Chapter 7 Rate and Ratio 7A p.2 7B p.9 7C p.24 Chapter 8 ...

1

Chapter 7 Rate and Ratio

7A p.2

7B p.9

7C p.24

Chapter 8 Angles in Triangles and Polygons

8A p.36

8B p.46

8C p.57

8D p.66

8E p.74

Chapter 9 Introduction to Deductive Geometry

9 p.83

For any updates of this book, please refer to the subject homepage:

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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

F2B: 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)

Consolidation Exercise


(Full Solution)

Maths Corner Exercise 7A Level 1


Teacher’s Signature

___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 7A Multiple Choice

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

___________ ( )

E-Class Multiple Choice Self-Test

○ Complete and Checked ○ Problems encountered ○ Skipped Mark:

_________

3

Book 2B Lesson Worksheet 7A (Refer to §7.1)

7.1 Rate

A rate is a comparison of two quantities of different kinds by division.

It usually has a unit.

e.g. (a) Patrick runs 8 m per second.

∴ Running speed = 8 m/s

(b) 3 cups of ice cream are sold at $60.

∴ Price rate =

cups 3

60$ = $20/cup

1. Express the following rates with the units given in brackets.

Situation Rate

(a) There are 10 beds in 2 rooms. (beds/room) rooms ) (

beds ) ( = ( ) beds/room

(b) 6 cans of coke are sold at $42. ($/can) ) (

) ( =

(c) A printer can produce 90 pages in 5 minutes.

(pages/min)

(d) After 4 months, a plant grows 60 cm taller.

(cm/month)

○○○○→→→→ Ex 7A 1–4

Example 1 Instant Drill 1

3 dozen eggs are sold at $54. Find the price

rate of the eggs in

(a) $/dozen eggs,

(b) $/egg.

Sol (a) Price rate =

eggsdozen 3

54$

= $18/dozen eggs

(b) Price rate =

eggsdozen 3

54$

=

eggs 123

54$

×

= $1.5/egg

A company receives 182 phone calls in

2 weeks. Find the frequency of phone calls in

(a) calls/week,

(b) calls/day.

Sol (a) Frequency =

weeks) (

calls ) (

=

(b) Frequency =

1 week = days

4

2. The selling price of 3 boxes of cereal is

$120. Each box is of 2 kg. Find the price

rate of the cereal in

(a) $/box, (b) $/kg.

3. A train travels 9 000 m in 15 minutes. Find

the speed of the train in

(a) m/min, (b) km/h.

○○○○→→→→ Ex 7A 10–14


The typing speed of Kelly is 60 words/min.

How many words can she type in 3 minutes?

Sol Number of words

= 60 × 3

= 180

Chicken wings are sold at $35/kg. Find the

selling price of a pack of 4 kg chicken wings.

Sol Selling price

= ( ) × ( )

=

4. Samuel works 8 hours a day. If his wage

rate is $46/h, find his wage in a day.

5. Mandy swims at a speed of 2.5 m/s. How

far does she swim in 1 minute?

○○○○→→→→ Ex 7A 6, 7

6. A machine produces chopsticks at a rate of

200 pairs/h. How many hours does it take

to produce 1 200 pairs of chopsticks?

7. Suppose the exchange rate between H.K.

dollars and U.S. dollars is 7.8 HKD/USD.

How many U.S. dollars can 117 H.K.

dollars be changed to?

○○○○→→→→ Ex 7A 8, 9

It means that 7.8 HKD can be changed to 1 USD.

5

�� ‘Explain Your Answer’ Questions

8. 4 red pens are sold for $35 and 10 blue pens are sold for $52. On average, which type of pens

is more expensive? Explain your answer.

Price rate of red pens =

Price rate of blue pens =

∵ $ ( > / = / < ) $

∴ On average, (red / blue) pens are more expensive.

9. The download count of software A in a week is 4 130 while the download count of software B

in 15 days is 8 550. Is the daily download rate of software A higher than that of software B?

Explain your answer.

�� Level Up Question

10. The water temperature in a cooking pot rises at a rate of 2.6°C/min.

(a) Find the time taken for the water temperature to rise by 65°C.

(b) By how many °C does the water temperature rise in half an hour?

6

New Century Mathematics (2nd Edition) 2B

7 Rate and Ratio

Level 1

Express the following rates with the units given in brackets. [Nos. 1−−−−4]

1. There are 36 balls in 3 bags. (balls/bag)

2. There are 200 needles in 5 boxes. (needles/box)

3. The total weight of 20 pencils is 180 g. (g/pencil)

4. Tom finishes 30 questions in 15 min. (questions/min)

5. Complete the table below.

Distance Time Speed

(a) 200 m 10 s

(b) 280 km 140 km/s

(c) 4 min 20 m/min

6. Some candies are sold at $36/kg. How much is 0.5 kg candies?

7. The salary of a worker is $42/hour. What is his salary after working for 5 hours?

8. Regina is reading a comic book and her reading rate is 2 pages/min. If there are 70 pages in the book,

how long will she take to finish reading the book?

9. The petrol consumption rate of a car is 20 km/L. How much petrol is consumed by the car in

travelling 50 km?

10. Two dozen eggs are sold at $48. Find the price rate in $/egg.

11. Henry can draw 30 circles in 20 seconds. Find his rate of drawing circles in circles/min.

12. The selling price of 3 bottles of 2L-lemon tea is $60. Find the price rate of the lemon tea in the

following units.

(a) $/bottle (b) $/L

13. A shop is open 6 days a week and 50 weeks a year. If the weekly profit of the shop is $6 000, find the

(a) daily profit, (b) annual profit.

14. A peak tram finishes a journey of 1.4 km in 8 minutes. Find its speed in the following units.

(a) m/min (b) km/h

Consolidation Exercise 7A ��

7

Level 2

15. A computer program can run 42 times in 12 minutes.

(a) How long does it take to run the program 7 times?

(b) How many times can the program run in 2 hours?

16. The price of 15 apples is $90.

(a) Find the price rate in $/apple.

(b) Sally has $150. Does she have enough money to buy two dozen apples? Explain your answer.

17. Robot T walks 1 980 m in 11 minutes.

(a) Find the walking speed of Robot T in

(i) m/s, (ii) km/h.

(b) Robot U walks 3 km in

4

1 hour. Which robot, T or U, has a higher walking speed? Explain your

answer.

18. In shop A, the price of 6 oranges is $21. In shop B, the price of 8 oranges is $32.

(a) Find the price rate of the oranges in each shop in $/orange.

(b) The oranges in shop B are now sold at a discount of 15%. Steven wants to buy oranges with the

lower price rate. Which shop, A or B, should he visit? Explain your answer.

19. There are 1 840 books in a library. A librarian starts to sort all the books. It is known that he has sorted

120 books in the first 30 minutes.

(a) Find his rate of sorting books in books/h.

(b) If he continues to sort books at this rate, can he finish the rest of the books in the next

7 hours? Explain your answer.

20. Suppose 202.5 Thailand Bahts are equivalent to 45 H.K. dollars.

(a) Find the exchange rate between Thailand Bahts and H.K. dollars in Thailand Bahts/HKD.

(b) How many Thailand Bahts can 140 H.K. dollars be changed to?

(c) How many H.K. dollars can 172.8 Thailand Bahts be changed to?

21. A ship sails 125 km in 150 minutes.

(a) Find the speed of the ship in km/h.

(b) How many hours does it take to travel 90 km?

(c) If the speed of the ship is decreased by 3 m/s, how many kilometres can it travel in

120 minutes?

8

Consolidation Exercise 7A (Answer)

1. 12 balls/bag

2. 40 needles/box

3. 9 g/pencil

4. 2 questions/min

5. Distance Time Speed

(a) 200 m 10 s 20 m/s

(b) 280 km 2 s 140 km/s

(c) 80 m 4 min 20 m/min

6. $18

7. $210

8. 35 min

9. 2.5 L

10. $2/egg

11. 90 circles/min

12. (a) $20/bottle

(b) $10/L

13. (a) $1 000

(b) $300 000

14. (a) 175 m/min

(b) 10.5 km/h

15. (a) 2 min

(b) 420

16. (a) $6/apple

(b) yes

17. (a) (i) 3 m/s

(ii) 10.8 km/h

(b) U

18. (a) shop A: $3.5/orange, shop B: $4/orange

(b) B

19. (a) 240 books/h

(b) no

20. (a) 4.5 Thailand Bahts/HKD

(b) 630 Thailand Bahts

(c) 38.4 HKD

21. (a) 50 km/h

(b) 1.8 h

(c) 78.4 km

9

F2B: Chapter 7B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 4


(Video Teaching)

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)

Book Example 10


(Video Teaching)

10

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)



(Full Solution)

Maths Corner Exercise 7B Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 7B Multiple Choice


___________ ( )



_________

11

Book 2B Lesson Worksheet 7B (Refer to §7.2)

7.2A Meaning of Ratio

(a) A ratio is a comparison of two quantities of the same kind by division.

It has no unit.

