°)

(ii) (6 , 270°)

(b) (i) school

(ii) museum

(c) (i) yes, cinema and park

(ii) no

82

F1B: Chapter 9A

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 1


(Video Teaching)

Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)

Book Example 4


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )



___________ ( )



Mark:

83

○ Skipped _________

84

Book 1B Lesson Worksheet 9A (Refer to §9.1A–B)

9.1A Figures with Reflectional Symmetry

If a plane figure can be folded along a straight line and the two sides of the figure

coincide completely, the figure is said to have reflectional symmetry. The straight

line is called the axis of symmetry.

e.g. The figure on the right has reflectional symmetry.

It has four axes of symmetry: L1, L2, L3 and L4.


Determine whether the figures below have

reflectional symmetry. If so, draw all the axes of

symmetry with dotted lines.

(a) (b)

Sol Only (a) is a figure that has reflectional

symmetry.

(a)


reflectional symmetry. If so, draw all the axes of

symmetry with dotted lines.

(a) (b)

Sol

(a) (b)

○○○○→→→→ Ex 9A 1

1. State the number of axes of symmetry for each of the following figures that have reflectional

symmetry. Also, draw all the axes of symmetry with dotted lines.

(a) (b) (c)

(d) (e) (f)

○○○○→→→→ Ex 9A 2, 3

L1 L2

L3

L4

85

9.1B Figures with Rotational Symmetry

If a plane figure coincides with its original figure n times (where n > 1) in one complete

revolution (i.e. 360°) about a fixed point, then it has n-fold rotational symemetry

(i.e. with order n) and the fixed point is called the centre of rotation.

e.g. The above figure has 2-fold rotational symmetry (or the order of rotational symmetry is 2).



rotational symmetry. If so, use the symbol ‘×’ to

indicate the centre of rotation.

(a) (b)

Sol Only (b) is a figure that has rotational

symmetry.

(b)


rotational symmetry. If so, use the symbol ‘×’ to

indicate the centre of rotation.

(a) (b)

Sol

(a) (b)

2. Each of the following figures has rotational symmetry. Use the symbol ‘×’ to indicate the centre of

rotation. Also, state the order of rotational symmetry.

(a) (b) (c)

(d) (e) (f)

○○○○→→→→ Ex 9A 4, 5

At the beginning

coincide (1st time)

coincide (2nd time)

rotate

180°

rotate

180° centre of

rotation ⊳ The dots show the relative positions only.

86


Refer to the following figures.

(i) Identify the figure(s) with reflectional

symmetry and draw the axes of symmetry with

dotted lines.

(ii) Identify the figure(s) with rotational symmetry

and use the symbol ‘×’ to indicate the centre of

rotation.

(a) (b)

Sol (a)

This figure has reflectional symmetry

but no rotational symmetry.

(b)

This figure has reflectional symmetry

and also rotational symmetry.

� It has four axes of symmetry and 4-fold rotational symmetry.

Refer to the following figures.

(i) Identify the figure(s) with reflectional

symmetry and draw the axes of symmetry with

dotted lines.

(ii) Identify the figure(s) with rotational symmetry

and use the symbol ‘×’ to indicate the centre of

rotation.

(a) (b)

Sol (a)

This figure has (reflectional /

rotational) symmetry but no

(reflectional / rotational) symmetry.

(b)

3. Refer to the following capital letters.

(a) Classify them according to the following.

(i) The letter has reflectional symmetry only.

(ii) The letter has rotational symmetry only.

(b) Are there any letters that have both reflectional and rotational symmetry?

(c) Draw the axes of symmetry for those letters that have reflectional symmetry.

○○○○→→→→ Ex 9A 7, 8

� It has one axis of symmetry.

87


4. Consider Fig. I–Fig. IV below.

Fig. I Fig. II Fig. III Fig. IV

(a) Draw the axes of symmetry for the figures that have reflectional symmetry with dotted lines. Use

‘×’ to indicate the centre of rotation for the figures that have rotational symmetry.

(b) Flora claims that only two of the figures have both reflectional and rotational symmetry. Do you

agree? Explain your answer.

(a)

Fig. I Fig. II Fig. III Fig. IV

(b) From the result of (a), only ____ figure(s) (has / have) both reflectional and rotational symmetry. ∴ The claim is (agreed / disagreed).


5. In each of the following figures, shade one square so that the dotted line becomes the axis of symmetry

of the figure.

(a) (b)

6. In each of the following figures, shade two squares so that the ‘×’ becomes the centre of rotation of the

figure.

