Chapter 10 Basic Properties of Circles 10A p.2 10B p

1

Chapter 10 Basic Properties of Circles

10A p.2

10B p.16

10C p.29

Chapter 11 More about Basic Properties of Circles

11A p.41

11B p.50

11C p.58

11D p.68

11E p.78

11F p.88

Chapter 12 Basic Trigonometry

12A p.94

12B p.106

12C p.116

12D p.121

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2

F4B: Chapter 10A

Date Task Progress

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Book Example 3


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Book Example 4


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Book Example 5


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Book Example 6


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Book Example 7


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3

Consolidation Exercise


(Full Solution)

Maths Corner Exercise

10A Level 1


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10A Multiple Choice

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4

4B Lesson Worksheet 10.0 (Refer to Book 4B P.10.3)

Objective: To review angles in intersecting and parallel lines, angles of a triangle and their properties,

congruent triangles, similar triangles and Pythagoras’ theorem.

Angles in Intersecting and Parallel Lines 1. In the figure, find x. 2. In the figure, AOD and COE are straight lines.

Find x and y. �Review Ex: 1–5

3. In the figure, find x. 4. In the figure, is AB parallel to CD?

Explain your answer. �Review Ex: 6, 7

Angles of a Triangle and Their Properties

5. In the figure, BCD is a straight line. 6. In the figure, find x. �Review Ex: 8–12

Find x and y.

5

Congruent Triangles and Similar Triangles

7. In the figure, △CBD ≅ △ABC. Find x and y. 8. In the figure, △ABC ~ △CBD. Find x and y.

�Review Ex: 13, 16

9. Are the triangles in the figure congruent? 10. Are the triangles in the figure similar?

Give reasons. Give reasons. �Review Ex: 14, 17, 18

Pythagoras’ Theorem

Find the unknowns in the following figures. [Nos. 11–12]

11. 12. �Review Ex: 20–22

��Level Up Question��

13. In the figure, △ABC ~ △DEC.

(a) Find ∠ABC.

(b) Is △DEC a right-angled triangle? Explain your answer.

6

4B Lesson Worksheet 10.2A (Refer to Book 4B P.10.12)

Objective: To understand the properties of the perpendiculars to chords of a circle.

Perpendiculars to Chords

(a) (b) (c)

If OM ⊥ AB, then AM = BM. If AM = BM, then OM ⊥ AB. If CM ⊥ AB and AM = BM, then

[perpendicular from centre to [line joining centre to mid-pt. CM passes through centre O.

chord bisects chord] of chord ⊥ chord] [ ⊥ bisector of chord passes

through centre]

[In this worksheet, O is the centre of a circle.]

In each of the following figures, PQ is a chord of the circle. Find the unknowns. [Nos. 1–3]

1. 2. 3.

RMS is a straight line.

The radius of the circle is 4 cm.

a = b = c =

Instant Example 1 Instant Practice 1

In the figure, AMB is a straight line. Find the

lengths of AM and OM.

∵ OM ⊥ AB (given) ∴ AM = BM

= 12 cm

In △OAM,

OM 2 + AM 2 = OA2 (Pyth. theorem)

OM = 22AMOA −

= 22 1213 − cm

= 5 cm

In the figure, PQR is a straight line. Find the length

of RQ and the radius OR.

∵ OQ ⊥ ( ) ∴ RQ = = cm ( )

(perpendicular from centre

to chord bisects chord)

7

4. In the figure, the chords PQ and ST meet at M. 5. In the figure, the chords PQ and RS meet at M.

Find θ. Find θ. �Ex 10A: 1–6

∵ PM = ( )

∴ ∠PMS =

( )

∠PMS = ∠ + ∠

=

6. In the figure, AB and CD meet at M. 7. In the figure, AB and CD meet at M.

Find the radius of the circle. Find the radius of the circle. �Ex 10A: 7, 8

∵ is the perpendicular bisector

of CD. ( )

∴ passes through the centre,

i.e. is a diameter.

( )


8. In the figure, the straight line PQ is the perpendicular bisector of CD.

AB ⊥ PQ and BM = 4 cm. Find the length of AM.

8

4B Lesson Worksheet 10.2B (Refer to Book 4B P.10.17)

Objective: To understand the properties of equal chords of a circle.

Equal Chords

(a) If AB = CD, then OM = ON. (b) If OM = ON, then AB = CD.

[equal chords, equidistant from centre] [chords equidistant from centre are equal]


In each of the following figures, PQ and RS are chords of the circle. Find the unknowns. [Nos. 1–2]

1. 2.

∵ PQ = (given) ∵ ON = (given)

∴ OM = ∴ RS =

( ) ( )

a = b =


In the figure, AM = 4 cm, BC = 8 cm and

OM = 6 cm. Find the lengths of AB and ON.

∵ OM ⊥ AB (given) ∴ AB = 2AM

= 2 × 4 cm

= 8 cm ∵ BC = AB = 8 cm ∴ ON = OM (equal chords, equidistant from

centre)

= 6 cm

In the figure, PM = 12 cm, RS = 24 cm and

OM = 5 cm. Find the lengths of PQ and ON.

∵ OM ⊥ ( ) ∴ PQ =

( )

= (perpendicular from centre to chord

bisects chord)

9

3. In the figure, PQ and RS meet at T. OM = ON, RT = 14 cm and PQ = 16 cm. Find

the lengths of RS and ST. �Ex 10A: 12–14

∵ ON = ( )

∴ RS = ( )

=

4. In the figure, OM and ON are the perpendicular bisectors of AB and CD

respectively. AB = 10 cm, CP = 2 cm and ∠OMN = ∠ONM. Find the lengths

of CD and NP.


5. In the figure, MON is a straight line perpendicular to the chords AB and CD.

OC = 10 cm and OM = ON = 6 cm.

(a) Find the length of CN. (b) Find the length of AB.

10

New Century Mathematics (Second Edition) 4B

10 Basic Properties of Circles

Consolidation Exercise 10A

[In this exercise, unless otherwise stated, O is the centre of a circle.]

Level 1

In each of the following figures, AB is a chord of the circle. Find the unknowns. [Nos. 1–6]

1. 2. 3.

4. 5. 6.

ONC is a straight line.

In each of the following figures, PQ and RS meet at T. Find the radius of the circle. [Nos. 7–9]

7. 8. 9.

C is the mid-point of PQ. C is the mid-point of PQ.

In each of the following figures, AB and CD are chords of the circle. Find the unknowns. [Nos. 10–12]

10. 11. 12.

O

A

B

M N

x cm

C

D 4 cm

O

C

D

10 cm

N

M

10 cm A

B

a cm

32 cm

O

C D

6 cm

N

M

x cm A

B

15 cm

15 cm

C R

S

4 cm

T

Q

3 cm 3 cm

P

C

R S

7 cm

T

Q

P

11 cm

R

Q

P

T

5 cm

S

O

A

30°

N B A B B

b

O

A B

N

M

P

x

O

A

B

N 3 cm

C

2 cm x cm

y cm

16 cm

A B

O

6 cm m cm

M

18 cm

O

A

B

M a cm

O

A

B

N

x cm

4 cm

11

O

A

B

E 4 cm

C

10 cm

D

N

M

E

A

B

C

D

O

M

N

O

A

B

15 cm P Q

16 cm

O

A

B C

P R

Q

O

A

B

M

Q

P

N

13. In the figure, AB and PQ are two equal and parallel chords. MN

passes through O and OM ⊥ AB. If the distance between AB and PQ is

9 cm, find the length of OM.

14. In the figure, AB = CD, OM ⊥ AB, BONE ⊥ CD, OB = 10 cm and

NE = 4 cm.

(a) Find the length of OM.

(b) Find the length of AB.

15. In the figure, OP ⊥ AB, OQ ⊥ BC, OR ⊥ AC and OP = OQ = OR.

Find ∠BAC.

16. In the figure, AED and BCD are straight lines. OM = ON, OM ⊥ AB,

ON ⊥ AE and AB ⊥ BC. It is given that AB = 12 cm, BC = 4 cm and

CD = 5 cm.

(a) Find the length of AE.

(b) Find the length of DE.

17. In the figure, OQ and AB intersect at P. OQ ⊥ AB, OP = 15 cm and

AB = 16 cm.

(a) Find the radius of the circle.

(b) Find the length of PQ.

12

O

D

A B

9 cm

12 cm 12 cm

C

B

O

M A

N

4 cm

3 cm

R

S

P Q T

24 cm

O

A

B

28°

C

D

O

A

B D

M N

C

18°

18. In the figure, COD and ACB are straight lines. OC = 9 cm and

AC = BC = 12 cm.

(a) Is OC the perpendicular bisector of AB? Explain your answer.

(b) Find the length of CD.

Level 2

19. In the figure, C is the mid-point of AB. COD is a straight line and

∠OBC = 28°.

(a) Find ∠BOC.

(b) Find ∠BDO.

20. In the figure, AB and CD are two equal chords. OM ⊥ AB and

ON ⊥ CD. If ∠MNO = 18°, find ∠MON.

21. In the figure, PQ and RS meet at T. PR = PS, ∠RPT = ∠SPT and

PQ = 24 cm.

(a) Show that �PRT ≅ �PST.

(b) Find the radius of the circle.