(b) A ratio of a quantity p to a quantity q can be written as p : q or

q

p,

where p, q ≠ 0 and they are called the terms of the ratio.

e.g. Ratio of the lengths of A to B

= 3 cm : 5 cm

= 3 : 5

5

3or

1. Determine whether each of the following is a ratio.

If yes, put a ‘�’ in the box and express the ratio in the form of p : q in the space provided.

If not, put a ‘�’ in the box.

(a) $11

8$ � (b)

s 4

L 9 �

(c) 50

cm 17 � (d)

g 1

kg 2 �

7.2B Properties of Ratio

(a) If k ≠ 0, then

(i) a : b = a × k : b × k (ii) a : b =

k

a :

k

b

(b) A ratio p : q is in its simplest form if both p and q are integers and

they have no common factor other than 1.

e.g. 3 : 2 is in its simplest form but 6 : 4 is not.


Simplify the following ratios.

(a) 21 : 12 (b) 3

1 :

2

1

Sol (a) 21 : 12

=

3

21 :

3

12

= 7 : 4

(b) 3

1 :

2

1

=

3

1 ×××× 6 :

2

1 ×××× 6

= 2 : 3


(a) 30 : 36 (b) 5

1 :

3

2

Sol (a) 30 : 36

=

) (

30 :

) (

36

=

(b) 5

1 :

3

2

=

5

1 × ( ) :

3

2 × ( )

=

� Note that

p : q ≠ q : p in general.

Divide each term by the H.C.F. of 21 and 12, i.e. 3.

Multiply each term by the L.C.M. of 3

and 2, i.e. 6.

Are the two quantities of the

same kind?

2 kg =

g

� Cancel out the units.

We call 21 : 12 and 7 : 4 two equal ratios.

A B

5 cm

3 cm

12

2. Simplify the following ratios.

(a) 8 : 40 =

) (

8 :

) (

40

=

(b) 20 : 15 =

(c) 2

1 :

7

1 =

2

1 × ( ) :

7

1 × ( )

=

(d)

4

1 :

8

5 =

(e) 3 :

5

2 =

(f) 3.3 : 2.7 = 3.3 × 10 : 2.7 × 10

= 33 : ( )

=

) (

33 :

) (

) (

=

(g) 0.6 : 0.12 = 0.6 × ( ) : 0.12 × ( )

=

○○○○→→→→ Ex 7B 1


Simplify 1 m : 50 cm.

Sol 1 m : 50 cm

= 100 cm : 50 cm

= 100 : 50

=

50

100 :

50

50

= 2 : 1

Simplify 20 s : 1 min.

Sol 20 s : 1 min

= 20 s : ( ) s

=

3. Simplify 2 kg : 800 g.

4. Simplify 1.5 cm : 65 mm.

○○○○→→→→ Ex 7B 2

Express both terms in the same unit first.

Cancel out the unit.

3 =

1

3

Convert both terms into integers first.

13


If x : 12 = 3 : 4, find x.

Sol x : 12 = 3 : 4

12

x =

4

3 � a : b =

b

a

x =

4

3 × 12

= 9

If 5 : 6 = 25 : y, find y.

Sol 5 : 6 = 25 : y

) (

5 =

) (

) (

=

Find the unknowns in the following ratios. [Nos. 5–6]

5. (y – 2) : 9 = 2 : 3

6. 5 : a = 20 : (a + 3)

○○○○→→→→ Ex 7B 4, 5

In each of the following, x and y are non-zero numbers. Find x : y. [Nos. 7–8]

7. 5x = 8y

8. 3

x – 4y = 0

○○○○→→→→ Ex 7B 6, 7

9. There are 80 pets in a pet shop, of which 32 are dogs and the rest are cats.

(a) Find the ratio of the number of dogs to the total number of pets.

(b) Find the ratio of the number of dogs to the number of cats.

○○○○→→→→ Ex 7B 11–13

Rewrite it as

‘?

?=

y

x’.

14

7.2C Three-term Ratio

(a) A ratio of three quantities a, b, c (a three-term ratio) can be expressed as

a : b : c.

e.g. Consider a : b : c = 5 : 2 : 4. We have:

(i) a : b = 5 : 2 � a : b : c = 5 : 2 : 4

(ii) b : c = 2 : 4, i.e. 1 : 2 � a : b : c = 5 : 2 : 4

(iii) a : c = 5 : 4 � a : b : c = 5 : 2 : 4

(b) If k ≠ 0, then

(i) a : b : c = a × k : b × k : c × k (ii) a : b : c =

k

a :

k

b :

k

c



(a) 2 : 10 : 4 (b) 4

1 :

3

1 :

2

1

Sol (a) 2 : 10 : 4

=

2

2 :

2

10 :

2

4

= 1 : 5 : 2

(b) 4

1 :

3

1 :

2

1

=

4

1 ×××× 12 :

3

1 ×××× 12 :

2

1 ×××× 12

= 3 : 4 : 6


(a) 16 : 8 : 4 (b) 3

2 :

6

1 :

9

2

Sol (a) 16 : 8 : 4

=

)(

16

:

)(

8

:

)(

4

=

(b) 3

2 :

6

1 :

9

2

=

3

2 × ( ) :

6

1 × ( ) :

9

2 × ( )

=

10. Simplify the following ratios.

(a) 20 : 25 : 40

=

)(

20

:

)(

25

:

)(

40

=

(b) 0.6 : 1.5 : 2.7

=

(c) 42 L : 21 L : 63 L

=

(d) 300 m : 3 km : 900 m

=

○○○○→→→→ Ex 7B 3

H.C.F. of 2, 10 and 4

2

Express all the terms in the

same unit first.

L.C.M. of 4, 3 and 2 = 12

15

We can combine two ratios to form a three-term ratio.

e.g. Suppose p : q = 1 : 3 and q : r = 3 : 4.

We have p : q : r = 1 : 3 : 4.

Example 5

If a : b = 4 : 3 and b : c = 2 : 1, find a : b : c.

Sol Method 1

a : b = 4 ×××× 2 : 3 ×××× 2 = 8 : 6

b : c = 2 ×××× 3 : 1 ×××× 3 = 6 : 3

∴ a : b : c = 8 : 6 : 3

Method 2

a : b = 4 : 3

b : c = 2 : 1

a : b : c = 2 × 4 : 2 × 3 : 1 × 3

= 8 : 6 : 3

Instant Drill 5

If a : b = 2 : 5 and b : c = 4 : 3, find a : b : c.

Sol Method 1

a : b = 2 × ( ) : 5 × ( ) = ( ) : ( )

b : c = 4 × ( ) : 3 × ( ) = ( ) : ( )

∴ a : b : c =

Method 2

a : b = 2 : 5

b : c = 4 : 3

a : b : c = ( ) : ( ) : ( )

=

11. If a : b = 5 : 6 and b : c = 9 : 2, find a : b : c.

Method 1

a : b = 5 × ( ) : 6 × ( ) =

b : c = 9 × ( ) : 2 × ( ) =

∴ a : b : c =

Method 2

a : b = 5 : 6

b : c = 9 : 2

a : b : c =

12. If a : b = 6 : 1 and a : c = 4 : 5, find a : b : c.

○○○○→→→→ Ex 7B 8

common term Make them the same by considering their L.C.M.

p : q : r 1 : 3 3 : 4

1 : 3 : 4

L.C.M. of 3 and 2 = 6

3 × ( ) = 6, 2 × ( ) = 6

�

a : b : c 4 : 3

2 : 1

a : b : c 6 : 1 4 : 5

×××× ××××

××××

×××× ××××

××××

L.C.M. of 5 and 4 =

Simplify the

ratio if possible.

a : b : c 2 : 5 4 : 3

L.C.M. of 6 and 9 =

16

7.2D Dividing a Quantity in a Given Ratio


(a) Divide 24 in the ratio of 1 : 2.

(b) Divide 40 m in the ratio of 4 : 3 : 1.

Sol (a)

First portion = 24 ×

21

1

+

= 24 ×

3

1

= 8

Second portion = 24 ×

21

2

+

= 24 ×

3

2

= 16

(b)


134

4

++ m

= 40 ×

8

4 m

= 20 m

Second portion = 40 ×

134

3

++ m

= 40 ×

8

3 m

= 15 m

Third portion = 40 ×

134

1

++ m

= 40 ×

8

1 m

= 5 m

(a) Divide 100 kg in the ratio of 3 : 2.

(b) Divide $63 in the ratio of 2 : 6 : 1.

Sol (a)


)()(

)(

+ kg

= 100 ×

)(

)(

kg

=

Second portion =

(b)

First portion = $63 ×

)()()(

)(

++

= $63 ×

)(

)(

=

Second portion =

Third portion =

13. Andrew and Belle share a 400 g steak in 14. Carl and Donna jointly donate a sum of

24

40 m

100 kg

$63

17

the ratio of 3 : 1. How much does Andrew

get?

Andrew’s share = 400 ×

)()(

)(

+ g

=

$3 500 in the ratio of 5 : 2. How much

does Carl donate?

