chapter 10 applications in trigonometry

1

Chapter 10 Applications in Trigonometry

10A p.2

10B p.15

10C p.26

Chapter 11 Coordinate Geometry of Straight Lines

11A p.40

11B p.48

11C p.59

11D p.73

11E p.83

Chapter 12 Introduction to Probability

12A p.89

12B p.98

12C p.111

12D p.118

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2

F3B: Chapter 10A

Date Task Progress

Lesson Worksheet

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Book Example 1

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Book Example 2


(Video Teaching)

Book Example 3


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Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)

Consolidation Exercise


(Full Solution)

Maths Corner Exercise 10A Level 1


Teacher’s Signature

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Maths Corner Exercise 10A Multiple Choice

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E-Class Multiple Choice ○ Complete and Checked Mark:

3

Self-Test ○ Problems encountered ○ Skipped _________

4

Book 3B Lesson Worksheet 10A (Refer to §10.1)

[In this worksheet, unless otherwise stated, give the answers correct to 3 significant figures if

necessary.]

10.1A Gradient of an Inclined Plane

Refer to the figure. If AB represents an inclined plane, then

gradient of AB =

distance horizontal

distance vertical =

AC

BC

e.g. Gradient of the inclined plane XY in the figure

=

XZ

YZ

=

m 10

m 5

=

2

1 (or 1 : 2)

1. In each of the following, find the gradient of the inclined road AB.

Vertical

distance AC

Horizontal

distance BC

Gradient of AB

(in the form of 1 : n)

(a) 5 cm 20 cm cm ) (

cm ) ( = 1 : ( )

(b) 0.4 m 12 m

(c) 400 m 60 000 m

○○○○→→→→ Ex 10A 1

Example 1 Instant Drill 1

A car travels down along an inclined road as shown. Find the value of d.

Sol Gradient of the road =

m

m 02

d

6

1 =

d

02

d = 20 × 6 = 120

May walks up along an inclined road as shown. Find the value of h.

Sol Gradient of the road =

) (

) (

=

Y

Z X

5 m

10 m

horizontal distance = d m

gradient =

6

1

vertical distance = 20 m

gradient = 1 : 10

horizontal distance = 70 m

vertical distance = h m

B

A

C

� Gradients are usually expressed in the

form of

n

1 or 1 : n, where n is an integer.

5

2. Kary walks along an inclined road with gradient 1 : 7. If the horizontal distance she walks is 17.5 m, find the vertical distance she walks.

Let d m be the vertical distance she walks.

Gradient of the road =

) (

) (

) (

) ( =

) (

) (

= ○○○○→→→→ Ex 10A 2, 3

3. When an ant crawls 85 cm up along a straight stick, the vertical distance that it crawls is 13 cm.

(a) Find the horizontal distance that the ant crawls.

(b) Find the gradient of the stick in the form of

n

1, where n is correct to

the nearest integer.

(a) Let AC = x cm. In △ABC,

x2 + ( )2 = ( )2 =

(b) Let

n

1 be the gradient of the stick.

Gradient of the stick =

AC

BC

n

1 =

) (

) (

= ○○○○→→→→ Ex 10A 4, 5

10.1B Gradient and Inclination Refer to the figure.

(a) The angle θ between the inclined plane AB and the horizontal (AC) is called the inclination of AB.

(b) Gradient of AB =

AC

BC = tan θ

e.g. Gradient of the inclined plane XY in the figure

= tan 38° = 0.781, cor. to. 3 sig. fig.

B

C

13 cm 85 cm

A

gradient =

)(

)(

_____ m

?

Use Pythagoras’ theorem to find x. Recall:

a2 + b2 = c2

a

b c

Z X

Y

38°

horizontal

(inclination)

6

4. In each of the following, find the gradient of the inclined plane AB. (a) (b) (c) In each of the following, find the gradient of an uphill road with the given inclination. [Nos. 5–6] (Express the answers in the form of 1 : n, where n is correct to the nearest integer.)

5. 7°

Let 1 : n

n

1 i.e. be the gradient of the road.

Gradient of the road = tan ( )

n

1 = tan ( )

=

∴ The gradient of the road is .

6. 4.4°

○○○○→→→→ Ex 10A 6


The gradient of a road is

4

1. Find its

inclination.

Sol Let θ be the inclination of the road.

tan θ =

4

1

θ = 14.0°, cor. to 3 sig. fig.

∴ The inclination of the road is 14.0°.

Find the inclination of a path with gradient 1 : 18.

Sol

7. In each of the following, find the

inclination θ of an inclined road with the given gradient.

(a) 3.7 (b) 8

5

8. Find the inclination of the inclined road

AB in the figure.

A

B

C 24°

horizontal

11°

A

B

C horizontal 50°

A

B

C horizontal

A

B

42 m

75 m horizontal

Find the gradient first.

7

○○○○→→→→ Ex 10A 8

○○○○→→→→ Ex 10A 9

9. The gradient of an inclined path is 0.62. (a) Find the inclination of the path. (b) Andy walks 100 m up along the path. Find the horizontal distance he walks. ○○○○→→→→ Ex 10A 10, 11

10.1C Gradient on Map

The figure shows a contour map. If AB represents an inclined straight path with horizontal distance 3 000 m, we have: (i) Since A lies on the contour line with label ‘400 m’,

A is 400 m above the sea level. Similarly, B is 100 m above the sea level.

(ii) Vertical distance between A and B = (400 − 100) m = 300 m

(iii) Gradient of AB =

m 000 3

m 300

=

10

1

inclination ( ) ?

100 m

Use ‘sin’, ‘cos’ or ‘tan’?

100 m

200 m

300 m

400 m A

B

� Recall:

Gradient =

distance horizontal

distance vertical

8


The figure shows a contour map, where PQ represents a straight road. It is given that the horizontal distance between P and Q is 900 m. Find (a) the vertical distance between P and Q, (b) the gradient of the straight road, and

express the answer as a fraction.

Sol (a) Vertical distance between P and Q = (350 – 250) m = 100 m (b) Gradient of the road

=

m 900

m 100

=

9

1

The figure shows a contour map, where MN represents a straight road. It is given that the horizontal distance between M and N is 3 600 m. Find (a) the vertical distance between M and N, (b) the gradient of the straight road, and

express the answer as a fraction.

Sol (a) Vertical distance between M and N = [( ) – ( )] m = (b)

10. The figure shows a contour map of a hill. A straight path is built from

point A to point B. It is given that the gradient of the path is

5

1. Find

(a) the vertical distance between A and B, (b) the horizontal distance between A and B. ○○○○→→→→ Ex 10A 13, 14

200 m

250 m

300 m

350 m P

Q

200 m

300 m

400 m 500 m

M

N

675 m

650 m 625 m

600 m

A

B

9

�� ‘Explain Your Answer’ Question

11. PQ and RS are two straight highways. It is given that the gradient of PQ is 1 : 7 and the

inclination of RS is 10°. (a) Find the gradient of RS. (b) Which highway is steeper, PQ or RS? Explain your answer.

(a) Gradient of RS = (b) Gradient of PQ =

∵ Gradient of PQ ( > / < ) gradient of RS

∴ Highway (PQ / RS) is steeper.

Level Up Question

12. In the figure, AB and BC represent two straight roads. ADC is a

horizontal line, BD ⊥ AC, BD = 2 m, AC = 30 m and the

inclination of AB is 8°. (a) Find the lengths of AD and DC. (b) Find the inclination of BC.

30 m

2 m 8°

A

B

C D

The greater the gradient, the steeper the

10

New Century Mathematics (2nd Edition) 3B

10 Applications in Trigonometry

Level 1

1. (a) AB is a straight road. The vertical distance between A and B is 5 m. The horizontal distance between A and B is 50 m. Find the gradient of the road AB.

(b) Jason travels up along a straight road. When he travels a horizontal distance of 16 m, he rises 6 m vertically. Find the gradient of the road.

2. In the figure, David runs from P to Q along a road with gradient 1 : 30.

If he runs a horizontal distance of 1 800 m, find the vertical distance travelled by him.

3. The figure shows a road AB with gradient

17

1.

If a car travels a vertical distance of 3.4 m, find the horizontal distance travelled by the car.

4. In the figure, the length of the lane XY is 25.7 m. The horizontal distance between X and Y is

25.5 m.

(a) Find the vertical distance between X and Y.

(b) Express the gradient of the lane in the form of

n

1, where n is correct to the nearest integer.

5. In each of the following, find the gradient of an uphill road with the given inclination.

(a) 8° (b) 2.3° ( Express the answers in the form of 1 : n, where n is correct to the nearest integer.)

3.4 m

A

B

1 800 m P Q

25.5 m

25.7 m

X

Y


10A

11

6. The figure shows a road AB with inclination θ. Find the gradient of the road in each of the following situations.

(a) tan θ = 0.06 (b) θ = 5.2° (c) sin θ = 0.4

(Express the answers in the form of

n

1, where n is correct to the nearest integer.)

7. In each of the following, find the inclination of a straight road with the given gradient.

(a) 1.5 (b) 3

7 (c) 1 : 18

(Give the answers correct to the nearest 0.1°.) 8. In the figure, a cable car travels along a straight cable. If it travels a horizontal distance of

960 m and a vertical distance of 128 m, find the inclination of the cable, correct to 3 significant figures.

9. In the figure, a cat walks down along a path with gradient 0.2.

(a) What is the inclination of the path? (b) If the cat walks 15 m along the path, find the horizontal distance the cat walks. (Give the answers correct to 3 significant figures.)

10. The gradient of a road PQ is

25

8. The inclination of another road QR is 16°. Wendy claims that

the road PQ is steeper than the road QR. Do you agree? Explain your answer.

gradient = 0.2

A

B θ

128 m

960 m

12

100 m

150 m

200 m

250 m

X

Y

P

Q

350 m

400 m

450 m 500 m

11. The figure shows a contour map, where XY represents a straight road of gradient 1 : 12.

(a) Find the vertical distance between X and Y. (b) Find the horizontal distance between X and Y. 12. The figure shows a contour map. PQ represents a straight

road and the horizontal distance between P and Q is 400 m. (a) Find the gradient of the road, and express the answer as

a fraction. (b) Find the inclination of the road. (Give the answer correct to the nearest degree.) Level 2

13. In the figure, Jenny drives at a speed of 14 m/s from P to Q along a straight road with

gradient

3

1. She rises 56 m vertically in the whole journey.

(a) Find the inclination of the road. (b) Find the time taken in the whole journey. (Give the answers correct to 3 significant figures.)

14. Fig. A shows a ladder PQ which leans against a vertical wall. The inclination of the ladder is

32° and the foot of the ladder is 1.4 m from the corner O of the wall.

(a) Find the length of the ladder. (b) Later, the top P slides down to X which is 0.7 m above O, and the foot Q slides to Y at the

same time, as shown in Fig. B. Find the new inclination of the ladder. (Give the answers correct to 3 significant figures.)

P

Q

32°

1.4 m

P

Q

X

Y

0.7 m

Fig. A Fig. B

13

200 m

250 m

300 m

350 m

400 m

450 m

X

Y Z

Scale 1 : 30 000

15. In the figure, XY and YZ represent two straight roads, where X and Z are two points on the

horizontal ground. N is a point on XZ such that YN ⊥ XZ. YN = 80 m and XZ = 650 m.

It is given that the inclination of the road XY is 13°. (a) Find the length of NZ, correct to 3 significant figures. (b) Which path, XY or YZ, is less steep? Explain your answer. 16. The figure shows two straight roads PQ and QR of lengths 2 000 m and 960 m respectively. The

gradient of PQ is 0.32. ONR is a horizontal line.

(a) Find the inclination of PQ. Hence, find the horizontal distance between P and Q. (b) It is given that the horizontal distance between P and R is 2 800 m. Find the gradient

of QR. (Give the answers correct to 3 significant figures.) 17. On a contour map of scale 1 : 8 000, AB represents a straight road. The vertical distance

between A and B is 50 m. If the inclination of the road is 9°, find the length of AB on the map correct to the nearest 0.01 cm.

18. The figure shows a contour map of scale 1 : 30 000. XY

and YZ represent the two parts of a hiking trail. The length of XY on the map is measured as 3 cm.

(a) Find the inclination of the path XY, correct to 3 significant figures.

(b) If the actual total length of the trail is 1.7 km, which path, XY or YZ, is steeper? Explain your answer.

650 m

80 m

X

Y

Z

P

Q

R

2 000 m 96 O N

14

Answer

Consolidation Exercise 10A

1. (a) 0.1

10

1or (b) 0.375

8

3or

2. 60 m

3. 57.8 m

4. (a) 3.2 m (b) 8

1

5. (a) 1 : 7 (b) 1 : 25

6. (a) 17

1 (b)

11

1 (c)

2

1

7. (a) 56.3° (b) 66.8° (c) 3.2°

8. 7.59°

9. (a) 11.3° (b) 14.7 m

10. yes

11. (a) 150 m (b) 1 800 m

12. (a) 4

1

(b) 14°

13. (a) 18.4° (b) 12.6 s

14. (a) 1.65 m (b) 25.1°

15. (a) 303 m (b) XY

16. (a) inclination = 17.7°,

horizontal distance = 1 900 m

(b) 0.387

17. 3.95 cm

18. (a) 9.46° (b) XY

15

F3B: Chapter 10B

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)



(Full Solution)

Maths Corner Exercise 10B Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 10B Multiple Choice

○ Complete and Checked ○ Problems encountered


___________

16

○ Skipped ( )

E-Class Multiple Choice Self-Test

○ Complete and Checked ○ Problems encountered ○ Skipped Mark:

_________

17

Book 3B Lesson Worksheet 10B (Refer to §10.2)


necessary.]

10.2 Angles of Elevation and Depression (a) When we look up at an object, the angle between the line of sight and the horizontal is called the angle of elevation. When we look down at an object, the angle between the line of sight and the horizontal is called the angle of depression.

e.g. In the figure,

the angle of elevation of P from A = 60°

the angle of depression of B from P = 40° (b) Angle of elevation of B from A = angle of depression of A from B


In the figure, BC is a lamp post of height 6 m. A is 2 m away from B. Find the angle of elevation of C from A.

Sol Let θ be the angle of elevation of C from A.

In △ABC,

tan θ =

m 2

m 6

θ = 71.6°, cor. to 3 sig. fig.