22. In the figure, AMB and MON are straight lines. OM ⊥ AB, AM = 4 cm

and OM = 3 cm.

(a) Find the radius OA of the circle.

(b) Find the length of BN, correct to 1 decimal place.

13

B

O

A N

D

8 cm

M

12 cm

C

B

O

A

D

24 cm

20 cm 14 cm

C

A

C

E

O

B

D

M

N

A B M

N

O

C

D

A B M

O

C

4 cm

16 cm

23. In the figure, ACB and MOCN are straight lines. OC ⊥ AB, AD // NM,

AB = 12 cm and OC = 8 cm.

(a) Find the radius OA of the circle.

(b) Find the length of AD.

24. In the figure, ABC and ODC are straight lines. CD = 14 cm,

AB = 24 cm and OA = 20 cm. Find the length of BC.

25. In the figure, O is the centre of two concentric circles. AMBC and

ANDE are straight lines. OM ⊥ AB, ON ⊥ AD and OM = ON. If

AB = 8 cm, BC = 3 cm and the radius of the smaller circle is 5 cm,

find

(a) the length of AE,

(b) the radius of the larger circle, correct to 1 decimal place.

26. In the figure, chords AB and CD are perpendicular to each other, and

they meet at N. OM ⊥ AB, AB = 8 cm and OM = 3 cm. N is the mid-

point of MB.


(b) Find the length of DN, correct to 1 decimal place.

27. In the figure, the chord AB and the radius OC meet at M, where

OC ⊥ AB. AB = 16 cm and CM = 4 cm. Find the radius of the circle.

14

B

O

A M

D N

C

A B M

O

C D N

A

B

P Q M

A B

O

C D

16 cm

12 cm

28. In the figure, AB and CD are two parallel chords. AB = 16 cm and

CD = 12 cm. The distance from O to CD is 8 cm.


(b) Find the distance between AB and CD.

29. In the figure, OM ⊥ AB and ON ⊥ AC. ON = 2 cm, AC = 12 cm and

AD : DN = 3 : 1.

(a) Show that �OND ~ �AMD.

(b) Find the length of AB.

30. In the figure, OM ⊥ AB, ON ⊥ CD and MON is a straight line. If

AB = 32 cm, CD = 24 cm and MN = 28 cm, find the radius of the

circle.

31. In the figure, P and Q are the centres of two circles. The two circles

intersect at A and B. M is a point on AB such that PM ⊥ AB. Does PM

produced pass through the centre Q? Explain your answer.

15

Answers


1. 4 2. 9

3. 10 4. x = 5, y = 8

5. 90° 6. 60°

7. 8 cm 8. 7 cm

9. 5 cm 10. 6

11. 32 12. 8

13. 4.5 cm

14. (a) 6 cm (b) 16 cm

15. 60°

16. (a) 12 cm (b) 3 cm

17. (a) 17 cm (b) 2 cm

18. (a) yes (b) 24 cm

19. (a) 62° (b) 31°

20. 144°

21. (b) 12 cm

22. (a) 5 cm (b) 8.9 cm

23. (a) 10 cm (b) 16 cm

24. 18 cm

25. (a) 11 cm (b) 7.6 cm

26. (a) 5 cm (b) 1.6 cm

27. 10 cm

28. (a) 10 cm (b) 2 cm

29. (b) 7.2 cm

30. 20 cm

31. yes

16

F4B: Chapter 10B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)

Book Example 10


(Video Teaching)

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)

Book Example 13


(Video Teaching)

Book Example 14


(Video Teaching)



(Full Solution)

17


10B Level 1


Teacher’s

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10B Level 2


Teacher’s

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10B Multiple Choice


Signature

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( )


Self-Test


_________

18


Objective: To understand the angle properties of a circle (angles at the centre and angles at the circumference).

Relationship between Angle at the Centre and Angle at the Circumference

x = 2y

[∠ at centre twice ∠ at circumference]



In the figure, ∠ABC = 40°.

Find x.

x = 2 ∠ABC (∠ at centre twice ∠ at circumference)

= 2 × 40°

= 80°

In the figure, reflex ∠AOC = 196°.

Find y.

y = ( )( ) ( )

=


1. 2. �Ex 10B: 1–3

Angle in a Semi-circle

(a) If AC is a diameter, (b) If ∠ABC = 90°,

then ∠ABC = 90°. then AC is a diameter.

[∠ in semi-circle] [converse of ∠ in semi-circle]

If p = 2q, then q = p.

19


In the figure, AOC is a diameter.

Find x.

∠ABC = 90° (∠ in semi-circle)

∠ABC + 50° + x = 180° (∠ sum of △)

90° + 50° + x = 180°

x = 40°

In the figure, AOC is a diameter.

Find y.

∠ABC = ( )

_______ + _____ + y = _____ ( )

=


3. 4. �Ex 10B: 4–6

∠ACB = ( )

5. In the figure, determine whether XY is a diameter of the circle. Explain your answer. �Ex 10B: 10, 11


6. In the figure, ∠ABD = 60° and ∠ADB = 30°.

(a) Determine whether BD is a diameter of the circle. Explain your answer.

(b) Hence, find y.

If ∠______ = 90°,

then XY is a diameter.

20


Objective: To understand the angle properties of a circle (angles in the same segment).

Angles in the Same Segment

x = y

[∠s in the same segment]


1. 2. �Ex 10B: 7

x = ∠_________ (∠s in the same segment)

=


In the figure, ∠BDC = 52° and

AC ⊥ BD. Find a.

∠BAC = ∠BDC (∠s in the same segment)

= 52°

In △ABE,

a + 90° + ∠BAC = 180° (∠ sum of △)

a + 90° + 52° = 180°

a = 38°

In the figure, ∠AEB = 105° and

∠ACD = 50°. Find b.


3. �Ex 10B: 8, 9

21

4. 5.

6. 7.


8. In the figure, AC is a diameter. Find x.

Consider ∠BCA.

22

O

B C

A

68°

b

D



Consolidation Exercise 10B


Level 1


1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

O A B

C

24° x

32°

D

O

A

B

C

20°

c

O A

B

C

y D

140°

O A B

C

4b b O

A B

C

60°

a

O B

C

A

y

D

E 66°

O

B

C

A

70°

x

O

A B

a

110°

C

O

A

B C z

260° O

A

B

C

y

65°

O

B

C

A

80°

x

23

A

B

C 35° 65° A

B

C 100°

50°

D

A

B

C

x + 10°

2x O

D

A

B

C

100°

140° O

13. 14. 15.

16. 17. 18.

19. In each of the following figures, determine whether AC is a diameter of the circle. Explain your

answer.

(a) (b)

20. In the figure, ∠AOC = 100° and ∠BOC = 140°.

(a) Find ∠AOB.

(b) Find ∠ACB.

21. In the figure, AOD and BDC are straight lines. ∠ABC = x + 10° and

∠COD = 2x. Find x.

O A

B

C

34°

y

D E

x

A

B C

b

86°

D

67°

E

A

B

C

a

50°

D

28°

E

A

B C

z

61°

D

80° E O B

C

A

30°

z D

A

B

C

y

25°

D

24

A

B

C

70° 127°

O D

A O

B

C D

40°

A

B

C

26° O

D

O

A

B

C 100°

A B

C

O

D

31° 64°

A B

C

50° 27° O

D

22. In the figure, AC and OB intersect at D. ∠ADB = 127° and

∠CBO = 70°. Find ∠CAO.

23. In the figure, O is the centre of the semi-circle ADCB. BD is the angle

bisector of ∠ABC. If ∠BAC = 40°, find ∠CAD.

24. In the figure, DB is a diameter of the circle. AO // DC and ∠ABD =

26°.

(a) Find ∠CDB.

(b) Find ∠CAO.

Level 2

25. In the figure, AC = BC and ∠AOB = 100°.

(a) Find ∠CAB.

(b) Find ∠CAO.

26. In the figure, ADC and AOB are straight lines. ∠BAC = 27° and

∠ABC = 50°. Find ∠ODC.

27. In the figure, AB is a diameter of the circle. If ∠BOC = 64° and

∠CAD = 31°, find ∠ABD.

25

A B

C

D

40°

A B

C

D

B O

C

D A

42°

A

B

C

D

25°

28. In the figure, diameter AC cuts BD at P. If ∠APD = 105° and AB =

BC, find ∠CBD.

29. In the figure, O is the centre of the semi-circle BADC. BA // OD and

∠ACB = 42°.

(a) Find ∠COD.

(b) Find ∠BAD.

30. In the figure, AC is a diameter of the circle. AD = BD and ∠ACB =

40°. Find ∠CAD.

31. In the figure, AC is a diameter of the circle. BD = CD and ∠ACD =

25°. Find ∠BDC.

32. In the figure, AB is a diameter of the circle. D is a point on AB such

that CD ⊥ AB. If AC = 8 cm and BC = 6 cm, find

(a) the length of CD,

(b) AD : DB.

P

A

B

C

105°

D

26

A

B

C

D

E

A B

C

O

D

50°

O B A

C

D

25°

A

B

C

D

84° E

A O

D

C B

36°

33. In the figure, BD is a diameter of the circle. BD cuts AC at E and

BD ⊥ AC.

(a) Write down a pair of similar triangles in the figure.

(b) If AC = 6 cm and AD = 5 cm, find

(i) the length of BC,

(ii) the radius of the circle.