15. The numbers of wins, losses and draws of

a football team are in the ratio of 7 : 2 : 3.

If the team played 36 matches altogether,

how many matches did the team draw?

○○○○→→→→ Ex 7B 17, 18, 22

16. A ribbon is divided into two parts in the

ratio of 7 : 3. If the larger part is 56 cm

long, what is the total length of the ribbon?

Let x cm be the total length of the ribbon.

x ×

)()(

)(

+ = 56

=

17. A drink is made by mixing honey and

green tea in the ratio of 2 : 13. If the drink

contains 40 mL of honey, find the total

volume of the drink.

18. Flora, Gina and Helen share a box of

cookies in the ratio 5 : 3 : 1. If Gina gets

18 cookies, find the total number of

cookies in the box.

○○○○→→→→ Ex 7B 19–21

�� ‘Explain Your Answer’ Question

x cm

larger part (56 cm)

18

19. Vivian is 25 years old and Wallace is 15 years old now.

(a) Find the ratio of Vivian’s age to Wallace’s age.

(b) Will the ratio of their ages change after 5 years? Explain your answer.

(a) The required ratio =

(b) 5 years later, Vivian will be _____ years old and Wallace will be _____ years old.

∴ The ratio (will / will not) change after 5 years.

�� Level Up Questions

20. If 2x + 4y = 7y – 3x, find x : y.

21. It is given that the ratio of Jason’s weight to Roy’s weight is 1 : 3, and the ratio of Sam’s

weight to Roys’s weight is 7 : 6.

(a) Find the ratio Sam’s weight : Jason’s weight : Roy’s weight.

(b) If Jason weighs 20 kg, find the total weight of Sam, Jason and Roy.


19

7 Rate and Ratio

Level 1

Complete the following table and give the answers in the simplest form. [Nos. 1−−−−3]

1. x y x : y

(a) 2 10

(b) 21 6

(c) 3

1

4

1

2. x y x : y

(a) 250 mL 1 L

(b) 0.8 kg 30 g

(c) 4

1 km 500 m

3. x y z x : y : z

(a) 5 45 30

(b) 40 12 28

(c) 14 cm 56 mm 35 mm

4. Find the unknowns in the following ratios.

(a) a : 3 = 27 : 9 (b) 5 : b = 10 : 20

5. Find the unknowns in the following ratios.

(a) 4 : (x − 1) = 32 : 8 (b) (y + 1) : 14 = (y + 6) : 21

6. In each of the following, a and b are non-zero numbers. Find a : b.

(a) 7a = b (b) 3a − 4b = 0

7. In each of the following, x and y are non-zero numbers. Find x : y.

(a) 5

x=

3

y (b)

9

2x−

4

y= 0

Consolidation Exercise 7B ��

20

8. In each of the following, find x : y : z.

(a) x : y = 1 : 2 and y : z = 2 : 5

(b) x : y = 1 : 9 and y : z = 3 : 1

(c) x : y = 4 : 3 and x : z = 8 : 1

(d) x : z = 3 : 2 and y : z = 7 : 6

In each of the following, divide the given quantity in the ratio as indicated. [Nos. 9−−−−10]

9. (a) Divide 35 in the ratio 1 : 6.

(b) Divide $20 in the ratio 3 : 2.

(c) Divide 180 g in the ratio 4 : 5.

10. (a) Divide 10 in the ratio 1 : 2 : 2.

(b) Divide 80 cm in the ratio 1 : 5 : 4.

(c) Divide 270 mL in the ratio 2 : 3 : 4.

11. The weight of a dog is 15 kg while the weight of a cat is 13.5 kg. Find the ratio of the weight of the

dog to that of the cat.

12. Among 150 towels in a box, 100 of them are black and the rest are white. Find the ratio of the number

of black towels to that of white towels.

13. In a test, the ratio of Joe’s score to Tom’s score is 5 : 2 and the ratio of Tom’s score to Ken’s score is 4 :

3. Find the ratio of Joe’s score to Tom’s score to Ken’s score.

14. In the figure, PQR is a straight wire.

Find the following ratios of lengths.

(a) PQ : QR

(b) PR : QR

(c) PR : QR : PQ

15. The height of building A is 52 m. Building B is 12 m lower than building A, while building C is 20 m

higher than building A. Find the ratio of the height of building A to that of building B to that of

building C.

16. A special drink is made by mixing orange juice and soda water in the ratio 2 : 1 by volume. Find the

volume of orange juice required to produce 750 mL of the special drink.

17. In a school, there are 1 044 students. If the ratio of the number of boys to that of girls is 5 : 7, how

many boys and girls are there in the school?

21

18. There are 168 red balls and some green balls in a bag. The ratio of the number of red balls to that of

green balls is 3 : 4. Find the total number of balls in the bag.

19. Eason, Fred and George share a pack of game cards in the ratio 5 : 9 : 2. If Eason gets 35 game cards,

find the total number of game cards in the pack.

20. In a triangle, the lengths of the three sides are in the ratio 2 : 3 : 4. If the length of the shortest side is

18 cm, find the perimeter of the triangle.

Level 2

21. A bottle of cleaning agent is formed by dissolving 2 g of bleach powder in 0.2 kg of water.

(a) What is the ratio of the weight of bleach to that of water in the bottle of cleaning agent?

(b) What is the ratio of the weight of bleach to that of the cleaning agent?

22. Find x : y in each of the following.

(a) 5x + 8y = 6x + 7y (b) (x + 4y) : (5x + 2y) = 1 : 2

23. Find x : y in each of the following.

(a) x

1:

y

1= 2 : 3 (b)

x2

1:

y

3= 1 : 2

24. If x : y = 4 : 3, find

(a) x

1:

y

1, (b) (x + 2y) : (2x − y).

25. If a : b = 1 : 2 and b : c = 1 : 3, find

(a) c : a, (b) (c − 2a) : (a + b).

26. In each of the following, find a : b : c.

(a) a = 2b and b = 4c (b) 10a = 5b = c

27. It is given that a : b : c = 1 : 6 : 2. Find the values of a and c in each of the following cases.

(a) b = 90 (b) a + c = 36

28. In a swimming race, the speeds of Leo and Nick are 150 m/min and 2 m/s respectively. Find the ratio

of the speed of Leo to that of Nick.

29. Winnie and Shirley share a pack of candies in the ratio 3 : 7. If Shirley gets 16 candies more than

Winnie does, find the total number of candies in the pack.

22

30. In the figure, ABCD is a rectangle and AB : AD = 7 : 6. If AB is 5 cm

longer than AD, find the area of ABCD.

31. There are 27 Chinese books and some English books on a bookshelf. The numbers of Chinese books

and English books are in the ratio 3 : 7.

(a) How many English books are there?

(b) Ben takes 18 Chinese books and 18 English books away from the bookshelf. He then claims that

the ratio of Chinese books to English books on the bookshelf is still 3 : 7.

Do you agree? Explain your answer.

32. Some cows are shared among 3 farmers A, B and C such that A’s share : B’s share = 3 : 2 and B’s

share : C’s share = 3 : 1.

(a) Find the ratio A’s share : B’s share : C’s share.

(b) It is given that B gets 48 cows. Find the total number of cows being shared.

33. The price of a green apple is $6 and the price of a red apple is $10. James buys some green and red

apples in the ratio 5 : 3. How many apples does he buy if he spends $360 in total?

34. △In the figure, the height of ABC is h cm with AB as the base.

D is a point on AB such that AD : DB = 2 : 5. Let AB = x cm.

(a) △Express the area of ABC in terms of x and h.

(b) △Find the ratio of the area of ADC △to that of DBC to △that of ABC.

23

Consolidation Exercise 7B (Answer)

1. x y x : y

(a) 2 10 1 : 5

(b) 21 6 7 : 2

(c) 3

1

4

1 4 : 3

2. x y x : y

(a) 250 mL 1 L 1 : 4

(b) 0.8 kg 30 g 80 : 3

(c) 4

1 km 500 m 1 : 2

3. x y z x : y : z

(a) 5 45 30 1 : 9 : 6

(b) 40 12 28 10 : 3 : 7

(c) 14 cm 56 mm 35 mm 20 : 8 : 5

4. (a) 9

(b) 10

5. (a) 2

(b) 9

6. (a) 1 : 7

(b) 4 : 3

7. (a) 5 : 3

(b) 9 : 8

8. (a) 1 : 2 : 5

(b) 1 : 9 : 3

(c) 8 : 6 : 1

(d) 9 : 7 : 6

9. (a) 5, 30

(b) $12, $8

(c) 80 g, 100 g

10. (a) 2, 4, 4

(b) 8 cm, 40 cm, 32 cm

(c) 60 mL, 90 mL, 120 mL

11. 10 : 9

12. 2 : 1

13. 10 : 4 : 3

14. (a) 3 : 2

(b) 5 : 2

(c) 5 : 2 : 3

15. 13 : 10 : 18

16. 500 mL

17. boy: 435, girl: 609

18. 392

19. 112

20. 81 cm

21. (a) 1 : 100

(b) 1 : 101

22. (a) 1 : 1

(b) 2 : 1

23. (a) 3 : 2

(b) 1 : 3

24. (a) 3 : 4

(b) 2 : 1

25. (a) 6 : 1

(b) 4 : 3

26. (a) 8 : 4 : 1

(b) 1 : 2 : 10

27. (a) a = 15, c = 30

(b) a = 12, c = 24

28. 5 : 4

29. 40

30. 1 050 cm2

31. (a) 63

(b) no

32. (a) 9 : 6 : 2

(b) 136

33. 48

34. (a) xh2

1 cm2

(b) 2 : 5 : 7

24

F2B: Chapter 7C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 13


(Video Teaching)

Book Example 14


(Video Teaching)

Book Example 15


(Video Teaching)

Book Example 16


(Video Teaching)

Book Example 17


(Video Teaching)

Book Example 18


(Video Teaching)



(Full Solution)

Maths Corner Exercise 7C Level 1



___________ ( )

25




___________ ( )




___________ ( )

Maths Corner Exercise 7C Multiple Choice


___________ ( )



_________

26

Book 2B Lesson Worksheet 7C (Refer to §7.3)

7.3A Similar Figures

In two similar figures, the ratios of the lengths of all the corresponding sides

are equal, i.e. they are proportional.