(a) (b)


88


9 Symmetry and Transformation

Level 1

1. Determine whether the figures below have reflectional symmetry. If so, draw all the axes of symmetry.

(a) (b) (c)

(d) (e) (f)

2. State the number of axes of symmetry for each of the following figures that have reflectional

symmetry. Also, draw all the axes of symmetry.

(a) (b) (c)

(d) (e) (f)


89

3. Refer to the following capital letters.

(a) Draw the axis (axes) of symmetry for each letter.

(b) Which letters have more than one axis of symmetry?

4. Consider the following figures. Identify the ones that have rotational symmetry and use the symbol ‘×’

to indicate the centre of rotation. Also, state the order of rotational symmetry.

(a) (b) (c)

(d) (e) (f)

5. Consider Fig. A–Fig. C below.

Fig. A Fig. B Fig. C

Identify the ones that have rotational symmetry and state the order of rotational symmetry for each of

them.

6. Fig. A and Fig. B shown on the right are

symmetrical figures.

(a) Which figure has reflectional symmetry?

(b) Which figure has rotational symmetry?

Fig. A Fig. B

90

7. Four district emblems are shown below.

Eastern District Kwun Tong District Sai Kung District Tuen Mun District

(a) Classify them according to the following.

(i) The emblem has reflectional symmetry only.

(ii) The emblem has rotational symmetry only.

(iii) The emblem has both reflectional and rotational symmetry.

(b) Draw the axes of symmetry for those emblems that have reflectional symmetry.

8. Five religious symbols are shown below.

Buddhism Christianity Islam Judaism Taoism

(a) (i) Which symbols have reflectional symmetry only?

(ii) Draw the axis (axes) of symmetry for each of the symbols in (a)(i).

(b) (i) Which symbols have both reflectional and rotational symmetry?

(ii) Draw the axis (axes) of symmetry for each of the symbols in (b)(i).

(iii) State the order of rotational symmetry for each of the symbols in (b)(i).

9. In each of the following figures, shade two squares so that the dotted line becomes the axis of

symmetry of the figure.

(a) (b) (c)

91

10. In each of the following figures, shade two squares so that the ‘×’ becomes the centre of rotation of the

figure.

(a) (b)

(c) (d)

11. The figure on the right is formed by five squares. Remove two squares from the

figure in two different ways so that the new figure has only one axis of

symmetry.

12. The figure on the right is formed by six squares. Add a square to the

figure in two different ways so that the new figure has reflectional symmetry.

13. The figure on the right is formed by seven squares. Move one square in the

figure in two different ways so that the new figure has rotational symmetry.

92

Level 2

14. Consider the following sentence:

(a) How many times do the letters with reflectional symmetry appear?

(b) How many times do the letters with rotational symmetry appear?

15. The word ‘ ’ is formed by letters with rotational symmetry only. Try to write down

two more English words that are also formed by letters with rotational symmetry only.

16. With the dotted lines as the axes of symmetry, complete the following figures that have

reflectional symmetry.

(a) (b) (c)

17. In each of the following figures, shade the minimum number of square(s) so that the dotted lines

become the axes of symmetry.

(a) (b)

93

18. In each of the following figures, shade the minimum number of square(s) so that the ‘×’ becomes the

centre of rotation of the figure.

(a) (b) (c)

19. In the figure, shade the number of square(s) listed below so that

the new figure has both reflectional and rotational symmetry.

(a) 1 (b) 2 (c) 3 (d) 4

20. Use the six identical squares in the figure to form a new figure

that has

(a) reflectional symmetry only,

(b) rotational symmetry only,

(c) both reflectional and rotational symmetry.

21. Use the four identical isosceles triangles in the figure to form

a new figure that has

(a) reflectional symmetry only,

(b) rotational symmetry only,

(c) both reflectional and rotational symmetry.

94


1. (a) yes

(b) no

(c) yes

(d) yes

(e) no

(f) yes

2. (a) 1

(b) 2

(c) 4

(d) 1

(e) 1

(f) 3

3. (b) H and X

4. (a) yes, 3

(b) no

(c) yes, 5

(d) no

(e) yes, 4

(f) yes, 6

5. Fig. B: 2, Fig. C: 4

6. (a) Fig. B

(b) Fig. A

7. (a) (i) Sai Kung District

(ii) Eastern District

(iii) Kwun Tong District

8. (a) (i) Christianity and Islam

(b) (i) Buddhism and Judaism

(iii) Buddhism: 8, Judaism: 6

14. (a) 11

(b) 8

15. SIX, SON (or other reasonable answers)

95

F1B: Chapter 9B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 5


(Video Teaching)

Book Example 6


(Video Teaching)

Book Example 7


(Video Teaching)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )

96

○ Skipped



___________ ( )



_________

97


9.2 Transformation

The process of changing the size, direction or position of a figure is called transformation.