34. In the figure, O is the centre of the semi-circle ABCD. If AB // OC and

∠COD = 36°, find ∠CDB.

35. In the figure, AC and BD intersect at E. AD // BC, AD = BD and

∠CED = 84°.

(a) Find ∠ADB.

(b) Find ∠ABD.

36. In the figure, OAB and BCD are straight lines. OC = BC and

∠ABC = 25°. Find ∠OAD.

37. In the figure, AB is a diameter of the circle. AD // OC and ∠BAD =

50°. Find ∠OCD.

27

A B

C

O

D

70°

D

A B

C

O

A

B

C

D

E F G

A

B

C

D

E

28°

A

B

C

D

I

E

38. In the figure, AB is a diameter of the circle. AC is the angle bisector of

∠DAO and OD // BC. Find ∠ABC.

39. In the figure, ∠CED = 28° and AD // BC. Find ∠ADB.

40. In the figure, two circles intersect at B and D. BFD and

AEFGC are straight lines. If ∠ABC = 145°, find ∠EDG.

41. In the figure, AO // DC and ∠AOD = 70°.

(a) Is AC the angle bisector of ∠DCO? Explain your answer.

(b) Find ∠ABC.

� 42. In the figure, I is the in-centre of �ABC. BID and CIE are straight

lines. If ∠ABC = 70° and ∠ACB = 48°,

(a) find ∠DCE,

(b) is it true that ∠DCE = ∠DIC? Explain your answer.

28

Answers


1. 40° 2. 130°

3. 130° 4. 140°

5. 146° 6. 20°

7. 33° 8. 30°

9. 18° 10. 50°

11. 70° 12. 34°

13. 25° 14. 30°

15. 39° 16. 78°

17. 19° 18. x = 34°, y = 56°

19. (a) no (b) yes

20. (a) 120° (b) 60°

21. 40° 22. 13°

23. 25°

24. (a) 52° (b) 26°

25. (a) 65° (b) 25°

26. 53° 27. 27°

28. 30°

29. (a) 48° (b) 114°

30. 20° 31. 50°

32. (a) 4.8 cm (b) 16 : 9

33. (a) �ADE ~ �BCE

(b) (i) 3.75 cm (ii) 3.125 cm

34. 18°

35. (a) 42° (b) 69°

36. 37.5° 37. 65°

38. 60° 39. 28°

40. 35°

41. (a) yes (b) 55°

42. (a) 59° (b) yes

29

F4B: Chapter 10C

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)


10C Level 1


Teacher’s

Signature

___________

( )


10C Level 2


Teacher’s

Signature

___________

( )


10C Multiple Choice


Signature

___________

( )


Self-Test


_________

30


Objective: To understand the equal relations among angles, arcs and chords of a circle.

Equal Relations among Angles, Arcs and Chords

(a) If x = y, then⌢

AB =⌢

CD .

[equal angles, equal arcs]

If⌢

AB =⌢

CD , then x = y.

[equal arcs, equal angles]

(b) If x = y, then AB = CD.

[equal angles, equal chords]

If AB = CD, then x = y.

[equal chords, equal angles]

(c) If AB = CD, then⌢

AB =⌢

CD .

[equal chords, equal arcs]

If⌢

AB =⌢

CD , then AB = CD.

[equal arcs, equal chords]

Note: ‘equal angles’ can

also mean ‘equal

angles at the

circumference’. If m = n, then x = y.



In the figure,⌢

BC =⌢

AB and

∠AOB = ∠COD = 70°.

Find x and y.

∵ ⌢

BC =⌢

AB (given) ∴ x = ∠AOB (equal arcs, equal angles)

= 70° ∵ ∠AOB = ∠COD (given) ∴ AB = CD (equal angles, equal chords)

y = 5

In the figure, AB = BC = CD,

⌢

AB = 6 cm and ∠CBD = 40°.

Find x and y.

∵ AB = ________ (given) ∴ x = ∠________ (equal , equal )

=


1. 2. �Ex 10C: 1–5

∵ ⌢

AB =⌢

BC (given) ∵ ∠______ = ∠______ (given)

∴ AB = (equal , equal ) ∴ ⌢

AB = ( )

x =

31

3. 4.

5. 6.


7. In the figure, AB = DE, ∠BOC = ∠COD = 75° and ∠AOE = 60°.

Find x and y.

EOB is a diameter. AOD is a diameter.

32


Objective: To understand the proportion relations between angles and arcs of a circle.

Proportion Relations between Angles and Arcs

(a) ⌢

AB :⌢

CD = m : n

[arcs prop. to ∠s at centre]

(b) ⌢

AB :⌢

BC = x : y

[arcs prop. to ∠s at

circumference]



In the figure,⌢

AB = 7 cm, ⌢

CD = 21 cm and

∠AOB = 55°. Find x.

In the figure,⌢

AB = 2 cm,

∠AOB = 50° and

reflex ∠BOC = 200°. Find x.

∵ ⌢

AB :⌢

CD = ∠AOB : ∠COD

(arcs prop. to ∠s at centre) ∴ 21

7 =

x

°55

x = 165°

∵ ⌢

AB :⌢

BC = :

( ) ∴ ) (

) (=

) (

) (

=


1. 2. �Ex 10C: 6, 7, 14

∵ ⌢

AB :⌢

CD = :

( )

∴

33


In the figure,⌢

AB = 10 cm, ⌢

CD = 5 cm and ∠ADB = 46°.

Find x.

In the figure,⌢

BC = 12 cm,

∠ADB = 18° and

∠BDC = 54°. Find x.

∵ ⌢

AB :⌢

BC = ∠ADB : ∠BDC

(arcs prop. to ∠s at circumference) ∴ 5

10 =

x

°46

x = 23°

∵ ⌢

AB :⌢

BC = : ( ) ∴

) (

) (=

) (

) (

=


3. 4. �Ex 10C: 8, 9, 15


5. In the figure, AOB is a diameter.⌢

AC = 8 cm and⌢

BC = 4 cm. Find x.

34

O A

B

D

74°

C



Consolidation Exercise 10C


Level 1


1. 2. 3.

4. 5. 6.

7. 8. 9.

10. In the figure, AD is a diameter of the circle. AB = CD and ∠BOC =

74°. Find ∠AOB.

6 cm

4 cm 3 cm

5 cm

x

70°

y 13 cm

y cm

65° x

12 cm

A B

C

O 100°

35°

15 cm

z cm A

B

C D

y

50°

c

18 cm

6 cm

d cm

9 cm

b cm

P

Q

R

S

20°

60° O

A B

C a

D

42°

O

P

S

Q

R

4 cm

z cm O

A

B

D

y

C

45°

115° O

A

B

C

x

100°

35

75°

A B

C

15π cm

60°

20°

A B

C

D

E

O

25° P Q

R 26 cm

40° A

B C

O

E

D

120°

A B

C

O 20 cm

24 cm

11. In the figure, AB = BC = CD and ∠AOB = 40°. Find ∠AED.

12. In the figure, PQ is a diameter of the circle. ∠OPR = 25° and ⌢

PR = 26 cm.

(a) Find ∠QOR.

(b) Find the length of⌢

RQ .

13. In the figure, ∠AOB = 120°,⌢

AB = 24 cm and⌢

BC = 20 cm.

(a) Find ∠BOC.

(b) Find the circumference of the circle.

14. In the figure, AC = BC, ∠BAC = 75° and⌢

BC = 15π cm.

(a) Find ∠ACB.

(b) Find the circumference of the circle in terms of π.

15. In the figure, chords AC and BD intersect at E. If ∠BEC = 60° and

∠BDC = 20°, find⌢

AD :⌢

BC .

36

D

A

B

C

x

A

B C

O

D 50°

20°

30°

A

B C

D

E

O

A

D

130° C

B

150° A

O

B

C

D

E

16. In the figure, chords AC and BD intersect at E.⌢

AD :⌢

BC = 3 : 2 and

∠ABD = 30°. Find ∠AED.

17. In the figure,⌢

AB :⌢

BC :⌢

CD :⌢

DA = 1 : 1 : 2 : 2. Let ∠ADB = x.

(a) Express ∠BAC and ∠BAD in terms of x.

(b) Is BD a diameter of the circle? Explain your answer.

(c) Are BD and AC perpendicular to each other? Explain your

answer.

Level 2

18. In the figure, AD = CD and ∠ABC = 130°.

(a) Find ∠AOD.

(b) Find ∠OAD.

19. In the figure, OABCD is a sector with centre O. AC cuts BD at E.

⌢

AB =⌢

BC =⌢

CD and ∠AOD = 150°.

(a) Find ∠OAE.

(b) Find ∠AED.

20. In the figure, AD is a diameter of the circle. ∠OBD = 20° and

∠COD = 50°. Find⌢

AB :⌢

BC :⌢

CD .

37

A

B

O

C

10°

70°

O

84°

C

B

D A

E

80° A

B

C

D

E

20° A B

C

D

E

21. In the figure, ∠ACO = 10° and ∠CBO = 70°.

(a) Find ∠AOB.

(b) Find⌢

AB :⌢

BC .

22. In the figure, AC is a diameter of the circle.⌢

BC :⌢

DE = 5 : 7 and

∠DOE = 84°.

(a) Find ∠BOC.