The diagram below shows two similar

rectangles. Find the unknown x.

Sol 3

x =

2

4

x =

2

4 × 3

= 6

The diagram below shows two similar

parallelograms. Find the unknown y.

Sol )(

y =

)(

)(

y =

In each of the following, the two given figures are similar. Find the unknowns. [Nos. 1–4]

1.

2.

3.

4.

○○○○→→→→ Ex 7C 1–4

5

x

15

21

10 6

a

9

12

27

16

y

y

20 16

25

x 4

3

2

x

4 8

5

4

y

27

7.3B Scale Drawings

(a) A scale drawing is a reduced or an enlarged drawing of a real object, where

the scale = length in the scale drawing : actual length.

(b) The scale is usually expressed in the form 1 : n:

(i) n > 1 stands for a reduction,

(ii) n < 1 stands for an enlargement.

e.g. Refer to the scale drawing of a tree in the figure.

Suppose the actual height of the tree is 3 m.

Scale = 2 cm : 3 m

=

m 3

cm 2

=

cm 1003

cm 2

×

=

150

1

= 1 : 150

5. In each of the following, find the scale of the drawing in the form 1 : n.

(a) Length in a scale drawing = 25 cm,

actual length = 800 m

Scale = ( ) : ( )

=

(b) Length in a scale drawing = 10 cm,

actual length = 1 mm

○○○○→→→→ Ex 7C 5, 6

6. The distance between two airports on a map is 4 cm. If the actual distance between them

is 200 km, find the scale of the map in the form 1 : n.

○○○○→→→→ Ex 7C 10

2 cm

1 km = 1 × 1 000 m

= 1 × 1 000 × 100 cm

i.e. A length of 1 cm in the drawing represents 150 cm (or 1.5 m) on the real tree.

28


The figure is a scale drawing

of a sofa. If the length of the

sofa in the drawing is 3 cm,

find its actual length in m.

Sol Let x cm be the actual length of the sofa.

3 cm : x cm = 1 : 90

∴ x

3 =

90

1

x = 3 × 90

= 270

∴ The required actual length is 270 cm,

i.e. 2.7 m.

The length of the ferry in the

photograph is 4.2 cm. Find

the actual length of the ferry

in m.

Sol Let x cm be the actual length of the ferry.

( ) : ( ) = 1 : 800

∴ =

7.

The length of a computer chip in the scale

drawing as shown is 9 cm. Find the actual

length of the computer chip in mm.

8. Tracy makes a scale drawing of a building

with a scale of 1 cm : 150 m. If the actual

height of the building is 420 m, find its

height in the drawing.

○○○○→→→→ Ex 7C 7–9

Scale 1 : 0.018

9 cm

Scale 1 : 800

4.2 cm

3 cm

Scale 1 : 90

29

9. The figure shows the floor plan of a meeting room. If the

actual length of the room is 18 m, find the length of the

meeting room on the floor plan in cm.

10. The figure shows a map with a scale of 1 : 6 000.

(a) Use a ruler to measure the distance between two bus stops

A and B on the map.

(b) Hence, estimate the actual distance (in m) between A and B.

○○○○→→→→ Ex 7C 11, 12


11. On a map with scale 1 : 400 000, the length of a cycling track is 4.5 cm.

(a) Find the actual length of the cycling track in km.

(b) Jason cycles at an average speed of 12 km/h along the cycling track. Can he finish the

whole journey in 2 hours? Explain your answer.

Scale 1 : 6 000

A

B

?

Scale 1 : 250

30


12. In the diagram, trapeziums ABCD and EFGH are

similar figures. Find the unknowns x and y.

13. The figure shows the floor plan of a rectangular hospital ward.

Its scale is 1 : 140. The dimensions of the floor plan are

4.5 cm × 3.5 cm. Find the actual area of the hospital ward in m2.

Find the actual length and the actual width first.

A

C D

B E F

G H x

x + 5

8

12

20 y

31


7 Rate and Ratio

Level 1

In each of the following, the two given figures are similar. Find the unknowns. [Nos. 1−−−−4]

1. 2.

3. 4.

5. The figure shows a photo of a dove. The height of the dove in

the photo is 3.2 cm. If the actual height of the dove is 28.8 cm,

find the scale of the photo in the form 1 : n.

Consolidation Exercise 7C ��

32

6. The figure shows a photo of a butterfly. The wingspan of the

butterfly in the photo is 4.8 cm. If the actual wingspan of the

butterfly is 0.32 cm, find the scale of the photo in the form n : 1.

7. The figure is a scale drawing of a fish. Its scale is 1 : 4. If the

length of the fish in the drawing is 5 cm, find the actual length

(in cm) of the fish.

33

8. The length of a cell in a scale drawing is 36 mm. If the scale of the drawing is 1 200 : 1, find the

actual length (in mm) of the cell.

9. Victor wants to make a scale drawing of a building with a scale of 1 cm : 9 m. If the actual height of

the building is 126 m, what should its height be in the drawing?

10. The distance between cities P and Q on a map is 5 cm. If the actual distance between them is 13 km,

find the scale of the map in the form 1 : n.

11. The distance between two places is 16 km. Find their distance apart (in cm) on a map with scale 1 :

250 000.

12. The scale of a map is 1 : 2 000 000. If the length of a path on the map is 3 cm, find its actual length (in

km).

13. The figure shows the floor plan of a rectangular room with a scale of

1 : 400.

(a) Use a ruler to measure the length and the width of the room on

the plan.

(b) Hence, estimate the actual area of the room.

Level 2

14. In the figure, quadrilateral WXYZ is formed by enlarging

quadrilateral PQRS. Find the perimeter of WXYZ.

15. In the figure, ABCD and FCDE are two similar rectangles. Find

the length of BF.

34

16. Patrick uses a photocopier to reduce Fig. A to

Fig. B. The height of Fig. A is 10 cm while the

height of Fig. B is 6 cm. It is given that the

perimeter of Fig. A is 63 cm.

(a) Find the perimeter of Fig. B.

(b) Patrick further reduces Fig. B so that the

perimeter of the new figure is 18.9 cm. Find the height of that new figure.

17. The figure shows a photo of a temple. It is given that the

height of the temple in the photo is 2.8 cm and the actual

height of the temple is 35 m.

(a) Find the scale of the photo in the form 1 : n.

(b) If the actual height of the entrance is 6 m, what is its

height (in cm) in the photo?

18. The distance between cities A and B on a map is 5 cm. It is given that the actual distance between

them is 300 km.

(a) Find the scale of the map in the form 1 : n.

(b) The distance between another two cities C and D on the same map is 11 cm. Find the actual

distance (in km) between them.

19. On a map, 1 cm represents an actual distance of 500 m.

(a) Find the scale of the map in the form 1 : n.

(b) If the actual length of a tunnel is 2 km, find the length of the tunnel on the map.

20. The figure shows the floor plan of a restaurant with scale

1 : 800. On the plan, VU = UT = RQ = QP = 1 cm and

TS = SR = 2 cm.

(a) Find the actual area (in m2) of the restaurant.

(b) If the cost of tiling each m2 of the floor in the restaurant is $250,

find the total cost of tiling the whole floor.

21. The figure shows a scale drawing of a rectangular board.

(a) Use a ruler to measure the length and the width of the board in the

drawing.

(b) Estimate the actual length and the actual width of the board.

(c) Carol claims that the ratio of the area of the board in the drawing to the

actual area of the board is 1 : 200. Do you

agree? Explain your answer.