The new figure formed from transformation is called the image of the original figure.

9.2A Translation

For translation:

(a) Every point on the original figure is moved through the same distance in the same

direction to form the image.

(b) The image obtained has the same shape, size and direction as the original figure.

e.g. In the figure on the right, AB is translated 5 units to the right to form the

image A′B′, where AA′ = BB′ = 5 units.


On the given graph paper, draw the image of the

line segment PQ when it is translated

6 units upwards.

Sol


line segment RS when it is translated

4 units to the left.

Sol

1. In the figure, draw the image of each of the following

line segments after translation.

(a) AB is translated 3 units downwards.

(b) CD is translated 5 units to the right.

○○○○→→→→ Ex 9B 2

P

Q

R

S

P′

Q′

P

Q

R

S

A

B C

D

Step 1111:

Translate P and Q 6 units upwards to

P′ and Q′ respectively. Step 2222:

Join P′ and Q′.

A′

B′

A

B 5 units

� PP′ = QQ′ = 6 units

RR′

= SS′ = ___ units

98


On the given graph paper, draw the image of △ABC when it is translated 4 units to the right.

Sol


rectangle EFGH when it is translated

3 units to the left.

Sol

2. On the given graph paper, draw the image of

the figure PQRS when it is translated

5 units upwards.

3. On the given graph paper, draw the image of

the figure WXYZ when it is translated

3 units downwards.

○○○○→→→→ Ex 9B 3, 4

A B

C

E F

G H

P

Q

R S

W X

Y

Z

A B A′ B′

C′

AA′ = BB′ = CC′ = 4 units

C

E F

G H

EE′ = FF′ = GG′ = HH′ = ___ units

99

9.2B Reflection

For reflection:

(a) The image obtained has the same shape and size as the original figure, but the position

is different and the corresponding parts are opposite to one another.

(b) The corresponding parts of the original figure and the image are equidistant from

the axis of reflection.

e.g. In the figure on the right,

AB is reflected in the dotted line ℓ

to form the image A′B′.


In the figure, draw the image △A′B′C′ after

reflecting △ABC in the straight line ℓ .

In the figure, draw the image E′F′G′H′ after

reflecting the quadrilateral EFGH in the straight

line ℓ .

Sol

Sol

A

B

C

E

F

G

H

A

B

C

A′

B′

C′

E

F

G

E′

H

Find the images (i.e. E′, F′, G′ and H′) of all the vertices of quadrilateral EFGH first.

Then join E′, F′, G′ and H′ in order.

To reflect point A in ℓ :

Step 1111: Draw AO ⊥ ℓ such that AO = 2 units.

Step 2222: Draw OA′ ⊥ ℓ such that OA′ = 2 units.

A O

2 units

A O

2 units 2 units

A′

A′

B′

A

B

(axis of reflection)

100

4. In the figure, draw the image △P′Q′R′ after

reflecting △PQR in the straight line ℓ .

5. In the figure, draw the image W′X′Y′Z′ after

reflecting the quadrilateral WXYZ in the

straight line ℓ .

○○○○→→→→ Ex 9B 6, 7

9.2C Rotation

For rotation:

(a) Every point on the original figure is rotated about the centre of rotation through the

same angle to form the image.

(b) The image obtained has the same shape and size as the original figure.

e.g.

OA is rotated clockwise about the point O through 90° to form the image OA′.


Draw the image P′of point P after rotating

anticlockwise about the point O through 90° on the

graph paper.

Sol

Draw the image A′ of point A after rotating

clockwise about the point O through 90° on the

graph paper.

Sol

P

Q

R

W

X

Y Z

A

O A′

A

O

rotate clockwise

through 90°

� ∠AOA′ = 90°,

OA = OA′.

A

O

To which direction? By how many

degrees?

P′

P O

A

O

P O

Rotate anticlockwise

through 90°.

101

6. Draw the image of point R on the given graph paper after each of

the following transformations.

(a) Rotate clockwise about point O through 270° to S.

(b) Rotate anticlockwise about point O through 90° to T.

7. Consider the image of a certain point after each of the following rotations about a point O. If their

positions are the same, join them up.

Rotating clockwise through 90° •



• Rotating anticlockwise through 90°




Draw the image of △ABC on the graph paper after

rotating about the point B through 180°.

Sol

� ∠ABA′ = 180°, AB = A′B.

B and B′ are the same point.

∠CBC′ = 180°, CB = C′B.

Draw the image of the rectangle PQRS on the graph

paper after rotating clockwise about the point P

through 90°.