∴ The required angle of elevation is

71.6°.

In the figure, DE is a building of height 40 m. F is 27 m away from E. Find the angle of elevation of D from F.

Sol Let θ be the angle of elevation of D from F.

In △DEF,

θ =

) (

) (

=

θθθθ

2 m A B

C

6 m θθθθ

D

E F 27 m

40 m

P 40°

60°

A B horizontal

horizontal

2 m A B

C

6 m

horizontal ground

F

D

E

40 m

27 m horizontal ground

18

1. In the figure, a hot-air balloon H is 110 m

vertically above B. Points A and B are 185 m apart. Find the angle of elevation of H from A.

2. The figure shows a road sign RS of height

5.3 m. Q is 4.6 m away from S. Find the angle of depression of Q from R.

Refer to the notations in the figure.

In △QRS,

θ =

) (

) (

=

∠PRQ = � alt. ∠s, PR // QS

∴ The required angle of depression is

. ○○○○→→→→ Ex 10B 1, 2

3.

In the figure, the tree TS casts a shadow PS of length 15 m on the ground. The angle of

elevation of the sun from P is 23°. Find the height of the tree.

In △PST,

23° =

) (

) (

=

4.

In the figure, a bird B is 600 m from a buoy A at sea level. The angle of

depression of A from B is 51°. Find the horizontal distance between A and B.

○○○○→→→→ Ex 10B 3–5

600 m

51°

B

A sea level

A B

110 m

185 m

H

horizontal ground

Q

5.3 m

S

R

4.6 m horizontal ground

23°

horizontal ground

T

P S 15 m

Q S

R

θ

5.3 m

4.6 m

P

19


In the figure, the height of the flagpole CD is 9.5 m. Ivan stands at point B. His eyes at A are 1.7 m above the ground. The angle of elevation

of C from A is 64°. Find the distance between Ivan and the flagpole.

Sol With the notations in the figure,

construct AE ⊥ CD. CE = CD – ED = CD – AB = (9.5 – 1.7) m = 7.8 m

In △ACE,

tan 64° =

AE

CE

AE =

°64tan

CE

=

°64tan

8.7 m

= 3.80 m, cor. to 3 sig. fig.

∴ The distance between Ivan and the

flagpole is 3.80 m.

In the figure, PQ and RS are two buildings on the horizontal ground, and their heights are 50 m and 84 m respectively. The angle of

elevation of R from P is 35°. Find the distance between the two buildings.


construct PT ⊥ RS. RT = ( ) – ( ) =

In △PRT,

35° =

) (

) (

=

5. In the figure, the top P of a table is 130 cm above the horizontal

ground. The angle of elevation of P from D is 28°. P and D are 200 cm apart. Find the height of D above the ground.

64° A

B

C

D

E 1.7 m

9.5 m

35°

50 m

T

S

P

Q

R

84 m

35° P

Q

R

S

50 m

84 m

28°

table P

Q

200 cm

D 130 cm

64°

1.7 m

9.5 m

A

B

C

D horizontal ground

20

6. In the figure, the heights of a spotlight (X) and the top of a display board (Y)

are 6 m and 3.3 m above the horizontal ground respectively. The horizontal distance between X and Y is 2.5 m. Find the angle of depression of Y from X.

○○○○→→→→ Ex 10B 6–9

Level Up Question

7. In the figure, A and B are two windows of a building on the horizontal ground. The angles of elevation of A and B from a point C on the ground

are 42° and 63° respectively. A, B and C lie on the same vertical plane. Find the distance between A and B.

42°

A

B

C

63°

9 m

6 m

3.3 m

display board

X

Y

2.5 m

21



[In this exercise, give the answers correct to 3 significant figures.]

Level 1 1. In the figure, XY represents a building of height 45 m. K is a point on the horizontal ground and

KY = 80 m. Find the angle of elevation of X from K.

2. In the figure, the bottom B of a balloon is tied to a point C on the horizontal ground by a straight

string of length 6.3 m. If the horizontal distance between B and C is 1.9 m, find the angle of depression of C from B.

3. In the figure, ST is a tower of height 90 m. The angle of elevation of T from a point U on the

horizontal ground is 54°. Find the horizontal distance between U and T.

X

Y K

80 m

45 m

6.3 m

C 1.9 m

B

90 m

54°

T

S

U


10B

22

Q

6 m

23° P

1.5 m

4. In the figure, the top M of a vertical flagpole is tied to a point R on the horizontal ground by a

straight rope of length 8.2 m. If the angle of depression of R from M is 37°, find the height of the flagpole.

5. In the figure, a cat stands at P on the horizontal ground and a light bulb is mounted at Q on the

horizontal celling. The horizontal distance between P and Q is 0.9 m. If the angle of depression

of P from Q is 70°, find the distance between P and Q.

6. In the figure, Paul’s eyes at P are 1.5 m above the

horizontal ground. He looks at the top Q of a tree. The

angle of elevation of Q from P is 23°. The horizontal distance between P and Q is 6 m. Find the height of the tree.

7. In the figure, AB and CD represent two buildings on the horizontal ground. The height of

building AB is 160 m and the angle of depression of C from A is 48°.

It is given that the distance between A and C is 170 m. Find the height of building CD.

8.2 m 37°

M

R

Q

P

48°

C

A

160 m

D B

1

23

7° 29°

P

Q

X R

8. In the figure, a hawk X and a squirrel Y are 10.8 m and 7.2 m above the horizontal ground respectively. The distance between X and Y is 9 m. Find the angle of depression of Y from X.

Level 2

9. In the figure, XY is a lamppost of height 5 m. P and Q are points on the horizontal ground and

PQ = 13 m. PYQ is a straight line. The angle of elevation of X from P is 40°.

(a) Find PY. (b) Find the angle of depression of Q from X. 10. In the figure, SN is a Christmas tree. P and Q are two points on the horizontal ground. The

angles of elevation of S from P and Q are 37° and 52° respectively. If S and P are 26 m apart, find the distance between N and Q.

11. In the figure, PQR is a vertical statue. X and R lie

on the horizontal ground. The angles of elevation of

P and Q from X are 29° and 7° respectively. It is given that the distance between X and Q is 96 m.

(a) Find the distance between X and R. (b) Find the distance between P and Q.

Y

X

7.2 m

10.8 m

9

5 m

13 m

P

Q

X

Y 40°

26 m

P Q 37°

S

52°

N

24

A

B

C

D

30 m

25 m

15°

23°

56°

P

X

Y

Z

47 m

12. In the figure, the height of a vertical cliff BC is 75 m. The angle of elevation of a bird A from C

is 16°. The horizontal distance between A and C is 120 m.

(a) Find the height of the bird above the horizontal ground. (b) Find the angle of depression of A from B. 13. In the figure, the height of a hill PN is 760 m. X and Y represent the two banks of a river. N, X

and Y lie on the same horizontal line. The angles of depression of X and Y from P are 38° and

29° respectively. Find the width of the river.

14. The figure shows two buildings AB and CD in a school. C

is tied to A and B with two straight ribbons. The height of building AB is 25 m and the length of ribbon AC is

30 m. The angle of depression of A from C is 15°. (a) Find the height of building CD. (b) Find the angle of elevation of C from B. 15. In the figure, XY is the lightning rod of a building YZ and XYZ

is a vertical line. The angles of elevation of X and Y from a

helicopter P are 56° and 23° respectively. The length of the lightning rod is 47 m.

(a) Find the horizontal distance between the helicopter and the building.

(b) If XZ = 150 m, find the vertical distance between P and Z.

75 m

120 m

16°

A

B

C horizontal ground

P

29°

X Y

38°

N

760 m

25

Answer

Consolidation Exercise 10B

1. 29.4°

2. 72.4°

3. 65.4 m

4. 4.93 m

5. 2.63 m

6. 4.05 m

7. 33.7 m

8. 23.6°

9. (a) 5.96 m (b) 35.4°

10. 12.2 m

11. (a) 95.3 m (b) 41.1 m

12. (a) 34.4 m (b) 18.7°

13. 398 m

14. (a) 32.8 m (b) 48.5°

15. (a) 44.4 m (b) 84.1 m

26

F3B: Chapter 10C

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)

Book Example 16


(Video Teaching)

Book Example 17


(Video Teaching)

27

Book Example 18


(Video Teaching)



(Full Solution)

Maths Corner Exercise 10C Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 10C Multiple Choice


___________ ( )



_________

28

N

55°

30°

A

C

O

N

30°

A

O E

S

W

N

55°

θ

C

O

N

65°

70°

D

F

O

N

70°

F

θ

O

Book 3B Lesson Worksheet 10C (Refer to §10.3)


necessary.]

10.3 Bearings

There are 8 major bearings used in daily life: east (E), south (S), west (W), north (N) north-east (NE), south-east (SE), south-west (SW), north-west (NW)

10.3A Compass Bearing

Compass bearing is expressed in the form of Nx°E, Nx°W, Sx°E or Sx°W, where

x° is measured from either the north or the south, and 0 < x < 90. e.g. The compass bearings of A, B, C and D from O are:

N40°E N45°W S50°E S55°W

Example 1 Instant Drill 1 Refer to the figure. Find the compass bearings of (a) A from O, (b) C from O.

Sol (a) The compass bearing

of A from O is N30°E. (b) With the notations in the figure,

θ = 90° – 55°

= 35°

∴ The compass bearing of C from O

is S35°W.

Refer to the figure. Find the compass bearings of (a) D from O, (b) F from O.

Sol (a) The compass bearing of D from O is .

(b) With the notations in the figure,

θ =

∴ The compass bearing of F from O

is .

It cannot be written

as W55°S.

29

N

68° K

N

α H

33°

N

P

Q

N Mark ‘ ’ at Q first.

N

40° A

N

θ B

D

C

N P

15°

22°

Q

O

N

81°

47°

R

T

O

N

N

?

G

F

Step 1111:

Step 2222: Mark ‘ ’ at F.

1. Refer to the figure. Find the compass bearings of

(a) P from O, (b) Q from O.

2. Refer to the figure. Find the compass bearings of

(a) R from O, (b) T from O.

○○○○→→→→ Ex 10C 1(a)(i), (b)(i), (c)(i), (d)(i)


In the figure, the compass

bearing of B from A is N40°E.

(a) Find θ. (b) Find the compass bearing of A from B.

Sol (a) θ = 40° � alt. ∠s, CA // DB (b) The compass bearing of A from B is

S40°W.

In the figure, the compass

bearing of K from H is S68°W.

(a) Find α. (b) Find the compass bearing of H from K.

Sol

3. In the figure, the compass

bearing of Q from P is

S33°E. Find the compass bearing of P from Q.

4. If the compass bearing of F from G is

N76°W, find the compass bearing of G from F.

○○○○→→→→ Ex 10C 2, 4

30

N

O

Q 76° θ

N

25°

O

Q

P

76°

N

94°

O

T

R

32°

N

O

T

32°θ

O

58° 40°

N

126° D

A

B

C

10.3B True Bearing

True bearing is expressed in the form of y°, where y° is measured from the north in a

clockwise direction and 0 ≤ y < 360. The integral part of y must be written in 3 digits. e.g. The true bearings of P, Q, R and S from O are:

009° 097.5° 220° 300.5°


Refer to the figure. Find the true bearings of (a) P from O, (b) Q from O.

Sol (a) The true bearing of P from O is

025°. (b) With the notations in the figure,

θ = 360° – 76°

= 284°

∴ The true bearing of Q from O

is 284°.

Refer to the figure. Find the true bearings of (a) R from O, (b) T from O.

Sol (a) The true bearing of R from O is .

(b) With the notations in the figure,

θ = 180° + ( ) =

∴ The true bearing of T

from O is .

5. Refer to the figure. Find the true bearings of (a) A from O, (b) B from O, (c) C from O, (d) D from O.

○○○○→→→→ Ex 10C 1(a)(ii), (b)(ii), (c)(ii), (d)(ii)

31

N

69° B

A

N

x

N

69° B

A

N

x

θ

N

130°

C N

D

y

N

130°

C N

D

y

θ

N

55°

X

Z

Y

200°

331°

Q

N

P 245° K

N

H


In the figure, the true bearing

of B from A is 069°. (a) Find x. (b) Find the true bearing of A from B.

Sol (a) x + 69° = 180°

x = 111° (b) With the notations in the figure,

θ = 360° – 111°

= 249°

∴ The true bearing of A from B

is 249°.

In the figure, the true bearing

of D from C is 130°. (a) Find y. (b) Find the true bearing of C from D.

Sol (a) y + = y = (b) With the notations in the figure,

θ =

∴ The true bearing of C from D

is . ○○○○→→→→ Ex 10C 3

6. In the figure, the true

bearing of Q from P is 331°. Find the true bearing of P from Q.

7. In the figure, the true

bearing of K from H is 245°. Find the true bearing of H from K.

○○○○→→→→ Ex 10C 5

8. Refer to the figure. Find (a) the true bearing of Y

from X, (b) the compass bearing

of Z from X.

○○○○→→→→ Ex 10C 6

9. If the true bearing of B from A is 226°, (a) find the true bearing of A from B, (b) find the compass bearing of A from B. ○○○○→→→→ Ex 10C 7

32

N

C

N

A

B

80 m

70 m

N

35°

T P 20 km

N

C

N

A

B

72 m

90 m

N

35°

T P 20 km

M

θ

N 152°

K H 50 m

N 152°

K H 50 m

β

L

T

d = shortest distance

between T and line L

L

10.3C Practical Problems Involving Bearings


In the figure, Sam walks 80 m due north from A to B, then walks 70 m due east from B to C. Find the true bearing of A from C.

Sol From the question, ∠ABC = 90°.

In △ABC,

tan ∠ACB =

m 70

m 80

∠ACB = 48.814°, cor. to 5 sig. fig.

∴ True bearing of A from C

= 270° – 48.814°

= 221°, cor. to 3 sig. fig.

In the figure, B is 72 m due north of A. B and C are 90 m apart. Find the compass bearing of C from B.

Sol

○○○○→→→→ Ex 10C 9, 10


In the figure, point P is 20 km due west of town T. If a car moves from P in the direction

N35°E, find the shortest distance between the car and town T.


construct TM ⊥ PM.

θ = 90° – 35° = 55°

In △PTM,

sin θ =

PT

TM

TM = PT sin θ

= 20 sin 55° km = 16.4 km, cor. to 3 sig. fig.