35

Consolidation Exercise 7C (Answer)

1. 14

2. 6

3. 9

4. a = 12, b = 32

5. 1 : 9

6. 15 : 1

7. 20 cm

8. 0.03 mm

9. 14 cm

10. 1 : 260 000

11. 6.4 cm

12. 60 km

13. (a) length: 4.5 cm, width: 3.5 cm

(b) 252 m2

14. 72 cm

15. 22.5 cm

16. (a) 37.8 cm

(b) 3 cm

17. (a) 1 : 1250

(b) 0.48 cm

18. (a) 1 : 6 000 000

(b) 660 km

19. (a) 1 : 50 000

(b) 4 cm

20. (a) 704 m2

(b) $176 000

21. (a) length: 3.6 cm, width: 3 cm

(b) actual length: 7.2 m, actual width: 6 m

(c) no

36

F2B: Chapter 8A

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)




___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 8A Multiple Choice


___________ ( )



_________

37

Book 2B Lesson Worksheet 8A (Refer to §8.1A)

8.1A Exterior Angles of a Triangle

Consider △ABC as shown. BC is produced to D.

BD and AC will form an angle c1 outside the triangle.

c1 is the exterior angle of △ABC.

a and b are the interior opposite angles of c1.

1. In each of the following, determine whether x is an exterior angle of △PQR.

(a) (b)

(c) (d)

QRS is a straight line. SRP and TRQ are straight lines.

2. Refer to the figure. Write down the two interior opposite

angles of each of the following exterior angles.

(a) p (b) q

An exterior angle of a triangle is equal to the sum of its

two interior opposite angles.

i.e. In the figure,

c1 = a + b

[Reference: ext. ∠ of △]

x

P Q

R

S

x

Q

R S

P

x

P

Q R S

x

P Q

R

S T

c

q

a

b

p

38


In the figure, ACD is a straight line. Find x.

Sol x = 42° + 78° �ext. ∠ of △

= 120°

In the figure, BCD is a straight line. Find y.

Sol y = ( ) + ( ) �ext. ∠ of △

=

3. In the figure, BCD is a straight line. Find x.

4. In the figure, ACD is a straight line. Find p.

100° = ( ) + ( ) �ext. ∠ of △

=

5. In the figure, ACD is a straight line. Find q.

○○○○→→→→ Ex 8A 1−6

6. In the figure, CBE and ACD are straight lines.

Find m and n.

m + ( ) = ( ) �adj. ∠s on st. line

=

7. In the figure, ABE and DACF are straight

lines. Find x.

8. In the figure, ACD and FBCE are straight

lines. Find y.

○○○○→→→→ Ex 8A 7−12

x A

B

C D

42°

78° y

A

B

C

D

55°

38°

B

C

D

122° 28°

A x

5q

A

B C

D

2q

B

C D

n

E

30°

104°

A

m

y A

B

C

80°

60° D

E

F

∠ABC and ∠BAC are the two interior

opposite angles of x.

∠ABC (i.e. m) and

∠______ are the adjacent angles on the straight line CBE.

B

C

D

x

E 77°

150°

A The two interior opposite angles of

x are ∠______ and

∠______. F

p

A

B

C D

100° 41°

39

9. In the figure, BCD and ECA are straight lines.

Find m and n.

10. In the figure, ADC and EBD are straight lines.

Find p and q.

In △ABC,

m = ( ) + ( ) �ext. ∠ of △

=

In △CDE,

○○○○→→→→ Ex 8A 13−15


11. In the figure, ADCG, BEC and DEF are straight lines.

Find p and q.

B

C

D

n

E

86°

47°

A

m

35°

B

C D

q

26°

68°

A

p 64°

E

B

C

A

m

C

D

n

E m

B

C D

q

58°

A

p

140°

E

F

G

40


8 Angles in Triangles and Polygons

Level 1

In each of the following figures, QRS is a straight line. Find the unknowns in the figures.

[Nos. 1−−−−6]

1. 2.

3.

4. 5. 6.

Find the unknown(s) in each of the following figures. [Nos. 7−−−−14]

Consolidation Exercise 8A ��

41

7. 8.

9. 10.

11. 12.

42

13. 14.

15. In the figure, ADB, AEC and DEF are straight lines.

Find x.

16. In the figure, QTR is a straight line. Find y.

Level 2


17. 18.

43

19.

20.

21. In the figure, UYZ and UXW are straight lines.

Is UZ parallel to VW ? Explain your answer.

22. Find x in the figure.

44

23. Refer to the figure. Express a in terms of x and y.

24. In the figure, MOL and NOK are straight lines.

Express d in terms of a, b and c.

25. In the figure, HAOED and BCOGF are straight lines. Find

the sum of all the marked angles.

45

Consolidation Exercise 8A (Answer)

1. 90° 2. 142°

3. 18° 4. 62°

5. 25° 6. 37°

7. a = 36°, b = 78°

8. 137° 9. 79°

10. 40° 11. 47°

12. 28°

13. p = 80°, q = 40°

14. x = 116°, y = 21°

15. 31° 16. 68°

17. 123°

18. x = 26°, y = 78°

19. x = 37°, y = 69°

20. x = 39°, y = 61°

21. yes 22. 49°

23. a = x + y

24. d = a + b − c

25. 360°

46

F2B: Chapter 8B

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)

Book Example 10


(Video Teaching)



(Full Solution)




___________ ( )

47




___________ ( )




___________ ( )

Maths Corner Exercise 8B Multiple Choice


___________ ( )



_________

48

Book 2B Lesson Worksheet 8B (Refer to §8.1B−C)

8.1B Isosceles Triangles

Names of different parts of an isosceles triangle:

Equal sides Vertical angle Base Base angles

I. Properties of Isosceles Triangles

The two base angles of an isosceles triangle are equal.

i.e. In △ABC, if AB = AC,

then ∠B = ∠C.

[Reference: base ∠s, isos. △]


Find x in the figure. Find a in the figure.

Sol ∵ PQ = QR

∴ x = 40° �base ∠s, isos. △

Sol ∵ DF =

∴ a = �

1. Find y in the figure.


3. In the figure, ACD is a straight line. Find p and

q.

4. In the figure, TP // QR. Find a and b.

○○○○→→→→ Ex 8B 1−4 ○○○○→→→→ Ex 8B 16−18, 20

R

40°

x P

Q

F

E D 30° a

A 3

67° 3

B

C

y

P Q 65°

x 2 2

R

A

q

p

D

C 140°

B

Find p first.

T

P

56° a

b R Q

A

B C

A

B C

49

In △ABC with AB = AC and D is a point on BC, if

any one of the conditions below is true, then the

other two conditions must also be true.

(i) ∠BAD = ∠CAD

(ii) BD = DC

(iii) AD ⊥ BC

[Reference: property of isos. △]


In the figure, D is a point on BC such that

BD = DC. Find p and q.

Sol In △ABC, ∵ AB = AC and BD = DC.

∴ p = 20° �property of isos. △

AD ⊥ BC �property of isos. △

i.e. q = 90°

In the figure, M is a point on PR such that

QM ⊥ PR. Find x and y.

Sol In △PQR,

∵ = and .

∴

5. In the figure, EGF is a straight line. DG is the

angle bisector of ∠EDF. Find x and y.

6. In the figure, M is the mid-point of YZ. Find p

and q.

○○○○→→→→ Ex 8B 10−12

B

p

C q

D

A

20°

R

4

M

y

Q

17° x P

x F 3

y

G E

10 10

D

Z

X 30°

q

p M

Y

i.e. ∠EDG =

∠

i.e. YM = MZ

50

II. Conditions for a Triangle to be Isosceles

In △ABC,

if ∠B = ∠C,

then AB = AC.

[Reference: sides opp. eq. ∠s]


Determine whether △ABC is an isosceles triangle.

Sol ∵ ∠A = ∠B = 63°

∴ CA = CB �sides opp. eq. ∠s

i.e. △ABC is an isosceles triangle.

Determine whether △ABC is an isosceles triangle.

Sol ∠A + ( ) + ( ) = ( )

∠A =

7. Find x and y in the figure.

8. In the figure, DBC is a straight line. Find p and

q.

○○○○→→→→ Ex 8B 5, 6

9. Refer to the figure. Find n, and hence determine whether △ABC is an

isosceles triangle.

○○○○→→→→ Ex 8B 19, 22

63°

63°

A

B C

A D

40° C

n

70°

B

Is △ABC

isosceles? If yes, which two sides are equal? q

D

113° B

A 67°

4 p C

B x

5 C

64°

y

A

58° 5.3

27° 126°

A

B

C

Find ∠A first.

Are there any equal angles?

51

8.1C Equilateral Triangles

Each of the three interior angles of an equilateral triangle is 60°.

i.e. In the figure,

if △ABC is an equilateral triangle,

then ∠A = ∠B = ∠C = 60°.

[Reference: property of equil. △]


Find x in the figure.

In the figure, △ABC is an equilateral triangle. Find

y.

Sol ∵ AB = BC = AC

∴ △ABC is an equilateral triangle.

∴ x = 60° �property of equil. △

Sol ∵ △ABC is an equilateral triangle.

∴ ∠ABC = _______ �property of equil. △

10. In the figure, △ABC is an equilateral triangle.

Find m and n.