Sol

B, B′

C

A′

C′

A Q R

S P

clockwise

Q R

S P

B C

R

O

Are S and T at the

same position?

Find the images of A, B and C (i.e. A′, B′ and

C′), and then join A′, B′ and C′.

A

P and P′ are the same point.

QP ⊥ Q′P, QP = Q′P.

RP ⊥ R′P, RP = R′P.

SP ⊥ S′P, SP = S′P.

102

8. Draw the image of each of the following

figures after rotation.

(a) Rotate clockwise about point A through

270°.

(b) Rotate about point F through 180°.

9. Draw the image of each of the following

figures after rotation.

(a) Rotate anticlockwise about point O

through 270°.

(b) Rotate about point P through 180°.

○○○○→→→→ Ex 9B 8

9.2D Enlargement/Reduction

For enlargement (or reduction):

(a) Each side of the image obtained will be enlarged (or reduced) by the same factor.

(b) The image retains the shape and the direction of the original figure.

e.g. (i) If A′B′C′D′ is the image of ABCD after enlargement,

then the enlargement factor =AB

BA ′′=

2

4= 2

(ii) If P′Q′R′S′ is the image of PQRS after reduction,

then the reduction factor =PQ

QP ′′=

4

2=

2

1

A B

C

O

A′

D

G

E

F

D

G

E

F

P

A

B C

A B B′ C

A′

C C′

D D′

Enlargement

P Q Q′ C

P′

R R′ S′

Reduction

S

103


In the figure, B′C′ is the image of BC after

enlargement.

(a) Draw the image of △ABC after

enlargement.

(b) Find the enlargement factor.

In the figure, D′E′ is the image of DE after

reduction.

(a) Draw the image of square DEFG after

reduction.

(b) Find the reduction factor.

Sol (a)

(b) Enlargement factor =BC

CB ′′

=3

6

= 2

Sol (a)

(b) Reduction factor =)(

ED ′′

=

10. In the figure, A′D′ is the image of AD after

enlargement.

(a) Draw the image of the trapezium ABCD

after enlargement.

(b) Find the enlargement factor.

(a)

(b)

11. In the figure, P′R′ is the image of PR after

reduction.

(a) Draw the image of △PQR after

reduction.

(b) Find the reduction factor.

(a)

(b)

○○○○→→→→ Ex 9B 12, 13

E

G

D E′ D′

F

A

C′ B′

A′ C B

P

QA

R

R′ P′

A B

C

D′

A′

D

E

G

D E′ D′

F

A

C B

C′ B′

104


12. (a) On the graph paper below, draw the image D′E′F′G′ of the rectangle DEFG after rotating

clockwise about the point D through 90°, then draw the image D″E″F″G″ of DEFG after rotating

anticlockwise about the point P through 90°.

(b) Steve claims that the images D′E′F′G′ and D″E″F″G″ coincide completely. Do you agree?


(a)

(b) From the figure, the images D′E′F′G′ and D″E″F″G″ (coincide / do not coincide)

completely. ∴ The claim is (agreed / disagreed).


13. Refer to the figure. If the figure ABCDE is enlarged so that the

length of the image of AB becomes 24 units,

(a) find the enlargement factor,

(b) calculate the length of the image of AE after enlargement.

14. Draw the image of the following figure when it is translated 8 units to the left and then 7 units

upwards.

E

G

D

F P

A E

B

D C

105



Level 1

1. In each of the following pairs of figures, one is the image of the other after a transformation

(translation, reflection, rotation, enlargement or reduction). State the kind of transformation for each

pair of figures.

(a) (b)

(c) (d)

(e) (f)

2. Draw the image of each of the following line segments after translation.

(a) Translated 7 units downwards. (b) Translated 5 units to the right.

(c) Translated 4 units to the left. (d) Translated 3 units upwards.


106

3. Draw the image of each of the following figures after translation.

(a) Translate 3 units to (b) Translate 2 units (c) Translate 4 units to

the right. upwards. the left.

4. Edmond translates the mouse 13 units to the right and then draws the image on the graph

paper. Describe the position of the mouse relative to the tree after the translation.

5. In each of the following, figure N is the image of figure M after reflection. Draw the axis of reflection

for each image.

(a) (b)

107

6. Draw the image of each of the following figures after reflection in the straight line ℓ .

(a) (b) (c)

7. The figure shows an English word formed by 4 letters.

(a) Reflect each of the letters in the dotted line. Are the images obtained identical to the letters in the

original word?

(b) Give another English word with 4 letters that has the same reflectional property as the one given

in (a).

8. Draw the image of each of the following after rotation.

(a) Rotate clockwise about point A (b) Rotate about point B through

through 90°. 180°.