∴ The shortest distance between the car

and town T is 16.4 km.

The figure shows a market K which is 50 m due east of bus stop H. If Jane walks from H in

the direction of 152°, find the shortest distance between Jane and market K.


construct KL ⊥ HL.

β =

○○○○→→→→ Ex 10C 11, 12

33

A

N

O

B

A

B N

N

C

N

160°

P

Q

N

R 63°

27° 135 km

110 km

10. Two ships A and B set out from the same pier O at 6:00 a.m. Ship A sails due south at a speed of 24 km/h while ship B sails due west at a speed of 30 km/h. Find, at 9:00 a.m. on the same day,

(a) the compass bearing of B from A, (b) the distance between A and B. ○○○○→→→→ Ex 10C 13

Level Up Questions

11. Refer to the figure. (a) Find the true bearing of A from B. (b) Find the compass bearing of B from C. 12. In the figure, ships Q and R are 135 km and 110 km from a pier P

respectively. The bearing of Q from R is N27°W and the bearing of P

from R is S63°W. Find the distance between the two ships.

34

N

N

50°

θ

P

Q

R

S

N

31°

43°

27°

18° O

N

N K

H

250° θ

N

115°

A

B



[ In this exercise, unless otherwise stated, give the answers correct to 3 significant figures if

necessary.]

Level 1

1. Refer to the figure. Find the (i) compass bearing, (ii) true bearing of each of the following points from O. (a) P (b) Q

(c) R (d) S

2. In the figure, the compass bearing of Q from P is N50°E.

(a) Find θ. (b) Find the compass bearing of P from Q.

3. In the figure, the true bearing of H from K is 250°.

(a) Find θ. (b) Find the true bearing of K from H.

4. In the figure, the true bearing of B from A is 115°. Find the true bearing of A from B.


10C

35

N

Q P

N

R

N

6 km

5 km

N

B

A

N

C

45 m 63°

X

N

144° 56°

Y

P

A

B C

N 1918U

N

7.5 km

320°

V

5. Refer to the figure. (a) Find the compass bearing of P from Y. (b) Find the true bearing of P from X. 6. In the figure, A is due west of B, and C is due south of B. It

is given that the compass bearing of C from A is S63°E and the length of BC is 45 m. Find the length of AC.

7. In the figure, P, Q and R are the positions of three houses. Q

is 5 km due east of P, and R is 6 km due north of P. Find the compass bearing of Q from R.

8. In the figure, A is due north of B, and C is due west of B.

If BC = 180 m and AC = 195 m, find the true bearing of C from A.

9. In the figure, V is 7.5 km due north of U. A car travels

from U in the direction 320° along a straight road. Find the shortest distance between the car and V.

36

N Ben

O

Calvin

N

N

63°

X

Y

Z

N

A B

C

96°

37°

N 109° 44°

P

Q

R 36°

10. In the figure, Ben and Calvin set out from the same point O at

the same time. Ben runs due east at a speed of 7.2 km/h and Calvin runs due south at a speed of 9.6 km/h. Find the compass bearing of Calvin from Ben after 4 hours.

Level 2

11. Refer to the figure. Find the compass bearing of Y from Z. 12. Refer to the figure. (a) Find the compass bearing of P from R. (b) Find the true bearing of R from Q.

13. In the figure, the true bearing of B from A is 096°,

AC = AB and ∠ABC = 37°. Find the true bearing of A from C.

37

14. In the figure, a race car travels 1.6 km due north from X, then travels 6 km due west. Finally, it

travels 4.8 km due south to Y.

(a) Find the compass bearing of Y from X. (b) Find the distance between X and Y.

15. In the figure, Tom walks 370 m from his home at F in the direction 148° to a cinema at G, then

walks 160 m in the direction 058° to a restaurant at H.

(a) Tom claims that FG ⊥ GH. Do you agree? Explain your answer. (b) How far does Tom walk along a straight road from the restaurant to his home? In what

direction does he walk?

4.8 km

6 km

1.6 km

N X

Y

N

N

148°

58°

F

G

H 370 m

160 m

38

N

72° 310°

Q

5 km P

R

16. In the figure, A and B are 3.9 km apart. O is the centre of a circular lake with diameter 4.5 km. It

is given that OA = OB, and the compass bearings of O from A and B are N40°E and N76°W respectively.

(a) Find the compass bearing of A from B. (b) Lily claims that A and B are both located inside the lake. Do you agree? Explain your

answer.

17. In the figure, P, Q and R represent three bus stations. A bus

travels 5 km from P in the direction 072° to Q, and then travels

in the direction 310° to R, which is due north of P. Find the distance between P and R.

18. In the figure, a car travels from S to T in the direction N53°W, then travels in the direction

S26°W to U, which is 140 km due west of S. Suppose the speed of the car is constant throughout the whole journey and the car takes 2 hours to travel from S to T. Find the time taken for the car to travel from T to U.

N

N

148°

58°

F

G

H 370 m

160 m

N

N

U

T

S

26° 53°

140 km

39

Answer

Consolidation Exercise 10C

1. (a) (i) N31°E (ii) 031°

(b) (i) S43°E (ii) 137°

(c) (i) S63°W (ii) 243°

(d) (i) N72°W (ii) 288°

2. (a) 50° (b) S50°W

3. (a) 70° (b) 070°

4. 295°

5. (a) S56°E (b) 324°

6. 99.1 m

7. S39.8°E

8. 247°

9. 4.82 km

10. S36.9°W

11. N27°W

12. (a) S80°E (b) 171°

13. 022°

14. (a) S61.9°W (b) 6.8 km

15. (a) yes

(b) distance: 403 m,

direction: 305° (or N55.4°W)

16. (a) S72°W (b) no

17. 5.54 km

18. 1.34 h

40

F3B: Chapter 11A

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 1


(Video Teaching)

Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )



___________ ( )



_________

41

Book 3B Lesson Worksheet 11A (Refer to §11.1)

[In this worksheet, unless otherwise stated, leave the radical sign ‘√’ in the answers if necessary.]

11.1 Distance between Two Points

For any two points A(x1 , y1) and B(x2 , y2) in a rectangular coordinate plane, the distance between A and B is given by:

AB =2

12

2

12 )()( yyxx −+−

Example 1 Instant Drill 1 In each of the following, find the length of the line segment AB.

(a) (b)

In each of the following, find the length of the line segment PQ.

(a) (b)

Sol (a) AB = 22 )25()15( −+− units

= 22 34 + units

= 5 units

Sol (a) PQ = 22 )]()[()]()[( −+− units

= 22 )()( + units

=

(b) AB = 22 )43()]3(2[ −−+−− units

= 22 )7(5 −+ units

= units 74

(b) PQ = 22 )]()[()]()[( −+− units

=

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

2. Find the length of the line segment CD in the figure.

x

y

O

A(1 , 2)

B(5 , 5)

y

x O

A(−3 , 4)

B(2 , −3)

y

x O

P(2 , 9)

Q(8 , 1)

y

x O

Q(−3 , 5)

P(−5 , −2)

y

x O A(1 , −1)

B(7 , −6)

y

x O

C(4 , 3)

D(−3 , −4)

O

B(x2 , y2)

A(x1 , y1)

y

x

42

In each of the following, find the distance between the two given points. [Nos. 3–4] 3. A(1 , 0), B(9 , 6)

4. C(−6 , 7), D(1 , −2)

○○○○→→→→ Ex 11A 1−8

5. Find the length of the line segment AB in

the figure.

6. Find the length of the line segment CD in the figure.

○○○○→→→→ Ex 11A 9−12

7. In the figure, A(1 , −2), B(8 , −2) and C(5 , 2) are the

three vertices of △ABC. Find the perimeter of △ABC,

correct to 2 decimal places.

AB = [8 − ( )] units = BC = AC =

∴ Perimeter of △ABC

= ○○○○→→→→ Ex 11A 13−15

y

x 0 −4 −3 −2 −1 1 2 3

6

5

4

3

2

1

−1

A

B

y

x 0 −8 −6 −4 −2 2 4 6

10

8

6

4

2

−2−4

D

C

Sketch the line segment

AB if you need help.

y

x O

Coordinates

= (−3 , ____)

B(8 , −2) A(1 , −2)

C(5 , 2)

y

x O

43


8. In the figure, A(0 , −2), B(6 , 4) and C(−7 , −3) are the three

vertices of △ABC. Is △ABC an isosceles triangle? Explain

your answer.

Level Up Questions

9. O(0 , 0), P(12 , 0) and Q(7 , 13) are the three vertices of △OPQ. Find the perimeter of

△OPQ, correct to 3 significant figures.

10. Find the area of square ABCD shown in the figure.

B(6 , 4)

A(0 , −2)

C(−7 , −3)

y

x O

y

x O

4

A

D

C

B(4 , 9)

44


11 Coordinate Geometry of Straight Lines

Level 1

In each of the following, find the distance between the two given points. [Nos. 1−−−−6] (Give the answers correct to 2 decimal places if necessary.)

1. A(−4 , 2), B(0 , 5) 2. C(1 , −1), D(6 , 11)

3. E(−7 , 4), F(2 , −8) 4. P(6 , −5), Q(−9 , 3)

5. R(−10 , 5), S(2 , 3) 6. T(−9 , −17), U(−1 , 0)

In each of the following, find the length of the line segment XY. [Nos. 7−−−−9] (Give the answers correct to 3 significant figures if necessary.)

7. 8. 9.

10. In the figure, A(−3 , 5), B(0 , −2) and C(7 , 3) are the three

vertices of △ABC. Find the perimeter of △ABC, correct to

2 decimal places.

11. In the figure, O(0 , 0), A(4 , −3), B(9 , 1) and C(5 , 7) are the four vertices of quadrilateral OABC. Find the perimeter of OABC, correct to 3 significant figures.


11A

y

x 0 1 2 3 4 5 6

6

5

4

3

2

1 X

Y

y

x 0 −3 −2 −1 1 2 3

4

3

2

1

−1

−2

X

Y

y

x 0 −4 −3 −2 −1 1 2 3 4

3

2

1

−1

−2

−3

−4

−5

X

Y

y

x O

A(−3 , 5)

B(0 , −2)

C(7 , 3)

y

x O

C(5 , 7)

A(4 , −3)

B(9 , 1)

45

12. In the figure, X(2 , 5), Y(−1 , −2) and Z(5 , −2) are the three vertices

of △XYZ. Is △XYZ an isosceles triangle? Explain your answer.

13. In the figure, A(−4 , 0), B(−2 , 4), C(5 , 9) and D(0 , 2) are the four vertices of quadrilateral ABCD. Is ABCD a kite? Explain your answer.

14. In the figure, the coordinates of P and R are (−7 , 2) and (5 , −3) respectively. If PQ is parallel to the x-axis and PQ = PR, find the coordinates of Q.

Level 2

15. A(5 , 3), B(−1 , 2) and C(2 , −3) are the three vertices of △ABC. Arrange the lengths of the

three sides of △ABC in ascending order.

16. P(4 , 8), Q(−7 , 5) and R(9 , −1) are the three vertices of △PQR. Which vertex, P, Q or R, is

closest to the origin O? Explain your answer.

17. A(−6 , 5), B(4 , 4), C(5 , −3) and D(−8 , −2) are the four vertices of quadrilateral ABCD. Which diagonal, AC or BD, is longer? Explain your answer.

18. P(a , 0), Q(2 , 4) and O(0 , 0) are the three vertices of △OPQ. If PQ = PO, find the value of a.

19. In the figure, A(1 , 1), B(10 , 7), C(6 , 0) and D(3 , −2) are the four vertices of a trapezium, where

AB // DC and AD ⊥ AB. Find the area of trapezium ABCD.

y

x O

X(2 , 5)

Y(−1 , −2) Z(5 , −2)

y

x O

P(−7 , 2) Q

R(5 , −3)

y

x O A(−4 , 0)

B(−2 , 4)

C(5 , 9)

D(0 , 2)

O A(1 , 1)

B(10 , 7)

C(6 , 0)

x

y

D(3 , −2)

46

20. In the figure, A(−3 , −2), B(1 , 2) and C(4 , −1) are the three

vertices of △ABC.

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

(b) Find the area of △ABC.

21. In the figure, the coordinates of P and R are (−1 , 0) and (8 , 6) respectively. Q is a point on the x-axis and PQ = QR.

(a) Find the coordinates of Q.

(b) Find the area of △PQR.

22. In the figure, the coordinates of Q and R are (8 , 9) and (5 , −4) respectively. P is a point on the y-axis and PQ = PR.

(a) Find the coordinates of P.

(b) Prove that △PQR is a right-angled triangle.

23. In the figure, ABCD is a square, where A is on the y-axis. The

coordinates of B and D are (7 , 11) and (−24 , −6) respectively. (a) Find the coordinates of A. (b) Find the perimeter and area of ABCD.

y

x O

A(−3 , −2)

B(1 , 2)

C(4 , −1)

y

x O

P(−1 , 0)

Q

R(8 , 6)

R(5 , −4)

P

Q(8 , 9)

y

x O

y

x O

A

B(7 , 11) C

D(−24 , −6)

47

Answer


1. 5 units

2. 13 units

3. 15 units

4. 17 units

5. 12.17 units

6. 18.79 units

7. 5 units

8. 5.39 units

9. 8.60 units

10. 26.42 units

11. 27.2 units

12. yes

13. yes

14. (6 , 2)

15. BC < AB < CA

16. Q

17. AC

18. 5

19. 26 sq. units

20. (b) 12 sq. units

21. (a)

0,

2

11 (b)

2

39 sq. units

22. (a) (0 , 4)

23. (a) (0 , −13)

(b) perimeter = 100 units,

area = 625 sq. units

48

F3B: Chapter 11B

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)



(Full Solution)




___________ ( )




___________

49

○ Skipped ( )




___________ ( )



___________ ( )



_________

50


11.2A Slope Formula

For any two points A(x1 , y1) and B(x2 , y2) (where x1 ≠ x2) in a rectangular coordinate plane, the slope m of the line AB is given by:

m =12

12

xx

yy

−

−

Note: The slope of a horizontal line is 0, while the slope of a vertical line is undefined.


Find the slope of the line AB in the figure.

Find the slope of the line PQ in the figure.