○○○○→→→→ Ex 8B 7−9

11. In the figure, △ABC is an equilateral triangle

and M is a point on AB. Find p and q.

B

7 cm

7 cm

x

7 cm

A C

B

n + 1

C

2n − 3

A m

B

M 8

A

q p

30°

C

� Conversely, for △ABC,

if ∠A = ∠B = ∠C = 60°,

then △ABC is an

equilateral triangle.

B y C

25°

A

Does △ABC have

the properties of

isosceles triangles?

52

12. In the figure, BDC and DAE are straight lines. △ABC is an equilateral triangle. Find p and q.

13. In the figure, AED and BEC are straight lines. △ABC is an equilateral triangle. Find x and y.

○○○○→→→→ Ex 8B 13−15 ○○○○→→→→ Ex 8B 21


14. Refer to the figure.

(a) Find x.

(b) Is △ABC an equilateral triangle? Explain your answer.

B

E

A q

40°

p

C

D

25° A C

x

y

D

E

B

D 75°

C

x

B 60°

A

Find ∠CBA first.

Consider the sizes of its interior angles.

53



Level 1


1. 2. 3.

4. 5. 6.

7. 8. 9.

Find the unknowns in each of the following figures. [Nos. 10−−−−13]

10. 11.

Consolidation Exercise 8B ��

54

12. 13.


14. 15.

16. 17.

18. In the figure, JKL is a straight line.

(a) Is △ KLM an isosceles triangle? Explain your answer.

(b) If a + b = 180°, is △ KLM an equilateral triangle?


Level 2


19. 20.

55

21. 22.

23. In the figure, QRS is a straight line, ∠QUR = 30°, UQ = US,

QP // ST and UR ⊥ QS. Find a.

24. In the figure, QUP, QRS, PTR and UTS are straight lines.

Is △ PQR an equilateral triangle? Explain your answer.

25. In the figure, RTP is a straight line, SR = SP and

ST // PQ.

(a) Find a and b.

(b) Is △ PRS an equilateral triangle? Explain your answer.

26. In the figure, ADB is a straight line and AD = BD = CD.

(a) Express ∠CDB in terms of x.

(b) Is AC perpendicular to BC? Explain your answer.

27. In the figure, △ ABC is an equilateral triangle.

(a) Find x.

(b) Is △ ADB an isosceles triangle? Explain your answer.

28. In the figure, ABFE is a square. △ BCF and △ EFD are

equilateral triangles. Find ∠FCD.

56

Consolidation Exercise 8B (Answer)

1. h = 42°, k = 96°

2. m = 71°, n = 38°

3. 66°

4. y = 50°, z = 3

5. 94°

6. a = 78°, b = 51°

7. 15 8. 2

9. 120°

10. h = 13, k = 32°

11. a = 90°, b = 38°

12. x = 8, y = 90°

13. y = 30°, z = 105°

14. 79° 15. 35°

16. a = 58°, b = 116°

17. x = 30°, y = 60°

18. (a) yes (b) yes

19. 66°

20. y = 104°, z = 294°

21. 48°

22. x = 70°, y = 95°

23. 15° 24. yes

25. (a) a = 45°, b = 30°

(b) yes

26. (a) ∠CDB = 2x (b) yes

27. (a) 15° (b) yes

28. 15°

57

F2B: Chapter 8C

Date Task Progress

Lesson Worksheet


(Full Solution)

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)




___________ ( )




___________ ( )




___________ ( )

58

Maths Corner Exercise 8C Multiple Choice


___________ ( )



_________

59

Book 2B Lesson Worksheet 8C (Refer to §8.2A)

8.2A Sum of Interior Angles of a Polygon

The sum of interior angles of an n-sided polygon is (n − 2) × 180°.

[Reference: ∠ sum of polygon]

1. Fill in the blanks.

(a) Sum of interior angles of a 14-sided polygon = ( − ) × 180° �∠ sum of

polygon

=

(b) sum of interior angles of a 22-sided polygon = _____________________

�______________

=

○○○○→→→→ Ex 8C 1−3


Find y in the figure.


Sol ∵ ABCD is a quadrilateral.

∴ °×−=°+°

+°+180)24(

6013090y

y + 280° = 360°

y = 80°

Sol ∵ ABCDE is a .

∴

2. Find m in the figure.

3. Find n in the figure.

E D

C

B

A

110°

110°

100°

x

A

B C

D 120°

130°

m

150°

110°

140°

F

E

A n

115°

140°

145°

136° B

113°

C D

E

F G

�This property holds for both convex and

concave polygons.

How many sides does this

polygon have?

�∠ sum of

polygon

B

C D

130° 60°

y

A

60

○○○○→→→→ Ex 8C 4−9

4. Find p in the figure.

5. In the figure, ABF is a straight line. Find θ.

○○○○→→→→ Ex 8C 10−12


The figure shows a regular decagon. Find x.

The figure shows a regular 16-sided polygon. Find

y.

Sol Sum of interior angles of a regular decagon

= (10 − 2) × 180° �∠ sum of polygon

= 1 440°

∵ All the interior angles of a regular

polygon are equal. ∴ x =10

4401 °

= 144°

Sol Sum of interior angles of a regular 16-sided

polygon

=

6. Find the size of each interior angle of a regular 20-sided polygon.

○○○○→→→→ Ex 8C 13−15

7. If the sum of interior angles of an n-sided

polygon is 1 620°, find the value of n.

8. If the sum of interior angles of an n-sided

polygon is 1 980°, find the value of n.

A

p 110°

120°

150° B

160°

C D

E

F

p A

126°

45°

97°

B

150° C

D

E

F θ

x

y

Set up an equation according to the

given conditions.

A decagon has _____ sides.

61

○○○○→→→→ Ex 8C 16−18


9. The figure shows a part of a regular polygon. Steven claims

that the polygon has 36 sides. Do you agree ? Explain your answer.

Sum of interior angles of a regular 36-sided polygon

=


10. Find the unknown(s) in each of the following figures.

(a) (b)


A

x B

275°

C

D

113°

E

2x 3x

A

142°

B 115°

C

D

F y

150°

x

E

G

104°

123°

170° 170° 170°

62


Level 1

Find the sum of interior angles of each of the following polygons. [Nos. 1−−−−3]

1. Nonagon 2. 14-sided polygon 3. 20-sided polygon

Find the unknown in each of the following figures. [Nos. 4−−−−9]

4. 5.

6. 7.

8. 9.

Consolidation Exercise 8C ��

63


10. 11.

12.

Find the size of each interior angle of the following regular polygons. [Nos. 13−−−−15]

13. Regular 10-sided polygon



In each of the following, the sum of interior angles of a polygon is given. Find the number of sides of the

polygon. [Nos. 16−−−−18]

16. 1 080° 17. 3 060° 18. 4 680°

Level 2


19. 20.

64

21. 22.

23. 24.

25. It is given that the sum of interior angles of an n-sided polygon is 4 times that of a heptagon. Find the

value of n.

26. Find the number of sides of a regular polygon if the size of each of its interior angles is

(a) 157.5°, (b) 165.6°, (c) 172°.

27. (a) Find the size of each interior angle of a regular pentagon.

(b) If the size of each interior angle of a regular n-sided polygon is 1.5 times that of a regular

pentagon, find the value of n.

28. Is it possible that the size of each interior angle of a regular polygon is 175°? Explain your answer.

29. In the figure, ABCDEF is a regular hexagon and AHGF is a square. Find

x and y.

65

Consolidation Exercise 8C (Answer)

1. 1 260° 2. 2 160°

3. 3 240° 4. 53°

5. 170° 6. 73°

7. 125° 8. 81°

9. 159° 10. 66°

11. 69° 12. 74°

13. 144° 14. 165°

15. 170° 16. 8

17. 19 18. 28

19. 24° 20. 58°

21. 58°

22. x = 36°, y = 25°

23. x = 62°, y = 123°

24. a = 63°, b = 98°

25. 22

26. (a) 16 (b) 25 (c) 45

27. (a) 108° (b) 20

28. yes

29. x = 75°, y = 45°

66

F2B: Chapter 8D

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 16


(Video Teaching)

Book Example 17


(Video Teaching)

Book Example 18


(Video Teaching)



(Full Solution)

Maths Corner Exercise 8D Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 8D Multiple Choice


___________ ( )



_________

67

Book 2B Lesson Worksheet 8D (Refer to §8.2B)

8.2B Sum of Exterior Angles of a Polygon

Each of the following shows a set of exterior angles of the given polygon.

Triangle Quadrilateral Pentagon

1. Draw a set of exterior angles of each of the following polygons.

(Part (a) has been done for you as an example.)

(a) (b)

(c) (d)

The sum of exterior angles of a convex polygon is 360°.

[Reference: sum of ext. ∠s of polygon]



Sol x + 80° + 60° + 90° = 360° x + 250° = 360°

x = 110°

Find y in the figure.

Sol y + =

=

A B

C

D

60° 80°

x

A B

C

D

65°

45°

y

50°

E

105°

�sum of ext. ∠s

of polygon

Step 1111: Extend a side of the polygon at each vertex.