(c) Rotate anticlockwise about point C (d) Rotate clockwise about point D

through 90°. through 270°.

108

9. In each of the following, the image is formed after rotating the original figure clockwise about the

centre of rotation through 90°. Use the symbol ‘×’ to indicate the centre of rotation.

(a) (b) (c)

10. Describe how the image P′Q′R′S′T ′U′ is formed after rotating the figure PQRSTU.

11. In each of the following pairs of △ABC and △A′B′C′, determine whether △A′B′C′ is the image of △ABC after one transformation. If so, state the kind of transformation.

(a) (b)

(c) (d)

109

12. Draw the image of each of the following figures after enlargement/reduction.

(a) The reduction factor is

4

1. (b) The enlargement factor is 2.

13. In each of the following figures, B′C′ is the image of BC after enlargement/reduction.

(i) Draw the image obtained after enlargement/reduction.

(ii) Find the enlargement/reduction factor of the transformation.

(a) (b)

(c) (d)

14. Refer to the figures M and N on the right.

(a) Suppose the figure N is the image of

the figure M after enlargement. Find

the enlargement factor.

(b) Suppose the figure M is the image of

the figure N after reduction. Find the

reduction factor.

110

Level 2

15. Draw the image of the following figures after translation.

(a) Translate 4 units downwards, and then (b) Translate 6 units to the right, and then

11 units to the left. 7 units upwards.

16. In the figure, P′Q′R′S′ is the image of the trapezium

PQRS after two translations. Describe the possible

translations.

17. Draw the image of each of the following figures after reflection in the straight line ℓ .

(a) (b)

(c) (d)

111

18. The figure on the right is formed by six identical right-angled

isosceles triangles. It is given that BF ⊥ GE and FB ⊥ AC. Write

down the image after each of the following rotations.

(a) Rotate △FDE about the point D through 180°.

(b) Rotate △BHF anticlockwise about the point F through 90°.

19. The rectangle PQTU on the right is formed by six identical

right-angled isosceles triangles. It is given that RW ⊥ PU, VS ⊥

QT and PQ = 1 unit. Write down the image after each of the

following transformations.

(a) Translate △PQW 2 units to the right.

(b) Reflect △WQR in the line segment WR.

(c) Rotate △WSV anticlockwise about the point S through

270°.

20. Draw the image of the figures after the following

transformations.

(a) Rotate △ABC clockwise about the point B through 90°.

(b) Reflect the image obtained in (a) in the straight line ℓ .

21. The figure shows a plane figure in the shape of the letter ‘H’.

(a) If the figure is enlarged so that QR becomes Q′R′,

(i) draw the image after enlargement on the graph paper,

(ii) find the enlargement factor.

(b) If the figure is reduced so that the length of the image of PQ become 3 units,

(i) find the reduction factor,

(ii) calculate the length of the image of QR after reduction.

112


1. (a) enlargement

(b) rotation

(c) translation

(d) reflection

(e) rotation

(f) reduction

4. The mouse is on the right-hand-side of the

tree.

7. (b) DICE (or other reasonable answers)

10. Rotate PQRSTU anticlockwise about the

point R through 90°. (or other reasonable

answers)

11. (a) yes, enlargement

(b) yes, translation

(c) no

(d) no

13. (a) (ii) 2

1

(b) (ii) 3

5

(c) (ii) 2

3

(d) (ii) 4

3

14. (a) 2

3

(b) 3

2

16. PQRS is translated 7 units upwards and is

then 10 units to the left.


18. (a) △CDB

(b) △EDF

19. (a) △VSU

(b) △WSR

(c) △UST

21. (a) (ii) 2

3

(b) (i) 2

1 (ii) 1 unit

113

F1B: Chapter 9C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 10


(Video Teaching)

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)

Book Example 13


(Video Teaching)

Book Example 14


(Video Teaching)

Book Example 15


(Video Teaching)



(Full Solution)




___________ ( )

114




___________ ( )




___________ ( )

Maths Corner Exercise 9C Multiple Choice


___________ ( )



_________

115

Book 1B Lesson Worksheet 9C (Refer to §9.3A–BI, C)

9.3 Transformations in a Rectangular Coordinate Plane

9.3A Translation

I. Translation to the Left or Right

For any point P(x , y),

(a) the coordinates of the image after P is translated n units to the right are (x + n , y),

(b) the coordinates of the image after P is translated n units to the left are (x – n , y).


If A(1 , 2) is translated 5 units to the right to a point

B, find the coordinates of B.

Sol

Coordinates of B

= (1 + 5 , 2)

= (6 , 2)

If P(5 , – 4) is translated 8 units to the left to a point

Q, find the coordinates of Q.