(b) Find⌢

AB :⌢

BC .

23. In the figure, AD and BE intersect at F. ∠AFB = 60° and ∠BDC = 24°.

If BE // CD, find⌢

AB :⌢

BC .

24. In the figure, AC and BD intersect at E. ∠AED = 80° and

⌢

AD :⌢

BC = 3 : 1. Find ∠ACD.

25. In the figure, ABC and AED are straight lines.⌢

BE :⌢

CD = 1 : 3 and

∠CAD = 20°. Find ∠ABD.

A

C

D

24°

60°

E

F

B

38

O A C

D

E

B

25°

E

A B

C

D

O

30°

O

A

B

C

D

A

B

C

E

D

36°

A

B

C

D E

A

B

C

D

26. In the figure, BD is a diameter of the circle. ∠OCD = 30° and

⌢

AD :⌢

BC = 2 : 1.

(a) Find ∠ABD.

(b) Is AB parallel to OC? Explain your answer.


AB :⌢

BC :⌢

CD = 1 : 2 : 2 and ∠BDC = 36°.

(a) Find ∠AED.

(b) Is AD a diameter of the circle? Explain your answer.

28. In the figure, AD and EC are diameters of the circle. AB // EC and

∠ADE = 25°. Find⌢

AB :⌢

DE .

29. In the figure, O is the centre of the semi-circle AEDB. AB and ED are

produced to meet at C. If OD = CD, find⌢

AE :⌢

BD .

30. In the figure, the two circles intersect at A and B. B is the mid-point of ⌢

AC of the larger circle. D is the mid-point of⌢

AB of the smaller circle.

ADC and EBC are straight lines. If ∠AED = 32°,

(a) find ∠BAD,

(b) find ∠ABC.


AC :⌢

BC = 1 : 2,⌢

CAD :⌢

CBD = 7 : 11 and ∠BAC = 80°.

(a) Find⌢

AC :⌢

AB :⌢

BC .

(b) Find⌢

AD :⌢

BD .

(c) Find ∠ACD.

39

A B O

M

C D

A

B C

D

E

30°

O

A

B C

D

E

32. In the figure, AB is a diameter of the circle. ADC and BEC are straight

lines. AB = AC and ∠BAC = 30°.

(a) Find ∠CBD.

(b) Find⌢

AD :⌢

DE .

(c) Are⌢

BE and⌢

ED equal arcs? Explain your answer.

33. In the figure, AB // DC, ∠ACB = 30° and ∠BDC = 40°. Find

⌢

BC :⌢

CD :⌢

DB .

34. In the figure, ∠BAD = 72° and⌢

BC :⌢

CD :⌢

DA = 2 : 4 : 5. Find

⌢

AB :⌢

CD .

35. In the figure, �ABC is an equilateral triangle. AD = BE and

⌢

BD = 2⌢

AD .

(a) Find ∠BAD.

(b) Find ∠DBE.

36. The figure shows a paper sheet in the shape of a semi-circle with

centre O. AB is the diameter and⌢

AM =⌢

MB . The paper sheet is folded

such

that M coincides with O.

(a) Find ∠AOD.

(b) Find ∠COD.

(c) Find⌢

AD :⌢

DC .

A B

C D 40°

30°

A

D

C

B

72°

40

Answers


1. 160° 2. 155°

3. 4 4. 84°

5. 3 6. c = 30°, d = 10

7. y = 70°, z = 10.5 8. x = 60°, y = 11

9. x = 110°, y = 50° 10. 53°

11. 60°

12. (a) 50° (b) 10 cm

13. (a) 100° (b) 72 cm

14. (a) 30° (b) 36π cm

15. 2 : 1 16. 50°

17. (a) ∠BAC = x, ∠BAD = 3x

(b) yes (c) yes

18. (a) 130° (b) 25°

19. (a) 40° (b) 130°

20. 4 : 9 : 5

21. (a) 120° (b) 3 : 1

22. (a) 60° (b) 2 : 1

23. 3 : 2 24. 60°

25. 150°

26. (a) 60° (b) yes

27. (a) 90° (b) yes

28. 8 : 13 29. 3 : 1

30. (a) 32° (b) 116°

31. (a) 2 : 3 : 4 (b) 1 : 1

(c) 30°

32. (a) 15° (b) 4 : 1

(c) yes

33. 4 : 7 : 7 34. 1 : 1

35. (a) 40° (b) 120°

36. (a) 30° (b) 120°

(c) 1 : 4

41

F4B: Chapter 11A

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)


11A Level 1


Teacher’s

Signature

___________

( )


11A Level 2


Teacher’s

Signature

___________

( )


11A Multiple Choice


Signature

___________

( )


Self-Test


_________

42


Objective: To understand the properties of cyclic quadrilaterals.

Opposite Angles of a Cyclic Quadrilateral

If PQRS is a cyclic quadrilateral,

then ∠P + ∠R = 180°

and ∠Q + ∠S = 180°.

[opp. ∠s, cyclic quad.]


In the figure, ∠ADC = 80°.

Find p.

p + 80° = 180° (opp. ∠s, cyclic quad.)

p = 100°

In the figure, ∠BAD = 105°

and ∠ACD = 35°. Find c.

∠______ + ∠______ = 180° ( )

=


1. 2. �Ex 11A: 1, 2, 7, 8

In △ABC,

∵ BA = BC (given)

∴ ∠BAC = ( )

Exterior Angles of a Cyclic Quadrilateral

If PQRS is a cyclic quadrilateral,

then x = y.

[ext. ∠, cyclic quad.]

∠_____ + ∠_____ = 180° ∠_____ + ∠_____ = 180°

43

In each of the following, write down the interior opposite angle corresponding to x. [Nos. 3–5]

3.

4.

5.

∠______ ∠______ ∠______


In the figure, ADE is a straight

line. ∠ABC = 110° and

∠BAD = 100°. Find x.

x = ∠ABC (ext. ∠, cyclic quad.)

= 110°

In the figure, CDE is a straight

line. ∠CBD = 50° and

∠ADE = 115°. Find b.

∠ABC = ∠_____ ( )

b + _____ = _____

=

In each of the following figures, CDE is a straight line. Find the unknowns. [Nos. 6–7]

6. 7. �Ex 11A: 3–6, 9–12


8. In the figure, AODE is a straight line. AOD is a diameter of the circle and

∠CFD = 20°. Find x and y.

44


11 More about Basic Properties of Circles



Level 1


1. 2. 3.

4. 5. 6.

ABF and CDE are straight lines. CBE is a straight line. ADE and CDF are straight lines.

7. 8. 9.

10. 11. 12.

BAG and DEF are straight lines. ADE is a straight line. ABE is a straight line.

A

E

D

C

B

b 120°

a

A

B

C

D

100°

20° c

A

B

C

D E

c

53°

27°

110°

37° z

D

C

E B A

112°

y

A G

B C

D

E

F

x

A b

D

C

B

70°

a

A B

C D

130°

r

87°

r

A

B C

D

F

E

101°

z 73°

A

B

D

C

E

88°

76°

y

x

A

C

B

F

E D

A

B C

D

d

f 140°

O

A

B

C

D

80° 122°

a

b

45

A

B

D

x +10°

x −10° 120° C

O

A

B C

D E

32°

76°

56°

38°

A

B

C

D

O

D

C B

A E

136° 70°

120° A

B E

C D

13. In the figure, ∠BAD = 120°, ∠ACB = x − 10° and ∠ACD = x + 10°.

Find x.

14. In the figure, ADE is a straight line. ∠OCB = 32° and ∠CDE = 76°.

(a) Find ∠OBC.

(b) Find ∠OBA.


BC =⌢

CD , ∠ACB = 56° and ∠CAD = 38°.

(a) Find ∠BAC.

(b) Find ∠ADC.

16. In the figure, AED and BEO are straight lines. ∠ABE = 70° and

∠BCD = 136°.

(a) Find ∠BED.

(b) Find ∠ADO.

17. In the figure, ABE is a straight line. AD = DC = CB and ∠CBE = 120°.

(a) Find ∠ACD.

(b) Is AB a diameter of the circle? Explain your answer.

46

100° 32° A

B

C D

E

F

O A

B C

D

E

20° 30°

A

B

C D

50°

25° A

B C

D

46°

A

B

C

D E

F

Level 2


AB =⌢

AD =⌢

CD and ∠CAD = 50°. Find ∠BAC.

19. In the figure, BC is a diameter of the circle.⌢

AB =⌢

AD and

∠ADB = 25°.

(a) Find ∠BAD.

(b) Find ∠CBD.

20. In the figure, ACD and DEF are straight lines. If ∠BAC = 32° and

∠BCF = 100°, find ∠CED.

21. In the figure, BD is a diameter of the circle. BDE and AFE are

straight lines. AD = DE and ∠ACD = 46°.

(a) Find ∠ADB.

(b) Find ∠EDF.

22. In the figure, AB is a diameter of the circle. AOBC and CDE are

straight lines. If ∠DAE = 20° and ∠BCD = 30°, find ∠ADE.

47

106°

100°

38°

A

B C

D

E

O B

C

D

A 40° 70°

87°

38° 45°

E

A B C

D

O

55°

A B C

D

E F

48° 35°

C

D

E

F

A

B

23. In the figure, ∠ABC = 100°, ∠BAE = 106° and ∠ADE = 38°. Find

∠BDC.