Step 2222: Mark the exterior angles.

� A polygon has more than one set of exterior angles.

e.g.

Quadrilateral

68

2. Find m in the figure.

3. Find n in the figure.

4. Find p in the figure.


○○○○→→→→ Ex 8D 1−9

6. Find the size of each exterior angle of a

regular octagon.

Sum of exterior angles =

∵ All the exterior angles of a regular

polygon are .

∴ Each exterior angle

=

7. Find the size of each exterior angle of a

regular 18-sided polygon.

○○○○→→→→ Ex 8D 10−12

8. If the size of each exterior angle of a regular n- 9. It is given that the size of an exterior angle of a

A

B

C

142°

m

83°

A

n

45°

122°

B

C

D

A

p

40° 81°

77°

B

84° C

D

E

F

p

A x

140°

B

82° D

y

116° C

An regular octagon has ____ sides and has (equal / unequal) exterior

69

sided polygon is 72°, find the value of n.

Sum of exterior angles =

n × ( ) =

=

regular polygon is 10°. Find the number of

sides of the polygon.

○○○○→→→→ Ex 8D 13−15




x

x

2x + 9°

46° 62°

79°

48° y x

110°

15°

29°

70



Level 1


1. 2.

3. 4.

5. 6.


7. 8.

9.

Consolidation Exercise 8D ��

71

Find the size of each exterior angle of the following regular polygons. [Nos. 10−−−−12]

10. Regular nonagon



In each of the following, the size of each exterior angle of a regular polygon is given. Find the number of

sides of the polygon. [Nos. 13−−−−15]

13. 30° 14. 24° 15. 8°

16. It is given that the size of each interior angle of a regular polygon is 135°.

(a) Find the size of each exterior angle of the regular polygon.

(b) Find the number of sides of the regular polygon.

Level 2


17. 18.

19.

20. In a regular polygon, if the size of each interior angle is 2.5 times the size of each exterior angle, find

the number of sides of the polygon.

21. In the figure, ABCD is a part of a regular polygon. AB and DC are

produced to meet at P.

(a) Find x.

(b) Find the number of sides of the regular polygon.

72

22. In the figure, PQRS is a part of a regular polygon. PQTU and

SRT are straight lines. If RT ⊥ QU, find the number of sides

of the regular polygon.

23. In the figure, ABCDEFGHIJKL is a regular 12-sided polygon.

BAOP and OLK are straight lines. Find ∠POL.

24. In the figure, ABCDE is a part of a regular n-sided polygon.

AB and ED are produced to meet at F such that BC = DC = FC.

If ∠AFE = 72°, find the value of n.

73

Consolidation Exercise 8D (Answer)

1. 92° 2. 142°

3. 63° 4. 41°

5. 80° 6. 46°

7. a = 71°, b = 63°

8. 111° 9. 152°

10. 40° 11. 20°

12. 10° 13. 12

14. 15 15. 45

16. (a) 45° (b) 8

17. 18° 18. 18°

19. a = 54°, b = 48° 20. 7

21. (a) 72° (b) 5

22. 8 23. 60°

24. 10

74

F2B: Chapter 8E

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 19


(Video Teaching)



(Full Solution)

Maths Corner Exercise 8E Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 8E Multiple Choice


___________ ( )



_________

75

Book 2B Lesson Worksheet 8E (Refer to §8.3A−C)

[In this worksheet, only compasses and a straight edge can be used for constructions.]

8.3A Construction of Angle Bisector

Consider a given angle ∠AOB.

If OS bisects ∠AOB, i.e. ∠AOS = ∠BOS,

then OS is the angle bisector of ∠AOB.

1. Follow the steps below to construct the angle bisector of ∠AOB in the figure.

2. Bisect each of the following marked angles.

(a) (b)

○○○○→→→→ Ex 8E 1, 2

D E

F

G

H J

B

O A

�

Step 1111: Using O as centre and a radius of suitable length, draw an arc to cut OA at a point P, and cut OB at another point Q.

Step 2222: Using P and Q as centres and a

radius longer than

2

1PQ, draw

two arcs to meet at a point S.

Step 3333: Join OS. The line OS obtained is

the angle bisector of ∠AOB.

B

O A

S

76

8.3B Construction of Perpendicular Bisector

Consider a given line segment AB.

If PQ is perpendicular to AB and bisects AB,

i.e. PQ ⊥ AB and AN = NB,

then PQ is the perpendicular bisector of AB.

3. Follow the steps below to construct the perpendicular bisector of the line segment AB in the

figure.

4. Construct the perpendicular bisector of each of the following marked line segments.

(a) (b)

○○○○→→→→ Ex 8E 3, 4

D

C

H

G

B A

B A

Q

P

N

Step 1111: Using A as centre and a radius

longer than

2

1AB, draw an arc on

each side of the line segment AB.

Step 2222: Using B as centre and the same radius as in 1111, draw an arc on each side of AB such that they cut the two arcs drawn in 1111 at two points P and Q.

Step 3333: Join PQ. The line PQ obtained is the perpendicular bisector of

AB.

77

8.3C Constructions of Some Special Angles

I. Construction of Angles of 90°°°° and 45°°°°

5. Follow the steps below to construct angles of 90° and 45°.

II. Construction of Angles of 60°°°° and 30°°°°

6. Follow the steps below to construct angles of 60° and 30°.

B A O

Step 1111: Draw a straight angle AOB.

(This step has been done for you.)

Step 2222: Bisect ∠AOB into ∠AOC and

∠BOC.

Then ∠AOC = ∠BOC = 90°.

Step 3333: Bisect ∠BOC into ∠BOD and

∠COD.

Then ∠BOD = ∠COD = 45°.

Which type of triangle can we get if we join BC?

Step 1111: Draw a line segment AB of suitable length.

Step 2222: Using A and B as centres and AB as radius, draw two arcs to meet at a point C.

Step 3333: Join AC.

Then we have ∠CAB = 60°.

Step 4444: Bisect ∠CAB into ∠CAD and

∠BAD.

Then ∠CAD = ∠BAD = 30°.

straight angle

= 180°

78


Construct an angle of 15° by using compasses

and a straight edge only.

Sol

In the figure, ∠BAE = 15°.

Construct an angle of 22.5° by using compasses

and a straight edge only.

Sol

In the figure, .

○○○○→→→→ Ex 8E 5−7


7. The figure below shows a line segment AB.

(a) Construct an angle ∠ABC of size 60° such that C is a point lying above AB.

(b) Construct an angle ∠ABD of size 45° such that D is a point lying below AB.

(c) Find the size of ∠CBD.

(a), (b)

(c) ∠CBD =

B A

15°

C

D

E

A B

30° ÷ 2 = 15°

Step 1111: Construct an angle of 60°.

Step 2222: Obtain an angle of 30° by bisecting

the angle of 60°.

Step 3333: Obtain an angle of 15° by bisecting

the angle of 30°.

( )° ÷ 2 = 22.5°

Step 1111: Construct an angle of ( )°.

Step 2222: Obtain an angle of ( )° by bisecting the angle in 1111.

Step 3333: Obtain an angle of 22.5° by bisecting the angle in 2222.

� ∠BAC = 60°,

∠BAD = ∠CAD = 30°,

∠BAE = ∠DAE = 15°

79



[In this exercise, use compasses and a straight edge only for constructions.]

Level 1

1. Bisect each of the following marked angles.

(a) (b)

(c)

2. Construct the perpendicular bisector of each of the following marked line segments.

(a) (b)

(c)

Consolidation Exercise 8E ��

80

Construct each of the following angles. [Nos. 3−−−−5]

3. 270° 4. 120° 5. 330°

6. In the figure, O is a point on AB. Construct a line passing through O and

perpendicular to AB.

7. (a) Construct the perpendicular bisectors of the line segments

PQ and QR in the figure respectively. Mark the point of

intersection of the two perpendicular bisectors as O.

(b) Use O as centre and OP as radius to draw a circle. Does the

circle pass through the points Q and R?

Level 2

8. Draw any line segment and divide it into 4 equal parts.

9. Draw any angle and divide it into 4 equal angles.

10. The figure shows a circle with centre O. Divide the circle into

8 equal parts.

11. (a) Construct a regular pentagon.

(b) Hence, construct a regular decagon.

12. (a) Construct any triangle ABC. Then, construct three line segments AA′, BB′ and CC′, where A′, B′

and C′ are the mid-points of BC, AC and AB respectively.

(b) Do the three line segments AA′, BB′ and CC′ constructed in (a) intersect at one point?


(a) Construct a line passing through O and perpendicular to AB.

(b) Hence, construct a line passing through O and parallel to AB.

81

Construct each of the following figures. [Nos. 14−−−−15]

(The sizes of the figures constructed can be different from the given figures.)

14.

15.