Sol Slope of AB =)4(2

41

−−

−

=6

3−

=2

1−

Sol Slope of PQ =)()(

)()(

−

−

=

1. Find the slope of the line RS in the figure.

2. Find the slope of the line CD in the figure.

y

x O

A(−4 , 4)

B(2 , 1)

O

B(x2 , y2)

A(x1 , y1)

y

x

y

x O

R(1 , 7)

S(6 , 2)

y

x O

D(−1 , 7)

C(−5 , 1)

y

x O

P(6 , 4)

Q(−2 , −3)

If a line slopes upwards from left to right, slope of the line > 0.

O

y

x

Slope = 0 Slope is

undefined.

51

3. Find the slope of the line PQ in the figure.

4. Find the slope of the line MN in the figure.

○○○○→→→→ Ex 11B 2, 3

5. Given that the slope of the line passing

through P(−8 , h) and Q(h − 8 , 6) is 5, find the value of h.

6. Given that the slope of the line passing

through A(1 , k) and B(3k + 1 , −2) is

3

2− ,

find the value of k. Slope of PQ = ( )

)()(

)()(

−

− = ( )

=

○○○○→→→→ Ex 11B 9, 10


Prove that the three points A(3 , 1), B(5 , 3) and C(9 , 7) lie on the same straight line.

Prove that the three points P(0 , 2), Q(2 , −2)

and R(5 , −8) lie on the same straight line.

Sol Slope of AB

=35

13

−

−

=2

2

= 1 Slope of BC

=59

37

−

−

=4

4

= 1

∵ Slope of AB = slope of BC

∴ The three points A, B and C lie on the

same straight line.

Sol Slope of PQ

=)()(

)()(

−

−

=

Slope of QR

=)()(

)()(

−

−

=

If a line slopes downwards from left to right, slope of the line < 0.

y

x O

A(3 , 1)

B(5 , 3)

C(9 , 7) same slope?

Sketch: y

x O

P(0 , 2)

Q(2 , −2)

R(5 , −8)

same slope?

Sketch:

We can also say that A, B and C are

y

x O M(2 , −1)

N(5 , −8)

y

x O

P(6 , 5)

Q(1 , −3)

52

7. Prove that the three points A(−8 , −4), B(0 , 0) and C(2 , 1) lie on the same straight line.

8. Prove that the three points R(2 , −1),

S(−6 , 1) and T(6 , −2) are collinear.

○○○○→→→→ Ex 11B 13, 14

11.2B Inclination

For a straight line ℓ with inclination θ :

slope of ℓ= tan θ

Note: If θ = 0°, the line is a horizontal line.

9. Complete the following tables. (Give the answers correct to 3 significant figures.)

(a) Inclination θθθθ Slope of ℓ (b) Slope of ℓ Inclination θθθθ

(i) 30° (i) 1.5

(ii) 40° (ii) 8

(iii) 75° (iii) 15.5

○○○○→→→→ Ex 11B 4, 5

O

ℓ

y

x θ (inclination)

Slope of ℓ = tan

θ

tan θ= 1.5

θ

For 0° < θ < 90°, the slope of ℓ (increases /

decreases) with θ.

� θ is the angle measured anticlockwise from the

positive x-axis to ℓ .

53


L is a straight line passing through two points

A(−4 , −3) and B(1 , 7). Find (a) the slope of L, (b) the inclination of L, correct to the nearest

degree.

L is a straight line passing through two points P(1 , 2) and Q(6 , 4). Find (a) the slope of L, (b) the inclination of L, correct to the nearest

degree.

Sol (a) Slope of L =)4(1

)3(7

−−

−−

=5

10

= 2

Sol (a) Slope of L =)()(

)()(

−

−

=

(b) Let θ be the inclination of L.

Slope of L = tan θ

2 = tan θ

θ = 63°, cor. to the nearest

degree

∴ The inclination of L is 63°.

(b) Let θ be the inclination of L.

10. L is a straight line passing through two

points R(2 , −4) and S(12 , 1). Find (a) the slope of L, (b) the inclination of L, correct to the

nearest 0.1°.

11. C(−8 , −10) and D(−4 , −3) lie on a straight line L. Find

(a) the slope of L, (b) the inclination of L, correct to the

nearest 0.1°.

○○○○→→→→ Ex 11B 6−8

54


12. In each of the following, which line has a greater slope? Explain your answer.

(a) line 1ℓ : passing through two points (2 , −7) and (7 , −1)

line 2ℓ : passing through two points (4 , 2) and (8 , 6)

(b) line 3ℓ : passing through two points (2 , −2) and (10 , −3)

line 4ℓ : passing through two points (−3 , −5) and (7 , −10)

Level Up Question

13. In the figure, A is a point on the y-axis. A straight line L

passes through A and B(−7 , −3). (a) Find the inclination of L. (b) Find the slope of L. (c) Find the coordinates of A.

y

x O

135°

A

B(−7 , −3)

L

We can compare the steepness of lines by considering their slopes:

(i) For lines with positive slopes, slope , steepness

(ii) For lines with negative slopes, slope , steepness

55

y

x O A(−4 , −3)

B(2 , 7)

y

x O

138°

L



Level 1

1. Name all the line segment(s) in the figure satisfying each of the following conditions.

(a) The slope is positive. (b) The slope is negative. (c) The slope is 0. (d) The slope is undefined. 2. Find the slope of the line AB in each of the following figures. (a) (b)

3. In each of the following, find the slope of the straight line passing through the two given points. (a) A(0 , 6), B(2 , 0)

(b) P(5 , −2), Q(9 , −1)

(c) X(−3 , 4), Y(1 , −4) 4. In each of the following, find the inclination of the line with the given slope, correct to the

nearest 0.1°.

(a) Slope = 5 (b) Slope = 0.5 (c) Slope =

4

9

5. In each of the following, find the slope of the line L, correct to 2 decimal places. (a) (b)


11B

y

x O

A

B

C

D E

y

x O

A(−6 , 4)

B(4 , 2)

y

x O

65°

L

56

6. L is a straight line passing through the points P(1 , −8) and Q(4 , 3). (a) Find the slope of L.

(b) Find the inclination of L, correct to the nearest 0.1°. 7. Find the inclination of the line XY in each of the following figures.

(Give the answers correct to the nearest 0.1°.) (a) (b)

C(−4 , 3 − 4n) 8. Given that the slope of the line passing through

and D(n + 3 , −5) is −2, find the value of n.

9. If P(−k , −5) and Q(3k , 3) lie on a straight line with inclination 45°, find the value of k. In each of the following, determine whether the three given points lie on the same straight line.

[Nos. 10−−−−11]

10. A(−6 , −5), B(0 , −1), C(3 , 1)

11. P(2 , 11), Q(4 , 7), R(10 , −3) 12. Refer to the figure. (a) Find the inclination of L2. (b) Find the slope of L2, correct to 2 decimal places.

13.Consider three points A(6 , −1), B(8 , 3) and C(−5 , 2). Among AB, BC and CA, which one has the greatest slope? Explain your answer.

Level 2

14. In each of the following, find the slope of the straight line passing through the two given points.

(a) A(−1.3 , 2.8), B(0.75 , 6.9)

(b) C

−

2

1,

7

11, D

2

5,

7

4

(c) E(4a , 3a), F(0 , −a), where a ≠ 0 15. In each of the following, find the inclination of the straight line passing through the two given

points.

(Give the answers correct to the nearest 0.1° if necessary.)

(a) P(−2.6 , −1.3), Q(4.4 , 3.7)

(b) R(−5 , 3 ), S(−4 , 32 )

(c) T(−c , −c), U(0 , c), where c ≠ 0

y

x O

Y(4 , 10)

X(−3 , −5)

y

x O

X(−4 , 1)

Y(5 , 9)

y

x O

140°

115°

L1

L2

57

16. In the figure, A is a point on the y-axis. If the slope of the straight line

passing through A and B(6 , −6) is

3

2− , find the coordinates of A.

17. The slope of a straight line L passing through (2 , 9) is 3. If L cuts the x-axis at A and cuts the

y-axis at B, find the coordinates of A and B.

18. In the figure, P(1 , −3), Q and R(9 , 1) are three points lying on the same straight line.

(a) If Q lies on the x-axis, find the coordinates of Q.

(b) Does the straight line pass through (6 , −1)? Explain your answer.

19. The inclination of a straight line L passing through (4 , 3 ) is 30°.

(a) Find the coordinates of the point where L cuts the x-axis.

(b) If P(a , 33 ) is a point lying on L, find the value of a.

20. Consider three points A(1 , k), B(5 , 7) and C(−2 , −5). It is given that the slope of AB is

2

3.

(a) Find the value of k. (b) Which line segment, AB, AC or BC, is the steepest? Explain your answer.

21. In the figure, L1 is a straight line passing through the points

A(−2 , −5) and B(9 , 6). The angle between the lines L1 and L2

is 20°. (a) Find the inclination of L1. (b) Find the slope of L2, correct to 2 decimal places. 22. In the figure, L1 and L2 are straight lines pass through A(4 , 1)

and B(−2 , 0) respectively. L1 and L2 intersect at C(8 , p). It is given that the slope of L1 is twice the slope of L2.

(a) Find the value of p. (b) Is the inclination of L1 twice that of L2? Explain your answer.

A B(6 , −6)

y

x O

y

x O

P(1 , −3)

Q

R(9 , 1)

y

x O

A(−2 , −5)

B(9 , 6)

L1 L2

20°

y

x O

L1

L2

A(4 , 1)

C(8 , p)

B(−2 , 0)

58

Answer


1. (a) CB, DA, EA (b) BA, CA

(c) DE (d) CD

2. (a) 3

5 (b)

5

1−

3. (a) −3 (b) 4

1 (c) −2

4. (a) 78.7° (b) 26.6° (c) 66.0°

5. (a) 2.14 (b) 0.90

6. (a) 3

11 (b) 74.7°

7. (a) 65.0° (b) 41.6°

8. −1

9. 2

10. yes

11. no

12. (a) 25° (b) 0.47

13. AB

14. (a) 2 (b) −3 (c) 1

15. (a) 35.5° (b) 60° (c) 63.4°

16. (0 , −2)

17. A(−1 , 0), B(0 , 3)

18. (a) (7 , 0) (b) no

19. (a) (1 , 0) (b) 10

20. (a) 1 (b) AC

21. (a) 45° (b) 2.14

22. (a) 5 (b) no

59

F3B: Chapter 11C

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)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )




___________

60

○ Skipped ( )



_________

61


11.3A Parallel Lines

In the figure, if m1 = m2, then L1 // L2.

1. In each of the following, determine whether the straight lines L1 and L2 are parallel. Slope of L1 Slope of L2 (a) 3 3 (yes / no)

(b) 2 −2 (yes / no)

(c) 5 5

1 (yes / no)

(d) −4 −4 (yes / no) 2. Refer to the figure.

(a) Find the slope of L1. (b) Determine whether L1 and L2 are

parallel. (a) Slope of L1 =

(b) ∵ Slope of L1 (= / ≠) slope of L2

∴ L1 and L2 (are / are not) parallel.

3. Refer to the figure.

(a) Find the slope of L1. (b) Determine whether L1 and L2 are

parallel.

○○○○→→→→ Ex 11C 1, 3

y

x O

L1

L2: slope = 1 (1 , 3)

(−3 , −1)

y

O x

L2: slope = −2

L1

(2 , −1)

(−4 , 3)

62

In the figure, if L1 // L2, then m1 = m2.


In the figure, L1 is parallel to L2.

(a) Find the slope of L2. (b) Find the slope of L1.

In the figure, L1 is parallel to L2.


Sol (a) Slope of L2 =)1(4

)4(6

−−

−−

=5

10

= 2

Sol (a) Slope of L2 =

(b) ∵ L1 // L2

∴ Slope of L1 = slope of L2

= 2

(b) ∵ L1 // L2

∴ Slope of L1 =

4. In the figure, L1 is parallel to L2.


5. In the figure, L1 is parallel to L2.

Find the slope of L1.

○○○○→→→→ Ex 11C 5, 7

y

x

(4 , 6)

(−1 , −4)

L2 L1

O

y

x O

L1 L2

(−5 , 5)

(1 , −4)

y

x O

L1 L2

(5 , −2)

(−2 , 7)

y

x O

L1 L2

(6 , 9)

Coordinates of the origin O

= (___ , ___)

63

6. In the figure, L1 is parallel to L2. (a) Find the slope of L2. (b) Find the value of a. ○○○○→→→→ Ex 11C 9, 12

11.3B Perpendicular Lines

In the figure, if m1 × m2 = −1,

then L1 ⊥ L2. 7. In each of the following, determine whether the straight lines L1 and L2 are perpendicular. Slope of L1 Slope of L2

(a) 4 −4 (yes / no)

(b) 1 −1 (yes / no)

(c) 2

1 2 (yes / no)

(d) 3 3

1− (yes / no)

8. Refer to the figure. (a) Find the slope of L1. (b) Determine whether L1 and L2 are perpendicular.

y

x O

(4 , −1)

(−2 , 2)

L2: slope = 2

L1

y

x O

L1 L2

A(0 , a)

B(5 , 6) C(−2 , 5)

D(−7 , −5)

64

In the figure, if L1 ⊥ L2,

then m1 × m2 = −1.


In the figure, L1 is perpendicular to L2.


In the figure, L1 is perpendicular to L2.


Sol (a) Slope of L1 =03

09

−

−

=3

9

= 3

Sol (a) Slope of L1 =

(b) ∵ L1 ⊥ L2

∴ Slope of L1 × slope of L2 = −1

3 × slope of L2 = −1

Slope of L2 =3

1−

(b) ∵ L1 ⊥ L2

∴ Slope of L1 × slope of L2 = ( )

=

9. In the figure, L1 is perpendicular to L2.


10. In the figure, L1 is perpendicular to L2. Find the slope of L2.

○○○○→→→→ Ex 11C 6, 8

y

x O

(3 , 9)

L2

L1 y

x O

L1

L2

(4 , 7)

(2 , −1)

y

x O

L1 L2

(4 , 7)

(−6 , −3)

y

x O L1

L2

(−8 , −5)

(2 , 1)

65

11. In the figure, L1 passes through C(1 , −3) and D(6 , 7). L2 cuts the y-axis at A and passes through B(6 , 2). It is given that L1 and L2 are perpendicular.