Sol

Coordinates of Q

=

1. If C(–6 , 4) is translated 2 units to the right to a

point D, find the coordinates of D.

2. If E(–3 , 9) is translated 4 units to the left to a

point F, find the coordinates of F.

○○○○→→→→ Ex 9C 1

A(1 , 2) B

5 units

Translated to the left: The x-coordinate (plus / minus) 8; the y-coordinate remains unchanged.

Q

8 units

P(5 , – 4)

� y-coordinate remains unchanged

after translating to the left or right.

� Translated to the right: The x-coordinate plus 5; the y-coordinate remains unchanged.

116

II. Translation Upwards or Downwards


(a) the coordinates of the image after P is

translated n units upwards are (x , y + n),

(b) the coordinates of the image after P is

translated n units downwards are (x , y – n).


If A(3 , 2) is translated 4 units upwards to a point B,

find the coordinates of B.

Sol Coordinates of B

= (3 , 2 + 4)

= (3 , 6)

If P(6 , 5) is translated 8 units downwards to a

point Q, find the coordinates of Q.

Sol

Coordinates of Q

=

3. If L(–6 , –9) is translated 5 units upwards to a

point M, find the coordinates of M.

4. If R(3 , –10) is translated 7 units downwards to

a point S, find the coordinates of S.

○○○○→→→→ Ex 9C 2

5. X(–5 , 7) is translated 2 units to the right to a point Y.

(a) Find the coordinates of Y.

(b) If Y is translated 9 units upwards to a point Z, find the coordinates of Z.

○○○○→→→→ Ex 9C 3

A(3 , 2)

B

4 units

Translated downwards: The x-coordinate remains unchanged, the y-coordinate (plus / minus) 8.

P(6 , 5) 8 units

Q

� x-coordinate remains unchanged after translating upwards or downwards.

� Translated upwards: The x-coordinate remains unchanged;

the y-coordinate plus 4.

P(x , y)

n units

n units

(x , y – n)

(x , y + n)

117

9.3B Reflection

I. Reflection in the x-axis or y-axis


(a) the coordinates of the image after P

is reflected in the x-axis are (x , –y),

(b) the coordinates of the image after P

is reflected in the y-axis are (–x , y).


(a) If A(1 , 5) is reflected in the x-axis to a point B,

find the coordinates of B.

(b) If C(– 4 , 2) is reflected in the y-axis to a point

D, find the coordinates of D.

Sol (a) The coordinates of B are (1 , –5).

(b) The coordinates of D are (4 , 2).

(a) If E(–5 , 4) is reflected in the x-axis to a point

F, find the coordinates of F.

(b) If G(2 , 3) is reflected in the y-axis to a point

H, find the coordinates of H.

Sol (a) The coordinates of F are ( , ).

(b) The coordinates of H are ( , ).

6. (a) If K(8 , –2) is reflected in the x-axis to a

point L, find the coordinates of L.

(b) If M(7 , –6) is reflected in the y-axis to a

point N, find the coordinates of N.

7. (a) If P(–1 , –3) is reflected in the x-axis to a

point Q, find the coordinates of Q.

(b) If R(–9 , –5) is reflected in the y-axis to a

point S, find the coordinates of S.

○○○○→→→→ Ex 9C 6, 7

C(– 4 , 2) D

A(1 , 5)

B

G(2 , 3)

H

F

E(–5 , 4)

� y-coordinate remains unchanged after

reflecting in the y-axis.

� x-coordinate remains unchanged after reflecting in the x-axis.

� x-coordinate of D = –(– 4) = 4

118

9.3C Rotation

When a point P(x , y) is rotated about the origin O through 90°, 180° and 270°, the

coordinates of the images obtained are as follows:

Direction and

angle of rotation

Coordinates of

the image

anticlockwise, 90°

(or clockwise, 270°) (–y , x)

anticlockwise, 180°

(or clockwise, 180°) (–x , –y)

anticlockwise, 270°

(or clockwise, 90°) (y , –x)


If A(1 , 3) is rotated anticlockwise about the origin

O through 90° to a point B, find the coordinates of

B.

Sol

Coordinates of B

= (–3 , 1)

If P(6 , –2) is rotated clockwise about the origin O

through 90° to a point Q, find the coordinates of Q.

Sol

Coordinates of Q

=

8. If E(7 , 4) is rotated about the origin O through

180° to a point F, find the coordinates of F.

9. If G(–5 , –1) is rotated anticlockwise about the

origin O through 90° to a point H, find the

coordinates of H.

10. If K(–6 , 3) is rotated anticlockwise about the

origin O through 270° to a point L, find the

coordinates of L.