24. In the figure, ABC is a straight line. If ∠ABE = 87°, ∠BEO = 38° and

∠ODC = 45°, find ∠BCD.

25. In the figure, AB is a diameter of the circle. ∠BAD = 70° and ∠OCD = 40°.

(a) Find ∠COB.

(b) Determine whether AB is parallel to DC. Explain your answer.

26. In the figure, ABC, AFE and CDE are straight lines. ∠BAF = 48°,

∠DBF = 35°, DB = DC and AB = AF.

(a) Find ∠DEF.

(b) Find ∠BFD.

27. In the figure, ABC and CDE are straight lines. AE // BD and

∠BFE = 55°. Find ∠BCD.

48

B

A

D

C E 5 cm

4 cm

4.5 cm x cm

A

B

C

D 100°

a

b

c B

A F

E

D C

A

B

C D

E O 70°

A

B

C

D E

30°

28. In the figure, ADC and BEC are straight lines. BE = 5 cm, EC = 4 cm

and CD = 4.5 cm.

(a) Write down a pair of similar triangles and give reasons.

(b) Find the value of x.


AB :⌢

BC :⌢

CD = 3 : 1 : 2 and ∠ABC = 100°.

(a) Find ∠BAD.

(b) Find

⌢

AB :⌢

AD .

30. In the figure, the sides of the hexagon ABCDEF are all different. Find

a + b + c.

31. In the figure, O is the centre of the smaller circle. AED is a straight

line. If ∠ABC = 70°, find ∠CDE.

32. In the figure, ABCDE is a pentagon. CD = DE, AB = BC = AE and

∠DAE = 30°.

(a) Find ∠ABC.

(b) Find ∠AEC.

49

Answers


1. a = 100°, b = 58° 2. 60°

3. d = 70°, f = 110° 4. x = 76°, y = 88°

5. 28° 6. 87°

7. a = 110°, b = 35° 8. a = 60°, b = 30°

9. 25° 10. x = 112°, y = 112°

11. 80° 12. 36°

13. 30°

14. (a) 32° (b) 44°

15. (a) 38° (b) 94°

16. (a) 114° (b) 26°

17. (a) 30° (b) yes

18. 30°

19. (a) 130° (b) 40°

20. 48°

21. (a) 44° (b) 112°

22. 50° 23. 44°

24. 100°

25. (a) 40° (b) yes

26. (a) 53° (b) 35°

27. 70°

28. (a) �ABC ~ �EDC (AAA)

(b) 3.5

29. (a) 60° (b) 1 : 1

30. 360°

31. 40°

32. (a) 100° (b) 80°

50

F4B: Chapter 11B

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)



(Full Solution)


11B Level 1


Teacher’s

Signature

___________

( )


11B Level 2


Teacher’s

Signature

___________

( )


11B Multiple Choice


Signature

___________

( )


Self-Test


_________

51


Objective: To understand the tests for four concyclic points and cyclic quadrilaterals.

Tests for Concyclic Points and Cyclic Quadrilaterals

(a) If ∠P + ∠R = 180°

or ∠Q + ∠S = 180°,

then P, Q, R and S are concyclic.

[opp. ∠s supp.]

(b) If y = x,

then P, Q, R and S are

concyclic.

[ext. ∠ = int. opp. ∠]

(c) If x = y,

then P, Q, R and S are concyclic.

[converse of ∠s in the same segment]


In the figure, determine whether

A, B, C and D are concyclic.



∠_____ + ∠_____

∠BAD + ∠BCD = 70° + 110° = _______ + _______ + _______

= 180° ∴ A, B, C and D are concyclic. (opp. ∠s supp.)

=

In each of the following, determine whether A, B, C and D are concyclic. If so, find the unknowns. [Nos. 1–2]

1. 2. �Ex 11B: 1, 4, 9, 10

CDE is a straight line.


In the figure, CDE is a straight line.

Determine whether A, B, C and

D are concyclic.

In the figure, CDE is a straight

line. Determine whether A, B,

C and D are concyclic.

∠ABC = ∠ADE = 100° ∠______ = ______ + ______ ∴ A, B, C and D are concyclic. (ext. ∠ = int. opp. ∠) =

Consider ∠BAD

and ∠BCD.

Consider ∠DAB

and ∠DCB.

�

52


3. 4. �Ex 11B: 2, 5, 7, 12

CDE is a straight line. CDE is a straight line.




∠CBD = ∠CAD = 36° ∴ A, B, C and D are concyclic.


P, Q, R and S are concyclic.

∠______ = ∠______ = ______


5. 6. �Ex 11B: 3, 6, 8, 11

ADE is a straight line.


7. In the figure, AB = BC = AC, ∠CAD = 45° and ∠ACD = 15°.

(a) Find ∠ADC.

(b) Determine whether A, B, C and D are concyclic.

Consider ∠DAC

and ∠DBC.

Consider ∠CBD

and ∠CAD.

Consider ∠ABC

and ∠ADE.

Consider ∠ABC

and ∠ADE.

(converse of ∠s in the same segment)

53



� Consolidation Exercise 11B

Level 1

In each of the following figures, determine whether A, B, C and D are concyclic. [Nos. 1–3]

1. 2. 3.

DAE is a straight line.

In each of the following, determine whether A, B, C and D are concyclic. If so, find the unknowns.

[Nos. 4–6]

4. 5. 6.

ADE is a straight line.

In each of the following, determine whether ABCD is a cyclic quadrilateral. [Nos. 7–10]

7. 8.

CDE is a straight line. ABE is a straight line.

9. 10.

ABE is a straight line.

37°

65° 76°

A B

E

D C

60°

40°

100°

D

B A

C

84° 36°

48° A

D

C

B E

A B

C D

103°

26°

B A

D

C

E 75°

40°

35°

80°

D

A

B

C E

40°

B

D

39° y

63°

42°

36° A

C

A B

C

D

75°

50° 50° z

100°

70°

30°

x

A 42°

B

C

D E

103°

A B

43°

35°

C

D

54

A B

C

D 30°

20°

86°

47°

A B

E

C D

A

B C

D

E F

G H

P

Q R

S T

G

A B

E

C

D

11. In the figure, DA ⊥ AC and DB ⊥ BC. If ∠ACD = 30° and ∠ADB =

20°, find ∠BAC.

12. In the figure, ABE is a straight line. AD = CD, ∠ACD = 47° and

∠CBE = 86°. Find ∠CBD.

13. In the figure, ADC and ABE are straight lines. AB = AD and BE = DC.

(a) Is it true that �ABC ≅ �ADE? Explain your answer.

(b) Determine whether B, E, C and D are concyclic.

14. In the figure, �ABC and �DEF are equilateral triangles. EF // BC

and D is a point outside �ABC. DE and DF cut BC at G and H

respectively. Determine whether each of the following quadrilaterals

is a cyclic quadrilateral.

(a) BDHE

(b) AEDF

(c) EGHF

Level 2

15. In the figure, PQR is an equilateral triangle. The two medians QS and

RT intersect at G.

(a) Find ∠SGT.

(b) Determine whether P, S, G and T are concyclic.

55

A

D

C E

F

O

B

84°

A

B

C

D E

F

P

D C

B A

I

65°

A B

C D H

G

F

E

A

B

F

D

E

C

G

H

36°

48° 114°

30°

C

D

E

A B

F

16. In the figure, O is the centre of the circle. AB is a diameter of the

circle. AE cuts BD at F. AD and BE are produced to meet at C.

(a) Find ∠AEC.

(b) Determine whether C, D, F and E are concyclic.

17. In the figure, AC and BD intersect at E. ABF is a straight line.

∠CAD = 36°, ∠ACD = 30°, ∠BAC = 48° and ∠CBF = 114°.

(a) Determine whether A, B, C and D are concyclic.

(b) Find ∠BDC.

18. In the figure, BCF and ADE are straight lines. AB // EF and

∠ABC = 84°.

(a) Find ∠CDE.

(b) Determine whether C, D, E and F are concyclic.

19. In the figure, ABCD is a square. DP = CP and ∠DPC = 90°. I is the

in-centre of �CDP.

(a) Find ∠CID.

(b) Determine whether A, D, I and C are concyclic.

20. In the figure, AGFD, EHFB, CGB and CHD are straight lines. It is

given that �ABC ≅ �EDC.

(a) Is it true that �ACD ≅ �ECB? Explain your answer.

(b) Determine whether G, H, D and B are concyclic.

21. In the figure, ABCD is a square. AFG, AEH and DEFB are straight lines.

�ABG ≅ �ADH and ∠AGB = 65°.

(a) Find ∠AGH and ∠DEH.

(b) Determine whether EFGH is a cyclic quadrilateral.

56

A

D

C

F

G

E B

50° 80° A B

O

E D

C

F

A B O

M

C

E

D

A

B

C F

E

D

25°

A B

C D

S

T

P

Q

R

O C

A

D

E B 20°

22. In the figure, ABC is a right-angled triangle with ∠BAC = 90°. ADC,

AEB and BFC are straight lines. CE and DF intersect at G. CDEF is a

rhombus.

(a) Determine whether ADGE is a cyclic quadrilateral.

(b) Steve claims that CDEB is a cyclic quadrilateral if AC = AB. Do

you agree? Explain your answer.