(a) Find the slope of L1. (b) Find the coordinates of A.

○○○○→→→→ Ex 11C 10, 13


12. L1 is a straight line passing through A(2 , −3) while L2 is another straight line with slope

4

9. If

L1 // L2, does the point B(−2 , 3) lie on L1? Explain your answer.

∵ L1 // L2

∴ Slope of L1 =

Slope of AB =

∵ Slope of AB (= / ≠) slope of L1

∴ The point B (lies / does not lie) on L1.

y

x O

C(1 , −3)

D(6 , 7)

B(6 , 2)

A

L1

L2

Step 1111: Find the slope of L1.

Step 2222: Check if slope of AB = slope of L .

66

Level Up Questions

13. O(0 , 0), B(2 , 3), C(6 , 1) and D(4 , −2) are the vertices of a quadrilateral. It is given that BC // OD. Determine whether OBCD is a parallelogram.

14. X(−9 , −2), Y(0 , −8) and Z(8 , 4) are the vertices of a triangle.

(a) Find the slopes of the three sides of △XYZ.

(b) Hence, prove that △XYZ is a right-angled triangle and state which angle is a right angle.

Sketch OBCD

first.

67

y

x O

(2 , 1)

(−3 , −4)

L1

L2 : slope = 1 y

x O

(3 , −5)

(−5 , 4)

(7 , 1)

(−6 , −2)

L1

L2

y

x 0 5

7

L1

L2 : Slope =

2

3

y

x 0 8

(5 , −6)

(−3 , −2)

L2

L1

y

x O

(7 , 2)

(−2 , −4)

L y

x O

(2 , 9)

(5 , −3)

L



Level 1

1. In each of the following, determine whether L1 and L2 are parallel. (a) (b)

2. In each of the following, determine whether L1 and L2 are perpendicular. (a) (b)

3. Consider four points A(−4 , 3), B(7 , 6), C(−6 , −9) and D(5 , −6). (a) Find the slopes of AB and CD. (b) What is the relationship between AB and CD?

4. Consider four points P(−4 , 0), Q(1 , 6), R(−8 , 7) and S(4 , −3). (a) Find the slopes of PQ and RS. (b) What is the relationship between PQ and RS?

In each of the following, find the slope of the line L. [Nos. 5−−−−6] 5. (a) (b)


11C

68

y

x O

(1 , 1)

(7 , 4)

L

y

x 0 −7

(1 , −10)

L

y

x O

(a , −3)

L1

L2 : slope = −3

1

y

x 0

−4 (c , −6)

L2 : slope = −2

L1

y

x O

(−3 , d)

(6 , −7)

L1

L2 : slope =

4

3

y

x O L1

L2 : slope =

5

2

(4 , 1)

(−6 , b)

6. (a) (b)

Find the unknown in each of the following figures. [Nos. 7−−−−8] 7. (a) (b)

8. (a) (b)

9. Consider four points P(−3 , −2), Q(n , 1), R(3 , 5) and S(1 , −1). Find the value of n in each of the following cases.

(a) PQ // RS

(b) PQ ⊥ RS

10. In the figure, L1 passes through A(−5 , −7) and B(3 , −4) while L2

passes through C(−2 , −3). It is given that L1 // L2. (a) Find the slope of L1. (b) Suppose L2 cuts the x-axis at P. Find the coordinates of P.

y

x O

C(−2 , −3)

A(−5 , −7)

B(3 , −4)

L1

L2

69

11. In the figure, L is perpendicular to the line passing through P(−1 , 6)

and Q(−4 , −1). R(−7 , −2) is a point lying on L. (a) Find the slope of PQ. (b) Find the coordinates of the point where L cuts the y-axis.

12. A(−5 , 2), B(7 , 6), C(8 , 4) and D(2 , 2) are the vertices of a quadrilateral.

(a) Find the slopes of AB, BC, CD and AD. (b) Name all the parallel sides of quadrilateral ABCD.

13. X(2 , 11), Y(5 , −1) and Z(9 , 0) are the vertices of a triangle.

(a) Find the slopes of the three sides of △XYZ.

(b) Hence, prove that △XYZ is a right-angled triangle and state which angle is a right angle.

Level 2

14. In the figure, two perpendicular lines L1 and L2 intersect at

A(−3 , −6). It is given that L1 cuts the y-axis at B(0 , −7), while L2 cuts the y-axis at C. (a) Find the coordinates of C.

(b) Find the area of △ABC.

15. The figure shows two parallel lines L1 and L2. L1 passes through

P(−1 , 7) and Q(1 , 2) while L2 passes through R(6 , 1). Does L2 passes through the point (4 , 5)? Explain your answer.

y

x O

L

P(−1 , 6)

Q(−4 , −1)

R(−7 , −2)

y

x O

L2

L1

A(−3 , −6) B(0 , −7)

C

y

x O

R(6 , 1)

P(−1 , 7)

Q(1 , 2)

L1 L2

70

16. In the figure, L1 passes through C(−1 , 8) and D while L2 passes

through A(−7 , 6) and B(1 , 2). D lies on the x-axis and L1 ⊥ L2. (a) Find the coordinates of D.

(b) If K(−2 , a) is a point on L1, find the value of a.

17. In the figure, P(3 , −2), Q(13 , 6) and R

7,

2

3 are the three

vertices of △PQR. PQ cuts the x-axis at T.

(a) Find the coordinates of T.

(b) Is RT the corresponding altitude of △PQR if PQ is taken as its

base? Explain your answer.

18. The figure shows three points A(4 , −4), B(10 , 4) and C(8 , −7).

Suppose D is a point on the y-axis such that AD ⊥ AB. (a) Find the coordinates of D.

(b) Prove that the three points D, A and C lie on the same straight line.

19. A(−5 , k), B(k , 1), C(6 , 5) and D(−3 , 8) are the vertices of a quadrilateral. It is given that AB // DC. (a) Find the value of k. (b) Is ABCD a parallelogram? Explain your answer.

20. In the figure, P(−3 , −1), Q(a , b), R(8 , 6) and S(3 , 9) are the four vertices of parallelogram PQRS.

(a) Find the values of a and b. (b) Is PQRS a rectangle? Explain your answer.

y

x O

L1

L2

A(−7 , 6) B(1 , 2)

C(−1 , 8)

D

y

x O

P(3 , −2)

Q(13 , 6)

T

R

7,

2

3

y

x O

A(4 , −4)

B(10 , 4)

C(8 , −7)

y

x O

P(−3 , −1)

Q(a , b)

R(8 , 6)

S(3 , 9)

71

21. In the figure, ABCD is a right-angled trapezium. The coordinates of

A, C and D are (2 , 0), (3 , 5) and

− 0,

2

9 respectively. It is given

that DC // AB and ∠DCB = 90°. Find the coordinates of B.

22. In the figure, OAB is a triangle. The coordinates of B are (−1 , 3 )

and the inclination of OA is 30°.

(a) Prove that △OAB is a right-angled triangle and state which

angle is a right angle. (b) If AB is parallel to the x-axis, find the coordinates of A.

(Leave the radical sign ‘√’ in the answer.)

y

x O A(2 , 0)

B

C(3 , 5)

D

− 0,

2

9

y

x O

A

30°

B(−1 , 3 )

72

Answer


1. (a) yes (b) no

2. (a) no (b) yes

3. (a) AB:

11

3, CD:

11

3 (b) AB // CD

4. (a) PQ:

5

6, RS:

6

5− (b) PQ ⊥ RS

5. (a) 3

2 (b) −4

6. (a) −2 (b) 5

4

7. (a) 9 (b) −3

8. (a) −4 (b) 5

9. (a) −2 (b) −12

10. (a) 8

3 (b) (6 , 0)

11. (a) 3

7 (b) (0 , −5)

12. (a) AB:

3

1, BC: −2, CD:

3

1, AD: 0

(b) AB // CD

13. (a) XY: −4, YZ:

4

1, XZ:

7

11−

(b) ∠Y

14. (a) (0 , 3) (b) 15 sq. units

15. no

16. (a) (−5 , 0) (b) 6

17. (a)

0,

2

11 (b) no

18. (a) (0 , −1)

19. (a) 4 (b) yes

20. (a) a = 2, b = −4 (b) yes

21. (a) (5 , 2)

22. (a) ∠AOB (b) (3 , 3 )

73

F3B: Chapter 11D

Date Task Progress

Lesson Worksheet


(Full Solution)

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 11D Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 11D Multiple Choice


___________ ( )



_________

74

Book 3B Lesson Worksheet 11D (Refer to §11.4)

11.4A Mid-point Formula

If M(x , y) is the mid-point of the line segment joining the points A(x1 , y1) and B(x2 , y2), then

x =2

21 xx + and y =

221 yy +

.


In the figure, M is the mid-point of the line segment AB. Find the coordinates of M.

In the figure, M is the mid-point of the line segment CD. Find the coordinates of M.

Sol Let (x , y) be the coordinates of M.

x =2

50 += 2.5

y =2

04 += 2

∴ The coordinates of M are (2.5 , 2).

Sol Let (x , y) be the coordinates of M.

x =)(

)()( +=

y =)(

)()( +=

∴ The coordinates of M are ( , ).

1. In the figure, M is the mid-point of the line

segment EF. Find the coordinates of M.

2. In each of the following, find the coordinates of the mid-point of the line segment AB.

(a) A(3 , 8), B(−3 , 6)

(b) A(−7 , 1), B(−5 , −3)

○○○○→→→→ Ex 11D 1−5

B(5 , 0)

A(0 , 4)

y

x O

M D(0 , 8)

C(−10 , 0)

M

y

x O

y

F(4 , 3)

M

E(−2 , −5)

x O

75

3. In the figure, P is the mid-point of AB. Find the values of a and b.

4. In the figure, P is the mid-point of AB. Find the values of a and b.

Consider the x-coordinate of P.

4 =2

)()( +

= Consider the y-coordinate of P.

( ) =2

)()( +

=

○○○○→→→→ Ex 11D 6, 7

11.4B Section Formula

If P(x , y) is a point on the line segment joining the points A(x1 , y1) and B(x2 , y2) such that AP : PB = r : s, then

x =sr

rxsx

+

+ 21 and y =sr

rysy

+

+ 21 .


In the figure, P is a point on AB. Find the coordinates of P.

In the figure, P is a point on AB. Find the coordinates of P.

Sol Let (x , y) be the coordinates of P.

x =21

)4(1)1(2

+

+=

3

42 += 2

y =21

)0(1)9(2

+

+=

3

018+= 6

∴ The coordinates of P are (2 , 6).

Sol Let (x , y) be the coordinates of P.

x =)()(

))(())((

+

+=

y =)()(

))(())((

+

+=

∴ The coordinates of P are ( , ).

P(4 , 5)

A(a , 1)

B(7 , b) y

x O

P(2 , 1)

B(−6 , b)

A(a , −5)

y

x O

� We say that P divides AB internally in the ratio r : s.

1 : 2

A(1 , 9)

B(4 , 0)

P

y

x O

3 : 1

B(6 , 5)

A(−6 , 1)

P

y

x O

76

5. In the figure, P is a point on the line segment AB such that AP : PB = 1 : 4. Find the coordinates of P.

6. If a point P divides the line segment

joining A(9 , 8) and B(−6 , −2) internally in the ratio 2 : 3, find the coordinates of P.

○○○○→→→→ Ex 11D 8−11

7. In the figure, P is a point on the line

segment AB such that AP : PB = 1 : 2. Find the value of b.

8. In the figure, P is a point on the line segment AB such that AP : PB = 3 : 1. Find the coordinates of B.

Consider the x-coordinate of P.

4 =)()(

)(1)(2

+

+

=

○○○○→→→→ Ex 11D 12, 13

y

x O

A(1 , 6)

P(4 , 4)

B(b , 0)

y

x O

A(6 , 5)

B

P(3 , −1)

Based on the given information, we can

sketch this: y

x O

We may do the checking by substituting the answer into the formula:

x-coordinate of P =)()(

)(1)(2

+

+

=

P

A(−8 , 0)

B(2 , −5)

y

x O

77

9. In the figure, P and Q are the points on AB such that they divide AB into three equal parts.

(a) Find AP : PB. (b) Find the coordinates of P. (c) Find the coordinates of Q. ○○○○→→→→ Ex 11D 14, 15


10. The figure shows two line segments APB and PQ. P is the mid-point of AB. (a) Find the coordinates of P. (b) Is PQ perpendicular to AB? Explain your answer.

(a) x-coordinate of P =

y-coordinate of P =

(b) Slope of AB =

Slope of PQ =

∴ PQ (is / is not) perpendicular to AB.

y

x O

B(3 , 5)

A(−3 , −4)

Q

P

B(3 , 5) A(−3 , −4) Q P

1 : 1 : 1

1 :1 + ( )

Recall:

If AB ⊥ PQ, then

slope of AB × slope of PQ = ( ).

y

x O

P

Q(6 , 6) A(−3 , 5)

B(7 , −3)

78

Level Up Questions

11. In the figure, a line segment runs from B(8 , 9) to cut the y-axis at M, and to cut the x-axis at A. If M is the mid-point of AB, find the coordinates of A and M.

12. P(3 , a) is a point on the line segment joining A(−1 , a + 1) and B(b , 2a). If P divides AB in the ratio 1 : 3, find the values of a and b.

y

x O

M

A

B(8 , 9)

79

y

x O

A(1 , 3)

B(5 , 1)

M

y

x O M

A(−7 , −8)

B(−3 , 6)

y

x O

A(−2 , −7) P(3 , −5)

B(r , s)

y

x O

A(−11 , r)

P(−2 , −3)

B(s , −10)



Level 1

1. In each of the following figures, M is the mid-point of the line segment AB. Find the coordinates of M.

(a) (b)

In each of the following, find the coordinates of the mid-point of the line segment XY. [Nos. 2−−−−4]

2. X(3 , −6), Y(11 , 0)

3. X(−7 , 2), Y(4 , −10)

4. X(−1.5 , 1.1), Y(−7.5 , 4.7) 5. In each of the following figures, P is the mid-point of the line segment AB. Find the values of r and s. (a) (b)

6. In each of the following, M is the mid-point of the line segment joining P and Q. Find the

coordinates of Q.