11. If M(2 , –8) is rotated clockwise about the

origin O through 270° to a point N, find the

coordinates of N.

○○○○→→→→ Ex 9C 9–12

P(6 , –2)

To which direction? By how many

degrees?

B

A(1 , 3)

90°

E(7 , 4)

G(–5 , –1)

119

�� ‘Explain Your Answer’ Questions

12. A(–10 , – 4) is translated 9 units to the right to a point B, and C(6 , 3) is translated 7 units downwards

to a point D. Does B coincide with D? Explain your answer. Coordinates of B =

Coordinates of D =

∵ The coordinates of B and D (are / are not) the same. ∴ B (coincides / does not coincide) with D.

13. P(–3 , 7) is rotated about the origin O through 180° to a point Q, and R(–7 , 3) is rotated anticlockwise

about O through 90° to a point S. Is QS a horizontal line? Explain your answer.


14. If a point A is translated 4 units to the right and then 10 units downwards to B(1 , –2), find the

coordinates of A.

Let (a , b) be the coordinates of A,

then

15. A(4 , –2) is reflected in the x-axis to a point B, and B is then rotated clockwise about the origin O

through 90° to a point C. Finally, C is reflected in the y-axis to a point D. Find the coordinates of D.

For points lying on the same horizontal

line, they have the same ___-coordinate.

Try to set up two equations to find the values of a and b.

Find the coordinates of B and D first.

120



Level 1

1. Refer to the figure. Find the coordinates of the image after each of

the following translations.

(a) The point A is translated 4 units to the left.

(b) The point B is translated 7 units upwards.

(c) The point C is translated 3 units to the right.

(d) The point D is translated 2 units downwards.

2. Refer to the figure. Find the coordinates of the image after each

of the following translations.

(a) The point P is translated 6 units upwards.

(b) The point P is translated 2 units downwards.

3. Refer to the figure. Find the coordinates of the image after each

of the following translations.

(a) The point Q is translated 5 units to the right.

(b) The point Q is translated 4 units to the left.

4. If A(4 , −5) is translated 2 units to the left to a point B, find the coordinates of B.

5. C(3 , 2) is translated 4 units downwards to a point D.

(a) Find the coordinates of D.

(b) If D is translated 7 units to the right to a point E, find the coordinates of E.

Consolidation Exercise 9C ��

121


(a) Write down the coordinates of P, Q and R.

(b) △PQR is translated 10 units upwards to form the image △P′Q′R′.

(i) Draw △P′Q′R′ on the graph.

(ii) Write down the coordinates of the vertices of △P′Q′R′.


(a) If a point S is translated 6 units upwards to the point T, find the

coordinates of S.

(b) If a point U is translated 9 units to the right to the point V, find the

coordinates of U.

8. Refer to the figure. If A and B are reflected in the x-axis to the

points A′ and B′ respectively, find the coordinates of A′ and B′.

9. Refer to the figure. If C and D are reflected in the y-axis to the

points C′ and D′ respectively, find the coordinates of C′ and D′.

10. If a point E(−3 , 7) is reflected in the x-axis to E′, and a point F(−5 ,

−8) is reflected in the y-axis to F′, find the coordinates of E′ and F′.

122


(a) Write down the coordinates of L, M and N.

(b) △LMN is reflected in the x-axis to form an image △L′M′N′.

(i) Draw △L′M′N′ on the graph.

(ii) Write down the coordinates of the vertices of △L′M′N′.


(a) Write down the coordinates of point A.

(b) For each of the following rotations, mark the position of the

image and find its coordinates.

(i) Rotate A clockwise about the origin O through 90°.

(ii) Rotate A about the origin O through 180°.

13. If C(7 , −2) is rotated anticlockwise about the origin O through 90° to a point D, find the coordinates

of D.


(a) Write down the coordinates of E, F and G.

(b) △EFG is rotated anticlockwise about the origin O

through 270° to form an image △E′F′G′.

(i) Draw △E′F′G′ on the graph.

(ii) Write down the coordinates of the vertices of △E′F′G′.

123

Level 2

15. The figure shows an image △P′Q′R′. It is formed by

translating △PQR 5 units to the right, and then translating

8 units upwards. Find the coordinates of the vertices of △PQR.

16. (a) If a point M is translated 4 units to the left and then 6 units upwards to N(−1 , 4), find the

coordinates of M.

(b) If M is translated twice to P(−6 , 2), describe the possible translations.

17. In the figure, ℓ is a line parallel to the y-axis and passing through the

point (−2 , 0).

(a) If B is the image of A(−4 , 5) when A is reflected in ℓ , find the

coordinates of B.