23. In the figure, O is the centre of the semi-circle ADEB. ADC, AFE,

BFD and BEC are straight lines. ∠AOD = 80° and ∠BOE = 50°.

(a) Determine whether C, D, F and E are concyclic.

(b) Find ∠DCE and ∠DFE.

24. In the figure, O is the centre of the semi-circle. ABCD is a rectangle.

M is the mid-point of AE.

(a) Determine whether BCMO is a cyclic quadrilateral.

(b) Find ∠BMO.

25. In the figure, O is the centre of the circle. AB and CD intersect at E. ⌢

AC =⌢

BD and ∠ABC = 20°.

(a) Is it true that BE = CE? Explain your answer.

(b) Find ∠AEC.

(c) Determine whether ACOE is a cyclic quadrilateral.

26. In the figure, BDEF is a parallelogram. ADE and BCF are straight

lines.

If ∠ACB = 25°,

(a) find ∠DEF,

(b) determine whether A, E, F and C are concyclic.

27. In the figure, ABCD is a square, where AP : PB = BQ : QC = CR : RD

= DS : SA = 1 : 2. PR and SQ intersect at T.

(a) Is it true that PQ = QR = RS = SP? Explain your answer.

(b) Is it true that ∠SPQ = 90°? Explain your answer.

(c) Determine whether PQRS is a cyclic quadrilateral.

57

Answers


1. no 2. yes

3. yes 4. yes, 138°

5. yes, 42° 6. yes, 55°

7. yes 8. no

9. yes 10. yes

11. 40° 12. 47°

13. (a) yes (b) yes

14. (a) yes (b) no

(c) yes

15. (a) 120° (b) yes

16. (a) 90° (b) yes

17. (a) yes (b) 48°

18. (a) 84° (b) yes

19. (a) 135° (b) yes

20. (a) yes (b) yes

21. (a) ∠AGH = 70°, ∠DEH = 70°

(b) yes

22. (a) yes (b) yes

23. (a) yes

(b) ∠DCE = 65°, ∠DFE = 115°

24. (a) yes (b) 45°

25. (a) yes (b) 40°

(c) yes

26. (a) 25° (b) yes

27. (a) yes (b) yes

(c) yes

58

F4B: Chapter 11C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 9


(Video Teaching)

Book Example 10


(Video Teaching)

Book Example 11


(Video Teaching)



(Full Solution)


11C Level 1


Teacher’s

Signature

___________

( )


11C Level 2


Teacher’s

Signature

___________

( )


11C Multiple Choice


Signature

___________

( )


Self-Test


_________

59


Objective: To understand the basic properties of tangents to circles.

Basic Properties of Tangents to Circles

(a) If PQ is the tangent to the circle at point T, then PQ ⊥ OT.

[tangent ⊥ radius]

(b) If PQ ⊥ OT, then PTQ is the tangent to the circle at point T.

[converse of tangent ⊥ radius]



In the figure, TAP is the tangent to the circle at A.

Find a.

∠OAT = 90° (tangent ⊥ radius)

a + 60° = 90°

a = 30°

In the figure, TAP is the tangent to the circle at A.

Find x.

∠OAP = ( ) ( ) x + ( ) = ( )

x = In each of the following figures, TAP is the tangent to the circle at A. Find the unknowns. [Nos. 1–2]

1. 2. �Ex 11C: 1–9

BOP is a straight line.

In each of the following, determine whether AB is a tangent to the circle. If yes, put a tick ‘�’ in the box.

[Nos. 3–5]

3.

4.

5.

�

60


In the figure, P is a point on the circle. APB is a

straight line. Determine whether AB is the tangent

to the circle at P.

OP2 + AP2 = (152 + 202) cm2 = 625 cm2

OA2 = 252 cm2 = 625 cm2 ∵ OP2 + AP2 = OA2 ∴ ∠OPA = 90° (converse of Pyth. theorem) ∴ AB is the tangent to the circle at P.

(converse of tangent ⊥ radius)

In the figure, P is a point on the circle. APB is a

straight line. Determine whether AB is the tangent to

the circle at P.

OP2 + BP2 =

OB2 = ∵ ∴ ∠OPB = ( ) ∴ ( )

In each of the following figures, P is a point on the circle. APB is a straight line. Determine whether AB is

the tangent to the circle at P. [Nos. 6–7]

6. 7. �Ex 11C: 10–12


8. In the figure, CA is a diameter. PAQ is a straight line.

(a) Find ∠BAC.

(b) Determine whether PQ is the tangent to the circle at A.

61



� Consolidation Exercise 11C


Level 1

In each of the following figures, TA is the tangent to the circle at A. Find the unknowns. [Nos. 1–12]

1. 2. 3.

4. 5. 6.

OBT is a straight line. BCD and OCA are straight lines. OBT is a straight line.

7. 8. 9.

AOB is a straight line. ABCD is a square. OBT is a straight line.

10. 11. 12.

BOA and OCT are straight lines. BOA and BCT are straight lines. BOA, BCD, DAT and OET

are straight lines.

O

A

B

T e

O

A T

B x

y 32°

O

A

C

B

T h

h

T

O

A

B C D

f 25°

O

A T

b c

B

A

B D

C

T

O

a

b

O

A

2 cm

k cm

6 cm

C

B

T

O

A T

B d

O

A

B

D

E

T

C y cm

8 cm 5 cm

13 cm O

A

B

C

T

30°

x cm

32 cm

O

A

70°T

B

b

a

O

A T

a

a

62

In each of the following, T is a point on the circle and PTR is a straight line. Determine whether PR is a

tangent to the circle. [Nos. 13–18]

13. 14. 15.

OSQ is a straight line. ABC, DBT and AOT are

straight lines.

16. 17. 18.

ABT and OBCR are straight lines. AOT and ABR are straight lines. OQR is a straight line.

19. In the figure, BT touches the circle at T. AT is a diameter of the circle.

ACB is a straight line and ∠COT = 68°.

(a) Find ∠BAT.

(b) Find ∠ABT.

20. In the figure, PR touches the circle at T. DBC is a straight line.

∠DBA = 70° and ∠CTR = 65°.

(a) Find ∠ATC.

(b) Find ∠ATO.

21. In the figure, PR touches the circle at T. AT is a diameter of the circle.

If ∠CTR + ∠ABC = 180°, find ∠ATC.

O

T

S

Q P R

50°

140°

O

T

A

D

B

C

R P 65°

25°

20°

O

T

A

P R

B C

36°

27° O

T

Q

R P 3a

2a

a

O

T P R

62°

28°

Q

O

A

C

B T

68°

O

D B

C

A

P R T

65°

70°

O

A

B

C

P T

R

O

T

P

R

A

B

12 cm

5 cm

13 cm

63

22. In the figure, TA is the tangent to the circle at A. ∠AOB = 130° and

∠BTA = 54°.

(a) Find ∠BAT.

(b) Find ∠ABT.

23. In the figure, TB is the tangent to the circle at T. If O, A, B and T are

concyclic,

(a) find ∠OAB,

(b) is AB the tangent to the circle at A?

24. The figure shows two concentric circles with a common centre O. BC

touches the larger circle at T. A is a point on the smaller circle such

that ∠BTA = ∠AOT. Determine whether AT is a tangent to the smaller

circle.

Level 2

25. In the figure, OABC is a parallelogram. BC touches the circle at B.

Find ∠BCO.

26. In the figure, T is a point on the circle. OAP and OBR are straight

lines. AP = BR and T is the mid-point of PR. Is PR a tangent to the

circle?

O A

B C

O

T

A

P

B

R

O

A

B T

O

A

B T

C

O

B

A T

130°

54°

64

27. In the figure, CB is a diameter of the circle. AD touches the circle at T.

CB is produced to meet AD at A and ∠CAT = 46°.

(a) Find ∠ACT.

(b) Find ∠CTD.

28. In the figure, BT touches the circle at T. ACB is a straight line. AB ⊥

BT and ∠BAT = 27°.

(a) Find ∠ATO.

(b) Find ∠ACT.

29. In the figure, CD is the tangent to the circle at T. AB // CD and

∠OTB = 25°.

(a) Find ∠ABT.

(b) Find ∠BAO.

30. In the figure, RS touches the circle at T. OQ // RS and ∠PTR = 70°.

(a) Find ∠OQT.

(b) Find ∠POQ.

31. In the figure, DE is the tangent to the circle at C. OA // CB and

∠ABC = 110°.

(a) Find ∠AOC.

(b) Find ∠BCE.

O

P

Q

R T

S 70°

O

A

B

E C

D

110°

O

A

C

B T

27°

O

C

B

A T

D

46°

O

A B

C T

D

25°

65

32. In the figure, AC is the tangent to the circle at B. OA = 18 cm, AC = 27

cm and AB : BC = 4 : 5.

(a) Find the length of OC.

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

(b) Is △OAC a right-angled triangle? Explain your answer.

33. In the figure, BC is a diameter of the circle. DT touches the circle at T.

BC is produced to meet DT at D. A is a point on the circle such that

AB = 6 cm and AC = 8 cm. If the areas of △ABC and △ODT are

equal, find the length of DT.

34. In the figure, CD is a diameter of the circle. AB touches the circle at B.

DC is produced to meet BA at A. If

⌢

CB :⌢

BD = 2 : 7, find ∠BAD.