(a) P(−8 , −3), M(−7 , 4) (b) P(−2 , 9), M

6,

2

5


11D

80

y

x O

A(−2 , −1)

P

B(10 , 7)

AP : PB = 1 : 3

y

x O

A(r , s)

P(2 , 3)

B(7 , 5)

AP : PB = 2 : 1

y

x O A(−4 , r)

P(−1 , −3) B(s , −7)

AP : PB = 1 : 4

y

x O

A(−5 , 4)

B(1 , −5)

P

AP : PB = 2 : 1

7. In each of the following figures, P is a point on AB. Find the coordinates of P. (a) (b) In each of the following, find the coordinates of a point P which divides the line segment XY

internally in the given ratio. [Nos. 8−−−−9]

8. X(−4 , 1), Y(8 , −5), XP : PY = 1 : 2

9. X(2 , −1), Y(9 , 6), XP : PY = 4 : 3 10. In each of the following figures, P is a point on AB. Find the values of r and s. (a) (b) 11. In each of the following, P is a point which divides the line segment CD internally in the given

ratio. Find the coordinates of D. (a) C(1 , 12), P(5 , 4), CP : PD = 4 : 5

(b) C(−3 , −5), P(9 , 1), CP : PD = 6 : 1 12. In the figure, X and Y are points on the line segment joining

A(−4 , −7) and B(14 , 5). If AX : XY : YB = 1 : 1 : 1, find the coordinates of X and Y.

y

x O

A(−4 , −7)

B(14 , 5)

X

Y

81

Level 2

13. The line segment joining A(−6 , k) and B(k + 3 , −5) cuts the x-axis at M. If M is the mid-point of AB, find

(a) the value of k, (b) the coordinates of M. 14. In the figure, Q is a point on the y-axis. The line segment joining

P(−16 , −5) and Q cuts the x-axis at M. If M is the mid-point of PQ, find the coordinates of Q and M.

15. P(2c − 1 , 2) is a point on the line segment joining A(9 , c + 4)

and B(3d , 1 − 3c), and P divides AB internally in the ratio 1 : 2. Find the values of c and d. 16. In the figure, B is a point on the x-axis. The line segment joining

A(−9 , 5) and B cuts the y-axis at P. If AP : PB = 3 : 2, find the coordinates of P and B.

17. In the figure, X and Y are points on the line segment joining

A(−5 , −15) and B(9 , 6). If AX : XY : YB = 2 : 1 : 4, find the coordinates of X and Y.

18.In the figure, the line segment joining A(3 , −10) and B(11 , 6) cuts the x-axis at P.

(a) Find AP : PB. (b) Using the result of (a), find the coordinates of P.

19. In the figure, A(6 , 7) is a vertex of △ABC. M(1 , 4) and N(5 , 1)

are the mid-points of AB and AC respectively. (a) Find the coordinates of B and C. (b) Suppose P is the mid-point of BC. Does P lie on the y-axis?

Explain your answer.

y

x O

M

Q

P(−16 , −5)

y

x O

P B

A(−9 , 5)

y

x O

X

Y

A(−5 , −15)

B(9 , 6)

y

x O

A(3 , −10)

B(11 , 6)

P

y

x O

A(6 , 7)

M(1 , 4)

N(5 , 1) B

C

82

Answer

Consolidation Exercise 11D

1. (a) (3 , 2) (b) (−5 , −1)

2. (7 , −3)

3.

−− 4,

2

3

4. (−4.5 , 2.9)

5. (a) r = 8, s = −3 (b) r = 4, s = 7

6. (a) (−6 , 11) (b) (7 , 3)

7. (a) (1 , 1) (b) (−1 , −2)

8. (0 , −1)

9. (6 , 3)

10. (a) r = −8, s = −1 (b) r = −2, s = 11

11. (a) (10 , −6) (b) (11 , 2)

12. X(2 , −3), Y(8 , 1)

13. (a) 5 (b) (1 , 0)

14. Q(0 , 5), M(−8 , 0)

15. c = 3, d = −1

16. P(0 , 2), B(6 , 0)

17. X(−1 , −9), Y(1 , −6)

18. 5 : 3 (b) (8 , 0)

19. (a) B(−4 , 1), C(4 , −5)

(b) yes

83

F3B: Chapter 11E

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 19


(Video Teaching)

Book Example 20


(Video Teaching)

Book Example 21


(Video Teaching)



(Full Solution)

Maths Corner Exercise 11E Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 11E Multiple Choice


___________ ( )



_________

84

Book 3B Lesson Worksheet 11E (Refer to §11.5B)

[In this worksheet, use analytic approach to complete the proofs.]

11.5B Prove Geometric Properties by Analytic Approach

Analytic approach in geometry: prove geometric properties by introducing a rectangular coordinate system


In the figure, OABC is a square. Let (0 , a) be the coordinates of A.

(a) Express the coordinates of B and C in terms of a.

(b) Prove that the two diagonals OB and AC are perpendicular to each other.

In the figure, OABC is a rectangle. Let (p , q) be the coordinates of B.

(a) Express the coordinates of A and C in terms of p and q.

(b) Prove that the two diagonals OB and AC bisect each other.

Sol (a) The coordinates of B are (a , a). The coordinates of C are (a , 0).

(b) Slope of OB =0

0

−

−

a

a= 1

Slope of AC =0

0

−

−

a

a= −1

∵ Slope of OB × slope of AC

= 1 × (−1)

= −1

∴ OB ⊥ AC

Sol (a) The coordinates of A are ( , ). The coordinates of C are ( , ). (b) Coordinates of the mid-point of OB =

Coordinates of the mid-point of AC

=

∵ The two diagonals OB and AC

(have / do not have) the same mid-point.

∴

1. Refer to Example 1. Prove that

∠BOC = 45°.

2. Refer to Instant Drill 1. Prove that the two diagonals are equal in length.

○○○○→→→→ Ex 11E 1−3

Check if m1 × m2 = −1.

Recall: For a straight line with

inclination θ, slope =

tan θ.

Distance =

212

212 )()( yyxx −+−

y

x O

A(0 , a) B

C

y

x O

A B(p , q)

C

85

3. In the figure, O(0 , 0), A(h , k) and B are the vertices of a triangle. M is the mid-point of OB and AM is a vertical line.

Prove that △OAB is an isosceles triangle.

○○○○→→→→ Ex 11E 6

Level Up Question

4. In the figure, B is a point on AO such that AB : BO = 1 : 2. C is a point on AD such that BC is a horizontal line. Prove that AC : CD = 1 : 2.

y

x O

A(a , p)

B C

D(a , 0)

Step 1111: Express the coordinates of B in terms of h.

Step 2222: Check if it has two equal sides by using

y

x O

A(h , k)

B M

86



Level 1

1. In the figure, ABC is a triangle. O is the mid-point of AC. Let (a , 0) and (0 , b) be the coordinates of A and B respectively. (a) Express the coordinates of C in terms of a.

(b) Hence, prove that △ABC is an isosceles triangle.

2. In the figure, O(0 , 0), A(0 , 2a), B(2a , 2a) and C(2a , 0) are

the vertices of a square. D and E are the mid-points of OA and AB respectively.

(a) Express the coordinates of D and E in terms of a.

(b) Hence, prove that OE ⊥ DC. 3. In the figure, O(0 , 0), P(a , 0), Q(a , b) and R(0 , b) are the

vertices of a rectangle. X and Y are the points on OP and RQ respectively such that OX = YQ. Let (c , 0) be the coordinates of X.

(a) Express the coordinates of Y in terms of a, b and c. (b) Hence, prove that RX // YP. 4. In the figure, O(0 , 0), D(0 , b), E(a , b) and F(a , 0) are the

vertices of a rectangle. M and N are the points on DO and EF respectively such that EN = NF and DE // MN // OF.

(a) Express the coordinates of N in terms of a and b. (b) Hence, prove that M is the mid-point of DO.


11E

y

x O A(a , 0)

B(0 , b)

C

y

x O X(c , 0) P(a , 0)

Q(a , b) Y R(0 , b)

y

x O

D(0 , b) E(a , b)

F(a , 0)

M N

y

x O

A(0 , 2a) B(2a , 2a)

C(2a , 0)

D

E

87

B

C

D

A W X

Y Z

5. In the figure, O(0 , 0), P(b , c), Q(a + b , c) and R(a , 0) are the

vertices of a parallelogram. X and Y are the mid-points of PO and QR respectively. Prove that PQ // XY by analytic approach.

Level 2

6. In the figure, O(0 , 0), P(b , c), Q(a + b , c) and R(a , 0) are the vertices of a quadrilateral, where OP = PQ. Prove that OR = RQ by analytic approach.

7. In the figure, O(0 , 0), A(6a , 0) and B(6b , 6c) are the vertices of

△OAB. P is the mid-point of AB. Y is a point on OP such that

OY : YP = 2 : 1. (a) Express the coordinates of Y in terms of a, b and c. (b) If Q is the mid-point of OA, prove by analytic approach that (i) BY : YQ = 2 : 1, (ii) B, Y and Q lie on the same straight line. 8. In the figure, ABCD is a kite, where AB = AD and CB = CD. W, X, Y

and Z are the mid-points of AB, AD, BC and DC respectively. Prove that WX // BD // YZ and BD = WX + YZ by analytic approach.

y

x O

P(b , c) Q(a + b , c)

R(a , 0)

X Y

y

x O

P(b , c) Q(a + b , c)

R(a , 0)

y

x O A(6a , 0)

B(6b , 6c)

P

Q

Y

88

Answer

Consolidation Exercise 11E

1. (a) (−a , 0)

2. (a) D(0 , a), E(a , 2a)

3. (a) (a − c , b)

4. (a)

2,

ba

7. (a) (2a + 2b , 2c)

89

F3B: Chapter 12A

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)




___________ ( )




___________ ( )




___________ ( )



___________ ( )



_________

90

Book 3B Lesson Worksheet 12A (Refer to §12.1B–C)

12.1B Possible Outcomes of an Activity and Outcomes Favourable to an Event 1. List ALL the possible outcomes of each of the following activities. (The first two have been done for you as examples.)

Activity All possible outcomes

(a) Toss a coin. head, tail

(b) Choose a letter from the word ‘MATHS’. ‘M’, ‘A’, ‘T’, ‘H’, ‘S’

(c) Record the gender of a person.

(d) Choose a digit from the number ‘630’.

2. Complete the table below. (The first one has been done for you as an example.)

Activity Event All possible outcomes

Outcome(s)

favourable

to the event

(a) Play the game ‘rock–paper–scissors’.

Paper is thrown. rock, paper, scissors paper

(b) Choose a letter from the word ‘SINE’.

‘I’ is chosen.

(c) Throw a dice to obtain a number.

An odd number is obtained.

12.1C Definition of Probability Suppose all the possible outcomes in an activity are equally likely to occur (i.e. they are equally likely outcomes). Then the probability of an event E, denoted by P(E), is defined as:

P(E) =

outcomes possible ofnumber total

to favourable outcomes ofnumber

E

Note: (a) For any event E, 0 ≤ P(E) ≤ 1. (b) If E is an impossible event, then P(E) = 0. (c) If E is a certain event, then P(E) = 1.

91


A letter is chosen randomly from the word ‘ADD’. Find the probability of choosing a letter ‘D’.

Sol There are 3 letters in the word ‘ADD’. There are 2 ‘D’s in the word.

∴ P(‘D’ is chosen) =

3

2

A letter is chosen randomly from the word ‘TERRY’. Find the probability of choosing each of the following letters. (a) ‘E’ (b) ‘R’

Sol (a)

There are letters in the word ‘TERRY’. There is/are ‘E’(s) in the word.

∴ P(‘E’ is chosen) =

) (

) (

(b)

There is/are ‘R’(s) in the word.

∴ P(‘ ’ is chosen) =

) (

) (

3. A digit is chosen randomly from the date

‘01-10-2019’. What is the probability that it is

(a) a ‘0’? (b) an odd number?

4. A fair dice is thrown. Find the probability of getting

(a) a ‘5’, (b) an even number.

○○○○→→→→ Ex 12A 1–4

‘Chosen randomly’ means that all the possible outcomes are equally likely to be chosen.

A D D

Total number of possible outcomes = 3

Number of favourable outcomes = 2

Total number of possible outcomes =

T E R R Y

Circle the favourable outcome(s), i.e. ‘E’.

Number of favourable outcomes =

T E R R Y

Circle the favourable outcome(s), i.e. ‘R’.

Number of favourable outcomes =

A fair dice means that it has a uniform weight. ∴ All the possible outcomes are equally

likely outcomes.

92


4# is a 2-digit number, where # is an integer from 0 to 9 inclusive. Find the probability that the 2-digit number is a multiple of 5.

Sol Total number of possible outcomes = 10 Only 40 and 45 are multiples of 5.

∴ Number of favourable outcomes = 2

P(a multiple of 5) =

10

2

=

5

1

3� is a 2-digit number, where � is an integer from 1 to 9 inclusive. Find the probability that the 2-digit number is a multiple of 3.

Sol Total number of possible outcomes = Only are multiples of 3.

∴ Number of favourable outcomes =

P( ) =

5. ♦6 is a 2-digit number, where ♦ is an integer from 2 to 7 inclusive. Find the probability that the 2-digit number is greater than 39.

6. In a school, the ages of 8 teachers are as follows:

24, 27, 29, 29, 33, 37, 38, 43 If a teacher is chosen at random from

them, find the probability that the age of the teacher is an odd number.

○○○○→→→→ Ex 12A 5–10

7. In a group of 20 students, 5 of them are boys. If a student is randomly selected from the group, what is the probability that the student selected is not a boy?

○○○○→→→→ Ex 12A 11

3� can be:

31 32

4# can be:

40 41 42 43 44

45 46 47 48 49

Multiples of 5

93

8. A box contains 6 apples, 4 oranges and 5 mangoes. If a piece of fruit is drawn at random from the box, find the probability that the fruit drawn is (a) an apple, (b) not a mango, (c) not a kiwi fruit, (d) a peach. ○○○○→→→→ Ex 12A 12–14

Level Up Question

9. In the figure, there are only white balls and black balls in a bag. A ball is randomly drawn from the bag. Find the probability that the ball drawn is

(a) a black ball or a white ball, (b) a white ball, (c) a green ball, (d) a white ball with a number.

1

2

3

94


12 Introduction to Probability

Level 1

1. A letter is chosen randomly from the word ‘EXPERIENCE’. Find the probability of choosing each of the following letters.