82

Consolidation Exercise 8E (Answer)

7. (b) yes 12. (b) yes

83

F2B: Chapter 9

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)

Book Example 5


(Video Teaching)

Book Example 6


(Video Teaching)

Book Example 7


(Video Teaching)

84

Book Example 8


(Video Teaching)



(Full Solution)

Maths Corner Exercise 9 Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 9 Multiple Choice


___________ ( )



_________

85

Book 2B Lesson Worksheet 9 (Refer to §9.2)

9.2A Proving Geometric Properties Relating to Intersecting and Parallel

Lines

(I) Intersecting lines

(a) (b) (c)

If POR is a straight line,

then a + b = 180°.

[Reference:

adj. ∠s on st. line]

If POQ and ROS are

straight lines,

then a = b and c = d.

[Reference: vert. opp. ∠s]

a + b + c = 360°

[Reference: ∠s at a pt.]


In the figure, PQR is a straight line. Prove that

x + y = 130°.

Refer to the figure. If x + y = 270°, prove that

AO ⊥ OC.

Sol x + 50° + y = 180°

x + y = 130°

adj. ∠s on st. line

Sol

x + y = 270°

∠AOC + ( ) + ( ) = ( )

∠AOC + ( ) = ( )

=

∴

given

__________

1. In the figure, AOB and COD are straight

lines. Prove x = y + 10°.

2. In the figure, POR is a straight line and

a = b. Prove that x = y.

○○○○→→→→ Ex 9 1−4

b a P

O R

Q

d

P

O a

c

Q

b

R

S

a b

c

O P

Q x

y

30°

20°

B

C

A

D

Q

R

S

P O

a

b

x

y

Consider the straight angles formed at O.

Facts given in

the question

T

R P Q

50°

y x

S

Statement Reason

State the reason(s)

here.

A

B

C O

y

x

86

If the sum of two adjacent angles ∠POQ and ∠ROQ is 180°,

then POR is a straight line.

i.e. In the figure,

if a + b = 180°,

then POR is a straight line.

[Reference: adj. ∠s supp.]


Refer to the figure. Prove that AOC is a straight

line.

Refer to the figure. Prove that WOZ is a

straight line.

Sol ∠AOC = ∠AOB + ∠BOC

= 142° + 38°

= 180°

∴ AOC is a straight line.

adj. ∠s supp.

Sol ∠WOZ

=

3. Refer to the figure. Prove that POS is a

straight line.

4. In the figure, AOB is a straight line.

(a) Find x.

(b) Prove that COD is a straight line.

○○○○→→→→ Ex 9 5, 6

B

C A O

142° 38°

W

X Y

O Z 55°

73° 52°

O

P U Q

R

S

T

y

x x y

B

C

A

D O x

5x 30°

b a P

O R

Q � This theorem can be

used to determine

straight lines.

87

(II) Parallel lines

(a) (b) (c)

If AB // CD,

then a = b.

[Reference:

corr. ∠s, AB // CD]

If AB // CD,

then a = b.

[Reference:

alt. ∠s, AB // CD]

If AB // CD,

then a + b = 180°.

[Reference:

int. ∠s, AB // CD]


In the figure, AFB and EFD are straight lines.

AB // CD. Prove that x + y = 360°.

In the figure, AB // CD and ADE is a straight

line. Prove that x + y = 180°.

Sol ∠EDC = x

∠EDC + y = 360°

∴ x + y = 360°

corr. ∠s, AB // CD

∠s at a pt.

Sol ∠ADC =

5. In the figure, BA // DC. Prove that

x + y = 180°.

6. In the figure, FDBG is a straight line.

AB // CD and EC // FG. Prove that x = y.

○○○○→→→→ Ex 9 7−10

A

D

a B

b C

A

C D

a B

b

A

C D

a B

b

A D

x B

y C

F E

A

D x

B

y C E

A

D x

B

y

C

E 2x

A

D

x B

y

C

E F

G

88

(a) (b) (c)

If a = b,

then AB // CD.

[Reference:

corr. ∠s equal]

If a = b,

then AB // CD.

[Reference:

alt. ∠s equal]

If a + b = 180°,

then AB // CD.

[Reference:

int. ∠s supp.]


In the figure, CDE and FAD are straight lines.

If x + y = 180°, prove that AB // CE.

In the figure, AF and CD intersect at E. If

x + y = 180°, prove that AB // CD.

Sol ∠FDE + y = 180°

∠FDE = 180° − y

x + y = 180°

x = 180° − y

∴ ∠FDE = x

∴ AB // CE

adj. ∠s on st. line

given

corr. ∠s equal

Sol

7. In the figure, BC // DE. If x + y = 360°,

prove that AB // CD.

8. In the figure, CD // EF. AQB and QRS are

straight lines. If x = y, prove that AB // CD.

○○○○→→→→ Ex 9 11−13

A

D

a B

b C

A

C D

a B

b

A

C D

a B

b

A

D

x B

y

C

F

E

A

D

x

B

y

C

E F

A

D

x B

y

C E

A

D

x B

y

C

F

E S

Q

R

89

9.2B Proving Geometric Properties Relating to Triangles

(a) (b)

a + b + c = 180°

[Reference: ∠ sum of △]

c = a + b

[Reference: ext. ∠ of △]


Refer to the figure. Prove that x + y = 80°.

In the figure, ACD is a straight line. Prove that

y − x = 90°.

Sol x + y + 100° = 180°

x + y = 80°

∠ sum of △ Sol x + ( ) = ( )

__________

9. In the figure, CDA is a straight line. Prove

that x + y = 90°.

10. In the figure, BCDF is a straight line.

Prove that AC // ED.

A x

y

100° C

B

C 50°

x

B

y

40° A

D

E

F

96°

D C B

52°

44°

A

x

B

A

C D

y

90

11. Refer to the figure. Prove that ACD is a

straight line.

12. In the figure, EDB is a straight line.

(a) Prove that EB ⊥ CB.

(b) Prove that EDBA is a straight line.

○○○○→→→→ Ex 9 14–19


13. In the figure, AB // ED. Prove that z = x + y.

A 53°

B

68°

C D 55°

66°

E

B 27°

A E D

130°

C

40° 63°

B

C

A

D

x y

E

z Add a suitable line to relate x, y and z.

91


9 Introduction to Deductive Geometry

Level 1

1. In the figure, RO ⊥ PO. If x = y, prove that SO ⊥ QO.

2. In the figure, AOE is a straight line, ∠BOC = 60° and

∠DOE = 30°. Prove that p + q = 90°.

3. In the figure, KOL and MON are straight lines.

If a + b + c = 180°, prove that POQ is a straight line.

4. In the figure, WO ⊥ OV. If a + b = 180°, prove that TO ⊥ UO.


(a) Find y.

(b) Prove that AOC is a straight line.

6. In the figure, ABCD is a straight line and AD // EF.

If x = y, prove that a = b.

7. In the figure, PUQ, RVS and TUVW are straight lines, and

PQ // RS. Prove that a = b.

Consolidation Exercise 9 ��

92

8. Refer to the figure. Prove that AB // DC.

9. In the figure, PQRS is a straight line. If q + r = 180°, prove TQ //

UR.

10. In the figure, KL // MN. Prove that LP // QM.

11. In the figure, ABC is a straight line and AC // DE. Prove that AC

// FG.

12. In the figure, JKL and MKN are straight lines.

If m + k = 90°, prove that JL ⊥ ML.

13. In the figure, PSQ and PTR are straight lines.

Prove that a = b + 10°.

14. In the figure, KLM is a straight line. Prove that JK // NM.

15. In the figure, BCD is a straight line. If d = a + b, prove that AC //

DE.

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Level 2

16. In the figure, AOD is a straight line. If ∠AOC = 129° and

∠BOD = 141°, prove that BO ⊥ CO.

17. In the figure, AOE, BOF and COG are straight lines,

∠AOB = ∠HOG and ∠BOC = ∠DOE. If GO ⊥ EO,

prove that HOD is a straight line.

18. In the figure, ∠NOM = 60°. Prove that JOL is a straight line.

19. In the figure, ABC and EBF are straight lines. AC // FD and EF //

CD. Prove that ∠ABE = ∠FDC.

20. In the figure, PQR and UVW are straight lines. It is given that

PR // ST // UW and QS // TV. Prove that a = b.

21. In the figure, BA // CE. If a = b, prove that BF // CD.

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22. In the figure, BCDE is a straight line. If a + b + e + f = 180°, prove

that AC // FD.

23. Refer to the figure. If e + f = g, prove that DE // HG.

24. Refer to the figure. Prove that QP // ST.

25. In the figure, KJ // MN. If k + m = 270°, prove that KL ⊥ ML.

26. In the figure, BA // CD. BE and CE are the angle bisectors of

∠ABC and ∠BCD respectively. Prove that BE ⊥ CE.

27. In the figure, QOT, POS and RST are straight lines.

(a) Prove that QP // RT.

(b) If QR // PS and ∠PQR = 148°, is ∠POT a right angle?


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Consolidation Exercise 9 (Answer)

5. (a) 36° 27. (b) no

Chapter 7 Rate and Ratio 7A p.2 7B p.9 7C p.24 Chapter 8 ...

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