35. In the figure, FG is the tangent to the circle at E. ∠BCE = 58°,

∠CDE = 90° and AB // EC. Find ∠AEF.

36. In the figure, ABCD is a square. DC touches the circle at E. If the

radius of the circle is 5 cm, find the area of ABCD.

O

A

B

C

D T

6 cm

8 cm

O

B

A C

D

O

A B

C D E

B

C

A

F E

58° D

G

O

A C B

27 cm

18 cm

66

37. In the figure, TC touches the circle at T. B is the mid-point of TC. A is

a point on the circle such that BA // TO. Find ∠CAB.

38. In the figure, O and B trisect AC. TC is the tangent to the circle at T

and DB // TC.

(a) Show that △ADB ≅ △OTC.

(b) If TC = 34 cm, find the length of DA.

39. In the figure, CD touches the circle at T. ABC is a straight line such

that AC ⊥ CD, AC = 25 cm and CT = 10 cm. Find the radius of the

circle.

40. In the figure, CE touches the circle at T. AT is a diameter of the circle.

B is a point on the straight line ADE such that OB // CE. Is B the

circumcentre of △ATE? Explain your answer.

41. The figure shows the right-angled triangle ABC, where ∠ABC = 90°.

AB = 8 cm and BC = 15 cm. The inscribed circle of △ABC touches

AB, BC and AC at X, Y and Z respectively. Let r cm be the radius of

the inscribed circle.

(a) Express the areas of △OBC, △OAB and △OAC in terms of r.

(b) Hence, find the value of r.

O

T

A

B C

A

B

C D

25 cm

10 cm T

O

A

B

C

D

E T

O

C

A B

Z

Y

X

15 cm

8 cm

O A

D T

C B

67

Answers


1. 45° 2. b = 60°, c = 30°

3. a = 20°, b = 140° 4. 45°

5. 65° 6. x = 61°, y = 29°

7. 60° 8. a = 45°, b = 45°

9. 30° 10. 4

11. 3 12. 4

13. yes 14. yes

15. no 16. yes

17. no 18. yes

19. (a) 34° (b) 56°

20. (a) 70° (b) 45°

21. 45°

22. (a) 65° (b) 61°

23. (a) 90° (b) yes

24. yes 25. 45°

26. yes

27. (a) 22° (b) 68°

28. (a) 27° (b) 117°

29. (a) 65° (b) 40°

30. (a) 45° (b) 130°

31. (a) 140° (b) 50°

32. (a) 59 cm (b) yes

33. 9.6 cm 34. 50°

35. 32° 36. 64 cm2

37. 60° 38. (b) 4 cm

39. 14.5 cm 40. yes

41. (a) area of △OBC =2

15rcm2,

area of △OAB = 4r cm2,

area of △OAC =2

17rcm2

(b) 3

68

F4B: Chapter 11D

Date Task Progress

Lesson Worksheet


(Full Solution)

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)



(Full Solution)


11D Level 1


Teacher’s

Signature

___________

( )


11D Level 2


Teacher’s

Signature

___________

( )


11D Multiple Choice


Signature

___________

( )


Self-Test


_________

69


Objective: To understand the properties of the tangents of a circle from an external point.

Tangents from an External Point

If two tangents drawn from an external point T touch the circle at

points P and Q, then

(a) TP = TQ,

(b) ∠TOP = ∠TOQ,

(c) ∠OTP = ∠OTQ.

[tangent properties]



In the figure, TA and TB are the tangents to the

circle at A and B respectively. Find the unknowns.

∠OTA = ∠OTB (tangent properties)

a = 30°

TB = TA (tangent properties)

b = 9

In the figure, TA and TB are the tangents to the circle

at A and B respectively. Find the unknowns.

∠TOB = ∠ ( )

=

TA = ( )

In each of the following figures, TA and TB are the tangents to the circle at A and B respectively. Find the

unknowns. [Nos. 1–6] �Ex 11D: 1–6

1. 2.

BTP is a straight line.

�

Consider the angle

sum of △ OAT. Consider ∠BAT.

70

3. 4.

5. 6.


7. In the figure, O is the centre of the inscribed circle of △ABC.

The circle touches AB, BC and CA at P, Q and R respectively.

If ∠OAC = 30° and ∠ACB = 42°, find x.

Consider the angle

sum of △ ABT.

Consider ∠BAT

and ∠ABT.

Consider the angle

sum of OATB.

Consider ∠BAC

and ∠ABT.

71



� Consolidation Exercise 11D


Level 1

In each of the following figures, TA and TB touch the circle at A and B respectively. Find the unknowns.

[Nos. 1–9]

1. 2. 3.

4. 5. 6.

7. 8. 9.

SOT is a straight line. SOT is a straight line.

In each of the following figures, two circles with centres A and B touch each other at P. Find the unknowns.

[Nos. 10–13]

10. 11.

66°

b

O

B

A

T

28°

f O

A

B

T

S

56°d

O A

BT

a

20°O

B

T

A

5 cmB

P

A

r cm 8 cm

BP A

3 cm

2 cm

x cm

72°

h

A

B

C

T

32°

g

O

A

B

T

S

52°

e

B

A

T

67°

c

B

A

T

9 cm x cm

A

B

T

72

12. 13.

14. In the figure, a circle is inscribed in △ABC, where P, Q and R are

points of contact. If AC = 13 cm, BC = 15 cm and BP = 8 cm, find the

length of AP.


points of contact. If AP = 4 cm, BP = 9 cm and BC = 14 cm, find the

length of AC.

16. In the figure, PA, AB and RB are the tangents to the circle at P, Q and

R respectively. If ∠APQ = 58° and ∠QBR = 76°,

(a) find ∠PQR,

(b) find ∠PSR.


If OB = OC and ∠BAC = 48°, find ∠ACO.

18. In the figure, TP and TS touch the circle at P and Q respectively. POS

is a straight line and ∠SOT = 123°.

(a) Find ∠OTP.

(b) Find ∠OST.

3 cm

7 cm B

P A

r cm

x cm6 cm

A

P

B

10 cm

13 cm

15 cm

A

B C

P R

Q

8 cm

A B

P R

S

58°

Q

76°

A

B

48°

C

O

T

P O

Q

123°

S

B C

P R

Q

9 cm

A

14 cm

4 cm

73

19. In the figure, two circles intersect at two points P and Q. R and S are

points on the larger circle such that PS and QR are the tangents to the

smaller circle at P and Q respectively. PS and QR intersect at T. If

∠PSR = 43°, find ∠PTQ.

20. In the figure, two circles with centres A and B touch each other

internally at P, where A lies on the smaller circle. M is a point on the

smaller circle and PM produced meets the larger circle at Q. If QM = 6

cm, find the length of PM.

Level 2

21. In the figure, TP and TQ are the tangents to the circle at P and Q

respectively. PB and QA are diameters of the circle. If ∠PTQ = 56°,

find ∠AOB.


If ∠OCB = 40°, find ∠AOB.

23. In the figure, TA and TB are the tangents to the circle at A and B

respectively. If ∠APB = 115°, find ∠ATB.

24. In the figure, AB, BQ and AC are the tangents to the circle at P, Q and

R respectively. If ∠PAR = 60° and ∠PBQ = 80°, find ∠CRQ.

T

P

S

Q

R

43°

A

Q

P

M

6 cm

B

115°

T

B

P

A

A

R Q

60° P

80°

C

B

56° T

B

O

A

Q

P

A

B

O

C 40°

74


respectively. AB is a chord such that BA // QP and QA // TP. QA and

BP intersect at R. If ∠ARP = 80°, find ∠PTQ.

26. In the figure, TA and TB are the tangents to the circle at A and B

respectively. SOT is a straight line and ∠SAB = 54°.

(a) Find ∠BOT.

(b) Find ∠ABT.


respectively. TOR is a straight line and ∠PTO = 45°. Find ∠TQR.


points of contact. If the perimeter of △ABC is 36 cm and CR = 8 cm,

find the length of AB.

29. In the figure, AD, BC and CD are the tangents to the circle at A, B and

E respectively. AOB is a straight line. The radius of the circle is 12 cm

and BC = 16 cm.

(a) Find the length of AD.

(b) Find the area of the quadrilateral ABCD.

30. In the figure, a circle is inscribed in the trapezium ABCD, where P, Q,

R and S are points of contact. AP = 9 cm, BP = 12 cm and QC = 3 cm.

(a) Find the diameter of the circle.

(b) Find the perimeter of the trapezium ABCD.

A

B

T Q

P

80°

R

B

O

A

T

S 54°

Q

O P

T

R

45°

B

O

A

C

D

E 12 cm

16 cm

D

A

C

B P 9 cm

3 cm

Q

R

S

12 cm

B

P

A

C Q

R

8 cm

75

31. In the figure, TP, TQ and AB are the tangents to the circle at P, Q and

R respectively, where A lies on TP and B lies on TQ. AT = 29 cm and

BT = 21 cm. The perimeter of △TAB is 70 cm.

(a) Find the length of AP.

(b) Find the radius of the circle.

32. In the figure, two circles touch each other internally at P. TP and TQ

are the tangents to the smaller circle at P and Q respectively. TR is the

tangent to the larger circle at R. If ∠PTR = 100°, find ∠PQR.


externally at T. PQ is a common tangent to the circles. If ∠BTQ = 40°,

find ∠ATP.