(a) ‘I’ (b) ‘E’ 2. Benny selects a digit from his staff number ‘39769’ at random. Find the probability that the

digit selected is an odd number. 3. A fair dice is thrown. Find the probability of getting (a) a ‘4’, (b) a number less than 4.

4. ☆9 is a 2-digit number, where ☆ is an integer from 1 to 9 inclusive. Find the probability that

the 2-digit number is (a) greater than 70, (b) a prime number, (c) an even number. 5. The scores of eight students in a test are 49, 25, 74, 36, 58, 43, 65 and 85 respectively. If a

student is chosen at random, find the probability that the score of the student chosen (a) is a square number, (b) has the tens digit 3 greater than the units digit. 6. A card is selected at random from the playing cards shown in the figure. What is the probability of selecting each of the following cards? (a) a ‘5’ (b) a face card (Note: A face card is a ‘J’, a ‘Q’ or a ‘K’.) (c) a diamond 7. There are 10 boys in a group of 16 students. If a student is chosen at random from the group,

find the probability that the student chosen is a girl.

8. An inspector visits a restaurant on a day at random in April. Suppose there are 9 public holidays

in April. Find the probability that the visit is not on a public holiday. 9. An integer is randomly selected from the 90 integers 1 to 90. What is the probability that the

integer selected is (a) a multiple of 9? (b) not a multiple of 9?


12A

95

10. A box contains 15 white chocolates and 25 dark chocolates. If a chocolate is chosen at random from the box, find the probability of each of the following events.

(a) A dark chocolate is chosen. (b) A white chocolate is chosen. 11. There are 5 oranges, 3 apples and 4 pears in a refrigerator. Kenneth takes out a piece of fruit at

random. Find the probability that the fruit taken out is (a) an apple, (b) not a pear, (c) a lemon. 12. Sam has n books, and 24 of them are storybooks. If he selects a book at random, the probability

of selecting a storybook is 0.3. Find the value of n. Level 2

13. A card is drawn randomly from a pack of 52 playing cards. What is the probability of drawing (a) a ‘10’, (b) a black ‘J’, (c) a diamond or a club, (d) any card with number from ‘3’ to ‘7’ inclusive. 14. There are 5 green stone marbles, 4 blue stone marbles, 2 blue glass marbles and 4 yellow glass

marbles in a bag. If a marble is drawn at random from the bag, find the probability of drawing (a) a green stone marble, (b) a blue marble, (c) a yellow marble or a stone marble, (d) a stone marble or a glass marble. 15. A number is randomly selected from the 40 integers 1 to 40. What is the probability that the

number selected is (a) an even square number? (b) a multiple of 7 or a multiple of 8? 16. The figure shows the distribution of the favourite fruit of a group of students.

If a student is selected at random from the group, what is the probability that the favourite fruit

of the student is (a) orange? (b) apple or peach? (c) not apple?

15

10

5

0 Apple Orange Mango Peach

Fruit

Favourite fruit of a group of students

Nu

mb

er o

f st

ud

ents

96

17. The stem-and-leaf diagram below shows the ages of the employees in a company.

Ages of the employees in a company

Stem(10) Leaf (1)

2 3 4 5 6 7 9 3 1 2 3 3 4 7 8 4 1 5 8 9 9 5 6 8

If an employee is randomly selected from the company, find the probability that the employee (a) is younger than 28, (b) is older than 43, (c) is aged between 35 and 57.

18. Janet has 120 books, in which 30 are cookery books, 36 are comic books, 42 are textbooks and

the rest are travel books. She takes a book at random. What is the probability that the book taken is

(a) a comic book? (b) a travel book? (c) neither a comic book nor a textbook? 19. In a group of people, the numbers of males and females who are smokers or non-smokers are

shown in the table below.

Smokers Non-smokers

Number of males 18 72

Number of females 12 108

(a) If a person in the group is chosen randomly, what is the probability that the person is (i) a male non-smoker? (ii) a smoker? (b) Henry claims that if a person is chosen randomly from the group, the probability of

choosing a female is more than that of choosing a male. Do you agree? Explain your answer.

97

Answer


1. (a) 10

1 (b)

5

2

2. 5

4

3. (a) 6

1 (b)

2

1

4. (a) 3

1 (b) 9

5 (c) 0

5. (a) 8

3 (b)

4

1

6. (a) 8

1 (b)

4

1 (c)

8

3

7. 8

3

8. 10

7

9. (a) 9

1 (b)

9

8

10. (a) 8

5 (b)

8

3

11. (a) 4

1 (b)

3

2 (c) 0

12. 80

13. (a) 13

1 (b)

26

1 (c)

2

1 (d)

13

5

14. (a) 3

1 (b)

5

2 (c)

15

13 (d) 1

15. (a) 40

3 (b)

4

1

16. (a) 30

13 (b)

10

3 (c)

5

4

17. (a) 4

1 (b)

10

3 (c)

5

2

18. (a) 10

3 (b)

10

1 (c)

20

7

19. (a) (i) 35

12 (ii)

7

1

(b) yes

98

F3B: Chapter 12B

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)




___________ ( )




___________

99

○ Skipped ( )




___________ ( )



___________ ( )



_________

100


12.2A Methods for Listing Possible Outcomes

I. Tree Diagram


A letter is randomly chosen from each of the two words ‘UP’ and ‘POP’. Find the probability of each of the following events. (a) Two ‘P’s are obtained. (b) Only one ‘P’ is obtained.

Sol The tree diagram below shows all the possible outcomes.

1st letter 2nd letter Outcome P . . . . . . UP

U O . . . . . . UO

P . . . . . . UP

P . . . . . . PP

P O . . . . . . PO

P . . . . . . PP


(a) Number of favourable outcomes = 2

∴ P(two ‘P’s are obtained)

=

6

2

=

3

1

(b) Number of favourable outcomes = 3

∴ P(only one ‘P’ is obtained)

=

6

3

=

2

1

Two fair coins are tossed. Find the probability of each of the following events. (a) Two heads are obtained. (b) Only one head is obtained.

Sol Let H stand for a head and T for a tail. The tree diagram below shows all the

possible outcomes. 1st coin 2nd coin Outcome H . . . . . .

H

. . . . . .

. . . . . .

T

. . . . . .


(a) Number of favourable outcomes =

∴ P(two heads)

=

(b) Number of favourable outcomes =

∴ P(only one head is obtained)

=

Which of them are favourable outcomes?

UP UO UP PP PO PP

Which of them are favourable outcomes?

UP UO UP PP PO PP

The favourable outcome is HH.

The favourable outcomes are .

101

1. Box X contains a red pen and a blue pen. Box Y contains a red pen, a green pen and a blue pen. Roy randomly chooses a pen from each box. Find the probability of each of the following events.

(a) Two blue pens are chosen. (b) The pens chosen are of the same

colour.

Let stand for a red pen, stand for a blue pen and stand for a green pen.

The tree diagram below shows all the possible outcomes.

Box X Box Y Outcome

Total number of possible outcomes = (a) Number of favourable outcomes =

∴ P(two blue pens)

=

(b)

2. There are three true or false questions in a quiz. If Donna answers each question by choosing ‘true’ or ‘false’ at random, find the probability that ‘true’ is chosen

(a) three times, (b) only once, (c) at least two times. ○○○○→→→→ Ex 12B 1–5

The favourable outcomes

are .

102

II. Tabulation


A box contains 3 balls marked with 1, 2 and 3 respectively. Two balls are randomly drawn from the box at the same time. Find the probability of each of the following events. (a) The numbers on the two balls are both odd

numbers. (b) One of the balls drawn is marked with 2.

Sol The table below lists all the possible outcomes.

Number on the 2nd ball

1 2 3

Nu

mb

er o

n

the

1st

ba

ll 1 (1 , 2) (1 , 3)

2 (2 , 1) (2 , 3)

3 (3 , 1) (3 , 2)


(a) Number of favourable outcomes = 2

∴ P(both are odd numbers)

=

6

2

=

3

1

(b) Number of favourable outcomes = 4

∴ P(one of the numbers is 2)

=

6

4

=

3

2

A bag contains two red blocks and two green blocks. Two blocks are randomly drawn from the bag at the same time. Find the probability of each of the following events. (a) Two red blocks are drawn. (b) The two blocks drawn are of different

colours.

Sol Let R1, R2 stand for the two red blocks, and G1, G2 stand for the two green blocks.

The table below lists all the possible outcomes. 2nd block

R1 R2 G1 G2

1st

blo

ck

R1 R1R2 R1G1

R2

G1

G2


(a) Number of favourable outcomes =

∴ P(two red blocks)

=

(b) Number of favourable outcomes =

∴ P(two blocks of different colours)

=

The favourable outcomes are:

(1 , 3) (3 , 1)

The favourable outcomes are:

(1 , 2) (2 , 1) (2 , 3) (3 , 2)

‘’ denotes that the outcome is impossible.

The favourable outcomes are .

Circle the favourable

outcomes in the table.

103

3. There are two boys and two girls in a room. Two of them are chosen at random.

(a) List all the possible outcomes in a table.

(b) Hence, find the probability of each of the following events.

(i) The two children chosen are of different genders.

(ii) No boys are chosen.

(a) Let stand for the two boys, and stand for the two girls.

The table below lists all the possible outcomes.

2nd child

1st

ch

ild

(b)

4. A letter is randomly chosen from each of the two words ‘SUN’ and ‘MOON’.

(a) List all the possible outcomes in a table.

(b) Hence, find the probabilities of the following events.

(i) Two ‘N’s are chosen. (ii) At least one ‘N’ is chosen. (iii) Two ‘O’s are chosen. ○○○○→→→→ Ex 12B 6–10

104

12.2B Geometric Probability

(a) The probability obtained by considering measures of geometric figures, such as lengths, areas or volumes, is called a geometric probability. (b) If an event E happens in a certain region of a geometric figure, then

P(E) =

figure whole theof measure same

happens in which region theof measure E

5. Refer to the line segment ABC on the right. If a point on AC is selected at random, what is the probability that the point lies on BC?

P(the point lies on BC) =

) ( oflength

) ( oflength

=

○○○○→→→→ Ex 12B 11


The figure shows a target formed by two squares. Ada throws a dart at random and it hits the target. Find the probability that the dart hits the smaller square.

Sol P(hitting the smaller square)

=

squarelarger theof area

squaresmaller theof area

= 22

22

cm 8

cm 4

=

4

1

The figure shows a target formed by two circles. Carol shoots an arrow at random and it hits the target. Find the probability that the arrow hits the smaller circle.

Sol P( )

=

6 cm 15 cm

Region in which the

event happens the smaller square

Whole figure the larger square

4 cm

8 cm

Do not write: P(hitting the smaller

square)

=

cm 8

cm 4

the region in which the event happens

the whole figure

2 cm 5 cm

A B C

105

6. The figure shows an equilateral triangular

target, in which all the small triangles are identical. Billy throws a dart randomly and it hits the target. Find the probability that the dart hits the shaded region.

7. The figure shows a circular lucky wheel. A player turns the wheel once at random. Find the probability that the pointer falls in the shaded sector.

○○○○→→→→ Ex 12B 12–14


8. Box X contains 2 gold coins and 1 silver coin. Box Y contains 1 gold coin and 2 silver coins. Teresa draws one coin from each box at random. She claims that the probability of drawing

one gold coin and one silver coin is greater than

2

1. Do you agree? Explain your answer.

80°

Consider the arc length of the

106

Level Up Questions

9. Two fair dice are thrown. Find the probability that the sum of the two numbers is 7. 10. The figure shows a target formed by three concentric circles.

Kevin throws a dart randomly and it hits the target. Find the probability of each of the following events.

(a) The dart hits the grey bullseye. (b) The dart hits the dotted region.

2 cm 4 cm

10 cm

107



Level 1

Use tree diagrams to solve the following problems. [Nos. 1–4] 1. James tosses a fair coin two times. Find the probability that he gets a head and then a tail. 2. David makes two basketball shots. Assume that the probabilities of the shots being successful

and unsuccessful are equal. Find the probability that he makes one unsuccessful shot only. 3. Patrick has a red tie, a blue tie and a green tie. He wears one of the three ties at random every

day. Find the probability that he wears the same tie in two successive days. 4. In a shop, there are apple juice, orange juice and pineapple juice only. Karen and Carman each

buy one kind of juice at random from the shop. What is the probability that (a) both of them buy pineapple juice? (b) only one of them buys orange juice? Use the method of tabulation to solve the following problems. [Nos. 5–8] 5. A letter is randomly selected from each of the two words ‘SUM’ and ‘US’. Find the probability

that the two letters selected are different. 6. There are three candidates P, Q and R in an election. Calvin and Ben each choose one candidate

at random. What is the probability that (a) they both choose candidate Q? (b) they choose different candidates? 7. There are two boxes X and Y. Each box contains one apple and two lemons. If a piece of fruit is

randomly taken out from each box at the same time, find the probability that an apple and a lemon are taken out.

8. There are three boys and four girls in a class. The ages of the boys are 3, 5 and 6 respectively,

and the ages of the girls are 3, 4, 5 and 7 respectively. If one boy and one girl are selected at random to answer a question, find the probability that they

(a) have the same age, (b) have an age difference of 2.


12B

108

X Y 8 cm N 12

129°

132°

Prize A

Prize B

Prize C

Prize D

60 cm

1 m

9. The figure shows a line segment XY of length 12 cm. N is a point on XY such that NY = 8 cm. If a point K is

selected randomly from XY, find the probability that K lies on NY. 10. The figure shows a square target of side 1 m. A circular region of diameter 60 cm is in the

middle of the target. Jacky shoots an arrow

at random and it hits the target. Find the probability, in terms of π, that the arrow hits the circular region.

11. The figure shows the circular wheel in a game. Fanny turns the wheel once at random and wins the prize indicated by the

pointer. Find the probability that she wins (a) prize C, (b) prize D. Level 2

12. George has a brown dog, a grey dog and two white dogs. He chooses two dogs at random. Find the probability that

(a) both dogs chosen are white, (b) one of the dogs chosen is brown. 13. A drawer contains two red socks and two green socks. Ken takes two socks randomly from the

drawer at the same time. Find the probability that (a) the colours of the two socks taken are different, (b) at most one green sock is taken.