(b) If C(1 , −4) is the image of a point D when D is reflected in ℓ , find

the coordinates of D.

18. In the figure, ℓ is a line parallel to the x-axis and passing through the

point (0 , −3).

(a) If F is the image of E(2 , −1) when E is reflected in ℓ ,

find the coordinates of F.

(b) If G(−4.5 , 0) is the image of a point H when H is reflected

in ℓ , find the coordinates of H.

19. It is given that M(m − 3 , −5) is reflected in the x-axis to N(−1 , 2 − n).

(a) Find the values of m and n.

(b) If M is translated to N, describe the translation.

20. It is given that R(r + 3 , −4) is reflected in the y-axis to S(2 , s − 1).

(a) Find the values of r and s.

(b) If R is rotated about the origin O through 180° to a point T, find the length of ST.

124

21. In the figure, ℓ is a vertical line passing through T(−3 , 0). It is

given that A(2 , a + 1) is reflected in the line ℓ to

B(b − 6 , 4).

(a) Find the distance of A from the line ℓ .

(b) (i) Find the values of a and b.

(ii) Find the coordinates of A and B.

22. In the figure, the point B is obtained when a point A is rotated

anticlockwise about the origin O through 270°.

(a) What are the coordinates of A?

(b) It is given that B is rotated clockwise about the origin O

through 90° to a point C.

(i) What are the coordinates of C?

(ii) Without finding any angle, determine whether the

three points A, O and C lie on the same straight line.


23. It is given that P(−4 , 1) is reflected in the y-axis and is then translated 7 units upwards to Q.

(a) Find the coordinates of Q.

(b) If the point P is translated 7 units upwards first and is then reflected in the y-axis, can we obtain

the same image? Explain your answer.

24. E(−1 , 2) is translated 3 units downwards to a point F, and F is then reflected in the y-axis to a point G.

(a) Find the coordinates of F and G.

(b) If E is translated twice to G, describe the possible translations.

(c) If F is rotated to G, describe the possible rotation.

25. M(7 , 3) is translated 2 units to the left to a point N, and N is rotated anticlockwise about the origin O

through 90° to a point P.

(a) Find the coordinates of N and P.

(b) M is rotated about the origin through 180° to a point Q. Describe the possible translation(s) from Q to

P.

125

Consolidation Exercise 9C (Answer)

1. (a) (−2 , −4)

(b) (4 , −1)

(c) (−1 , −5)

(d) (5 , −5)

2. (a) (−3 , 2)

(b) (−3 , −6)

3. (a) (4 , 2)

(b) (−5 , 2)

4. (2 , −5)

5. (a) (3 , −2)

(b) (10 , −2)

6. (a) P(−10 , −10), Q(−5 , −15), R(10 , −15)

(b) (ii) P′(−10 , 0), Q′(−5 , −5), R′(10 , −5)

7. (a) (4 , −1)

(b) (−9 , −8)

8. A′(−8 , 4), B′(−4 , −1)

9. C′(3 , −6), D′(−4 , 5)

10. E′(−3 , −7), F′(5 , −8)

11. (a) L(−4 , 1), M(2 , 0), N(−3 , 4)

(b) (ii) L′(−4 , −1), M′(2 , 0), N′(−3 , −4)

12. (a) (−3 , 4)

(b) (i) (4 , 3)

(ii) (3 , −4)

13. (2 , 7)

14. (a) E(−2 , 2), F(3 , 1), G(2 , 3)

(b) (ii) E′(2 , 2), F′(1 , −3), G′(3 , −2)

15. (b) P(−10 , −1), Q(−5 , −6), R(2 , 1)

16. (a) (3 , −2)

(b) M is translated 9 units to the left and is

then 4 units upwards to P.


17. (a) (0 , 5)

(b) (−5 , −4)

18. (a) (2 , −5)

(b) (−4.5 , −6)

19. (a) m = 2, n = −3

(b) M is translated 10 units upwards to N.

20. (a) r = −5, s = −3

(b) 8 units

21. (a) 5 units

(b) (i) a = 3, b = −2

(ii) A(2 , 4), B(−8 , 4)

22. (a) (−3 , 5)

(b) (i) (3 , −5)

(ii) yes

23. (a) (4 , 8)

(b) yes

24. (a) F(−1 , −1), G(1 , −1)

(b) E is translated 2 units to the right and is

then 3 units downwards to G.


(c) F is rotated anticlockwise about the

origin O through 90° to G.


25. (a) N(5 , 3), P(−3 , 5)

(b) Q is translated 4 units to the right and is

then 8 units upwards to the point P.


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Chapter 7 Directed Numbers 7A p.2 Chapter 8 Using Algebra ...

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