14. A fair coin is tossed three times. Find the probability of getting (a) three heads, (b) at most one tail. 15. Two fair dice are thrown. Find the probability of each of the following events. (a) The sum of the two numbers obtained is less than 5. (b) The difference of the two numbers obtained is a multiple of 2.

109

30 cm

20 cm

Region A

Region B

Region C

16. Amy has four cards and Ben has three cards as shown below. Each of them draws one of his/her

own cards at random.

Find the probability of each of the following events. (a) The cards drawn are in the same suit. (b) The sum of the numbers on the cards drawn is at least 10.

17. There are 5 pens in a case including 2 different red pens, 1 blue pen and 2 different green pens. A pen is drawn at random and put back into the case. Then, a pen is drawn at random from the case again. Find the probability that (a) a red pen and a blue pen are drawn, (b) only one of the pens drawn is green, (c) the pens drawn are different. 18. The figure shows a target formed by three concentric circles.

The diameter of smallest circle is 80 cm, and the widths of the rings of regions B and C are 30 cm and 20 cm respectively. Paul throws a dart at random and it hits the target. He can win a prize according to the region where the dart hits.

Region A Region B Region C

Prize A coupon of

$50 A pen

A roll of toilet paper

(a) Find the probability that Paul wins a coupon. (b) Susan claims that the probability that Paul wins a roll of toilet

paper is lower than that of a pen. Do you agree? Explain your answer.

Amy’s cards Ben’s cards

110

Answer


1. 4

1

2. 2

1

3. 3

1

4. (a) 9

1 (b)

9

4

5. 3

2

6. (a) 9

1 (b)

3

2

7. 9

4

8. (a) 6

1 (b)

3

1

9. 3

2

10. 100

π9

11. (a) 30

11 (b)

40

1

12. (a) 6

1 (b)

2

1

13. (a) 3

2 (b)

6

5

14. (a) 8

1 (b)

2

1

15. (a) 6

1 (b)

3

1

16. (a) 4

1 (b)

3

2

17. (a) 25

4 (b)

25

12 (c)

5

4

18. (a) 81

16 (b) yes

111

F3B: Chapter 12C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )



___________ ( )



_________

112


12.3 Experimental Probability (a) A probability found by deductive reasoning is called a theoretical probability. A probability based on relative frequencies found from experiments is called an experimental probability.

(b) Experimental probability of an event E

=

trialsofnumber

happens event that timesofnumber E

(c) When the number of trials is very large,

experimental probability ≈ theoretical probability


A coin is tossed many times and the results are as follows:

Outcome Head Tail

Frequency 15 35

Find the experimental probability of getting (a) a head, (b) a tail.

Sol Total frequency = 15 + 35 = 50 (a) Experimental probability of getting a

head

=

50

15

=

10

3

(b) Experimental probability of getting a tail

=

50

35

=

10

7

A drawing pin is thrown many times and the results are as follows:

Outcome The tip

points up The tip lands on the ground

Frequency 52 28

Find the experimental probability that the tip (a) points up, (b) lands on the ground.

Sol Total frequency = ( ) + ( ) = (a) Experimental probability that the tip

points up

=

(b) Experimental probability that the tip

lands on the ground

=

� In Lesson Worksheets 12A and 12B, all probabilities are

theoretical probabilities.

113

1. A group of students is randomly chosen from a school. The numbers of hats they have are recorded as follows:

Number of hats 1 2 3

Number of students 27 13 5

If a student is randomly chosen from the school, find the experimental probability that the student has

(a) 3 hats, (b) at least 2 hats.

2. A group of customers is randomly chosen from a restaurant. The set meals they ordered are recorded as follows:

Set meal A B C D

Frequency 38 21 29 32

If a customer is randomly chosen from the restaurant, find the experimental probability that the customer ordered

(a) set meal B, (b) set meal A or D.

○○○○→→→→ Ex 12C 1–6

Level Up Question

3. 300 books are selected at random from a library. Their languages are recorded as follows:

Language Chinese English Japanese Others

Frequency 126 x 55 14

(a) Find x. (b) If a book is selected at random from the library, find the experimental probability that it is (i) an English book, (ii) not a Chinese book.

(b) Frequency of ‘A’ = Frequency of ‘D’ = Frequency of ‘A or D’ = ( ) +

( ) =

114



Level 1

1. The test results of 24 students chosen randomly from S3 students in a school are shown below.

Result Fail Pass

Frequency 15 9

If a student is chosen randomly from S3 students in the school, what is the experimental probability that the student

(a) fails the test? (b) passes the test? 2. The genders of a group of people chosen randomly from a city are shown below.

Gender Male Female

Frequency 84 66

If a person is selected randomly from the city, what is the experimental probability that the person is a female?

3. Five brands of rice cookers are sold in a shop. The following table shows the brands of rice

cookers bought by 240 customers chosen randomly from the shop.

Brand A B C D E

Frequency 33 63 27 72 45

If a customer buying a rice cooker in the shop is chosen randomly, find the experimental probability that the rice cooker bought by the customer

(a) is brand C, (b) is not brand E. 4. 500 adults are chosen randomly from a city. The numbers of credit cards owned by them are as

follows.

Number of credit cards 0 1 2 3 4 5

Frequency 72 145 100 83 64 36

If an adult is chosen randomly from the city, find the experimental probability that the adult (a) does not have any credit card, (b) has at most 3 credit cards. 5. A dice is thrown many times and the results are as follows.

Number obtained 1 2 3 4 5 6

Frequency 19 58 37 21 44 71

Find the experimental probability of getting (a) an odd number, (b) a number less than 5.


12C

115

6. 50 students are chosen at random from a school. Their favourite sports are recorded as follows.

Favourite sport Football Basketball Swimming Others

Frequency 12 13 n 18

(a) Find the value of n. (b) If a student is chosen at random from the school, find the experimental probability that the

student’s favourite sport is basketball or swimming. Level 2

7. Three coins are tossed together 400 times. In 6% of the times, no heads are obtained, and in

25

7

of the times, only one head is obtained. (a) Find the number of times that (i) no heads are obtained, (ii) only one head is obtained. (b) If the three coins are tossed together once more, find the experimental probability that at

least two heads are obtained. 8. A sample of students from a school is chosen randomly and asked about their favourite

countries. The results are shown in the following bar chart.

If a student is chosen at random from the school, find the experimental probability that the

student’s favourite country is (a) Japan, (b) USA or others.

9. Yesterday, a scientist caught 76 tortoises from a lake. After making a mark on the shell of each

tortoise, they were put back into the lake. Today, he catches 285 tortoises from the lake and finds that 31 of them have marks on their shells. Estimate the number of tortoises in the lake, correct to the nearest integer.

10. There are 483 gold coins, silver coins and copper coins altogether in a bag. Daniel draws a coin

from the bag at random, records the result and then puts it back into the bag. He repeats the process 150 times and the results are shown in the table below.

Result Gold coin Silver coin Copper coin

Frequency 60 72 18

(a) When a coin is drawn from the bag at random, find the experimental probability that the coin drawn is not a copper coin.

(b) Daniel guesses that there are about 190 gold coins in the bag. Is his guess reasonable? Explain your answer.

30

20

10

0 China Japan USA Others

Countries

Favourite countries of a sample of students

Nu

mb

er o

f st

ud

ents

116

11. Jessie throws a dice 800 times and the results are as follows.

Number obtained 1 2 3 4 5 6

Frequency 185 88 172 94 129 132

(a) Jessie claims that the dice is fair. Is her claim reasonable? Explain your answer. (b) Suppose Nick throws the dice 96 more times. Estimate the number of times that he obtains

an even number, correct to the nearest integer.

117

Answer


1. (a) 8

5 (b)

8

3

2. 25

11

3. (a) 80

9 (b)

16

13

4. (a) 125

18 (b)

5

4

5. (a) 5

2 (b)

50

27

6. (a) 7 (b) 5

2

7. (a) (i) 24 (ii) 112

(b) 50

33

8. (a) 5

2 (b)

3

1

9. 699

10. (a) 25

22 (b) yes

11. (a) no (b) 38

118

F3B: Chapter 12D

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 13


(Video Teaching)

Book Example 14


(Video Teaching)

Book Example 15


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 12D Multiple Choice


___________ ( )



_________

119

Book 3B Lesson Worksheet 12D (Refer to §12.4)

12.4A Expected Number of Occurrences If the probability of an event is p, then we expect after n trials, this event will occur np times.


A fair coin is tossed once. (a) Find the probability that a head is obtained. (b) If the fair coin is tossed 500 times,

estimate the number of times of getting a head.

Sol (a) P(a head) =

2

1

(b) Estimated number of times

= 500 ×

2

1

= 250

A fair dice is thrown once. (a) Find the probability of getting a ‘2’. (b) If the fair dice is thrown 60 times, estimate

the number of times of getting a ‘2’.

Sol (a) P(getting a ‘2’) =

(b) Estimated number of times

= ( ) × ( )

=

1. The figure shows a circular wheel which is

divided into 5 equal sectors. If it is turned 200 times, find the expected number of times that the pointer stops at the shaded sectors.

○○○○→→→→ Ex 12D 1–3

2. The probability that a candidate passes a public examination is 0.71. This year, 30 000 candidates sit in the examination. Estimate the number of candidates who

(a) pass the examination, (b) fail the examination.

○○○○→→→→ Ex 12D 4–6

120

12.4B Concept of Expected Values Consider an activity with n possible outcomes, and the values obtained from the

possible outcomes are x1, x2, …, xn respectively. If the probabilities of the

occurrences of these possible outcomes are p1, p2, …, pn respectively, then

expected value for the activity = x1p1 + x2p2 + … + xnpn


A purse contains ten $1 coins, six $2 coins and four $5 coins. A coin is randomly drawn from the purse. (a) Complete the following table.

Coin Probability

$1

$2

$5

(b) Find the expected value of the coin drawn.

Sol (a) Total number of coins in the purse = 10 + 6 + 4 = 20 All the possible outcomes and the

corresponding probabilities are as follows:

Coin Probability

$1 2

1

20

10=

$2 10

3

20

6=

$5 5

1

20

4=

(b) Expected value of the coin drawn

=

×+×+×

5

15

10

32

2

11$

= $2.1

3 cards are marked with ‘2’, 5 cards are marked with ‘3’ and 7 cards are marked with ‘6’. A card is drawn from them randomly. (a) Complete the following table.

Number on the card Probability

2

3

6

(b) Find the expected value of the number on the card drawn.

Sol (a) Total number of cards = All the possible outcomes and the

corresponding probabilities are as follows:

Number on the card Probability

2

3

6

(b) Expected value of the number on the card drawn

=

121

3. There are 4 pencils A, B, C and D in a case. Their prices are $10, $5, $3 and $1 respectively. Tommy chooses a pencil randomly from the case. Find the expected price of the pencil chosen.

4. Box M contains 2 black balls and 1 white ball. Box N contains 2 white balls. Wilson randomly draws a ball from each box. If the two balls drawn are of the same colour, he will get $20; otherwise he will lose $5. Find the expected amount that Wilson can obtain.

○○○○→→→→ Ex 12D 7–9

Level Up Question

5. In the lucky draw of a bookshop, the probabilities of winning a $500 coupon, a $100 coupon and a $10 coupon are 0.04, 0.1 and 0.5 respectively. Anna takes part in the lucky draw 7 times. Find the expected total value of coupons won by Anna.

List all the possible outcomes and find their corresponding theoretical probabilities first.

You may use a tree diagram or a table to list all the possible outcomes.

122



Level 1

1. A fair dice is thrown 120 times. Estimate the number of occurrences of each of the following events.

(a) The number 5 is obtained. (b) A number less than 4 is obtained. 2. In a school, students are assigned randomly to the Red, Yellow, Green, Blue and Purple

Houses. If there are 780 students in the school, estimate the number of students assigned to Blue House or Purple House.

3. On a farm, the probability that an egg contains double yolks is 0.002 5. If the farm produces 1 600 eggs today, estimate the number of double-yolk eggs produced. 4. In a city, the probability that a newborn baby being a boy is 0.48. If there are 75 000 newborn

babies this year, estimate the number of newborn baby girls. 5. Suppose the probability that a kettle produced by a factory being defective is 0.045. If it is

expected that 144 kettles produced today are defective, estimate the total number of kettles produced today.

6. In a lucky draw, the probabilities that Alex wins a coupon of value $1 000, $200 and $40 are

0.03, 0.22 and 0.75 respectively. Find the expected value of the coupon that he wins. 7. Peter draws a card at random from a pack of 52 playing cards. If he can get $4, $2, $5 and $1

for a club, a diamond, a heart and a spade drawn respectively, find the expected amount he can get.

Level 2

8. A bag contains one red ball, three green balls and six blue balls. Hilary draws a ball at random from the bag. If she can get $4, $1 and nothing for a red ball, a green ball and a blue ball drawn respectively, find the expected amount she can get.

9. In a game, Zoe throws a fair dice once. Four points will be awarded if a factor of 6 is obtained,

while five points will be deducted for other results. Find the expected value of the points obtained by throwing the dice once.

10. A bag contains a $1 coin, a $5 coin and a $10 coin. Two coins are drawn randomly from the bag

at the same time. (a) Use a tree diagram or the method of tabulation to list all the possible outcomes. (b) Hence, find the expected total value of the coins drawn.


12D

123

11. In a game, a participant draws a card at random from a pack of 52 playing cards. A prize will be awarded according to the following table.

Card drawn a face card an ‘A’ any other card

Prize $7.8 $32.5 $1.3

If Tom plays the game 15 times, find the expected value of the total amount he will get. (Note: A face card is a ‘J’, a ‘Q’ or a ‘K’.) 12. The target in the figure is formed by three concentric circles. The radius of the smallest circle is

10 cm, and the widths of the two rings B and C are 10 cm and 20 cm respectively.

John pays $16 for throwing a dart once. A prize is awarded according to the following table.

Region A B C

Prize $100 $20 $4

If John throws a dart randomly and it hits the target, is the game favourable to John? Explain your answer.

A

B

C

124

Answer

Consolidation Exercise 12D

1. (a) 20 (b) 60

2. 312

3. 4

4. 39 000

5. 3 200

6. $104

7. $3

8. $0.7

9. 1

10. (b) 3

32$

11. $78

12. no

chapter 10 applications in trigonometry

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