34. In the figure, two circles touch each other externally at T. PQ is a

common tangent to the circles. QT produced meets the larger circle at

R. If ∠OQT = 35°, find ∠PRQ.

35. In the figure, two circles C1 and C2 with centres A and B respectively

touch each other externally at P. AT is the tangent to C2 at T, where

AT = 15 cm. If the radius of C1 is 9 cm, find the radius of C2.

P

Q

A

29 cm

B T

R

21 cm

15 cm

9 cm A

P B

T

C1

C2

A

P B

T

Q

40°

P

OT

Q

R

35°

P

Q

T

R

76


internally at P. AQ is the tangent to the smaller circle at T, where

AT = 6 cm. If the radius of the larger circle is 18 cm, find the radius of

the smaller circle.


externally at T. PQ is a common tangent to the circles. If the radius of

the larger circle is 4 cm and PQ = 4 cm, find the length of AB.

38. In the figure, the centre A of a circle lies on another circle with centre

B. PQ is a common tangent to the circles. If the radius of the circle

with centre B is 3 cm, find the diameter PR of the circle with centre A.

39. In the figure, a circle with centre O is inscribed in the sector TAB. If

∠ATB = 60° and the radius of the sector is 3 cm, find the radius of the

circle.


internally at P. Q is a point on the larger circle such that ∠QAB = 90°.

If the radius of the smaller circle is 7 cm and BQ = 13 cm, find the

radius of the larger circle.

7 cm A

P

13 cm

B

Q

A P

B

T

Q 4 cm

4 cm

A

B

R

Q

P

3 cm

18 cm

A

T 6 cm

B

P

Q

3 cm

A

B

O

T 60°

77

Answers

Consolidation Exercise 11D

1. 9 2. 70°

3. 24° 4. 113°

5. 34° 6. 116°

7. 118° 8. 26°

9. 36° 10. 5

11. 3 12. 4

13. 10 14. 6 cm

15. 9 cm

16. (a) 70° (b) 110°

17. 33°

18. (a) 33° (b) 24°

19. 94° 20. 6 cm

21. 124° 22. 130°

23. 50° 24. 70°

25. 100°

26. (a) 72° (b) 72°

27. 112.5° 28. 10 cm

29. (a) 9 cm (b) 300 cm2

30. (a) 12 cm (b) 56 cm

31. (a) 6 cm (b) 14 cm

32. 130° 33. 50°

34. 35° 35. 8 cm

36. 8 cm 37. 5 cm

38. 4 cm 39. 1 cm

40. 12 cm

78

F4B: Chapter 11E

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 17


(Video Teaching)

Book Example 18


(Video Teaching)

Book Example 19


(Video Teaching)



(Full Solution)


11E Level 1


Teacher’s

Signature

___________

( )


11E Level 2


Teacher’s

Signature

___________

( )


11E Multiple Choice


Signature

___________

( )


Self-Test


_________

79


Objective: To understand the angles in the alternate segments.

Angle in the Alternate Segment

If PQ is the tangent to the circle at point A,

then x = y.

[∠ in alt. segment]

In each of the following, AB is the tangent to the circle at X. Write down the angle in the alternate segment

corresponding to x. [Nos. 1–3]

1. 2. 3.

∠ ∠ ∠


In the figure, AB is the

tangent to the circle at X.

∠XYZ = 65° and

∠XZY = 32°. Find a.

In the figure, PQ is the

tangent to the circle at A.

∠CAP = 80° and

∠BAQ = 55°. Find b.

a = ∠XZY (∠ in alt. segment)

= 32°

b = ∠ ( )

=

In each of the following figures, PQ is the tangent to the circle at A. Find the unknowns. [Nos. 4–5]

4. 5. �Ex 11E: 1–8

∠BAQ = ∠__________ ( )

= __________

�

BOC is a diameter.

80

If x = y,

then PQ is the tangent to the circle at point A.

[converse of ∠ in alt. segment]

Note: If x ≠ y, then PQ is NOT the tangent to the circle at point A.

Instant Example 2

Instant Practice 2

Determine whether MN in

the figure is the tangent to

the circle at A.

Determine whether MN

in the figure is the

tangent to the circle at A. ∵ ∠ABC = ∠CAM = 40° ∴ MN is the tangent to the circle at A.

(converse of ∠ in alt. segment)

∵ ∠ = ∠ = ∴

( )

In each of the following figures, determine whether MN is the tangent to the circle at A. [Nos. 6–7]

6. 7. �Ex 11E: 9, 10

BOC is a diameter.


8. In the figure, QS is the tangent to the circle at R. QPT is a straight

line. ∠PQR = 30° and ∠TRS = 62°. Find x.

81



� Consolidation Exercise 11E

Level 1

In each of the following figures, AB is the tangent to the circle at C. Find the unknowns. [Nos. 1–12]

1. 2. 3.

FG is the tangent to the

circle at F.

4. 5. 6.

AD is the tangent to the DE is a diameter of

circle at E. the circle.

7. 8. 9.

DEB is a straight line. AED is a straight line. AED is a straight line and

ED

is a diameter of the circle.

10. 11. 12.

CEF is a straight line. ADF is the angle bisector of

∠EFC.

C

A

B

D

E

F G

c

35° 40°

A B C

D

E F

m

70°

35°

n

A B C

D

E F

q 105°

p

C A B

D

E

F m

140°

C A B

D

E

f

54°

C A B

D

E

F e

64° C

A

B

D

E d

63°

C A B

D

E

h

28°

C A B

D

E

k 32° C

A B

D

E

g

80°

C

A

B

D

E

b

40°

C A B

D

E

a

25°

82

13. In the figure, BF and DF are the tangents to the circle at C and E

respectively. AE = CE and ∠CEF = 52°.

(a) Find ∠ACE.

(b) Find ∠AEC.

14. In the figure, D is a point on the circle. CDE is a straight line.

∠ABD = 113° and ∠ADC = 67°. Determine whether CE is the tangent

to the circle at D.

15. In the figure, C is a point on the circle. BED is a straight line. A is a

point on the major arc BE. BC = DC, ∠EAB = 74° and ∠EBC = 24°.

Determine whether DC is the tangent to the circle at C.

16. In the figure, C is a point on the circle. EA // CB, CA = CB, ∠ECD =

40° and ∠ABC = 70°. Determine whether DC is the tangent to the

circle at C.

17. In the figure, CE touches the circle at D. AB // CE and ∠ADB = 60°.

(a) Find ∠ABD.

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

D

A

B C E

67°

113°

C

A

B

D

E

24°

74°

C

A B

D

E

40°

70°

D

A B

C E

60°

E

A

B

C

D F 52°

83

18. In the figure, the two circles touch each other internally at A. AF is the

common tangent to the two circles. BCA and DEA are straight lines.

Let ∠BAF = x.

(a) Express ∠AEC and ∠ADB in terms of x.

(b) Determine whether BD is parallel to CE.

19. In the figure, CD touches the circle at D. ABC is a straight line.

CB = 3 cm and BA = 9 cm.

(a) Is it true that △BCD ~ △DCA? Explain your answer.

(b) Find the length of CD.

20. In the figure, BD is the tangent to the circle at C. CA // DE, CA = 8

cm and DE = 2 cm.

(a) Name a triangle which is similar to △ACE.

(b) Find the length of CE.

Level 2

21. In the figure, CE and GF are the tangents to the circle at D and F

respectively. BA // DF and ∠BDC = 28°. Find ∠AFG.

22. In the figure, AB and FG are the tangents to the circle at B and F

respectively. BE and CF intersect at D. ∠BDF = 80° and ∠EFG = 64°.

Find ∠ABC.

A

B

C D

E

F

D

A

B

C

3 cm

9 cm

C

A

B D

E

8 cm

2 cm

D

A

B

C E

F

G

28°

F

A

B

C

D

E

G

80°

64°

84

23. In the figure, AC and EG are the tangents to the circle at B and F

respectively. BD // EG and ∠DFG = 65°.

(a) Find ∠ABD.

(b) Is AC parallel to DF? Explain your answer.

24. In the figure, DF is the tangent to the circle at E. AB is a diameter of

the circle. AF and BE intersect at C. ∠AED = 35° and ∠BCF = 76°.

(a) Find ∠BAC.

(b) Find ∠AFE.

25. In the figure, the two circles touch each other externally at C. ACE

and BCD are straight lines. Determine whether AB is parallel to DE.

26. In the figure, CD touches the circle at D. ABC is a straight line.

AB = 10.5 cm and CD = 5 cm.

(a) Name a triangle which is similar to △ACD.

(b) Find the length of CB.

27. In the figure, CG and EG are the tangents to the circle at D and F

respectively. AB // DF, ∠ADC = 51° and ∠ADB = 27°.

(a) Find ∠FDG.

(b) Determine whether BD is parallel to EG.

A

B

C

D

E

E

A

B

C

D F

76° 35°

D

A

B

C

10.5 cm

5 cm

G E F

C

D

A

B

65°

C

B

A

D

E

F

G 51°

27°

85

28. In the figure, EA and EG are the tangents to the circle at B and F

respectively. BC = CD, ∠BCD = 130° and ∠CDF = 83°.

(a) Find ∠ABC.

(b) Find ∠AEG.

29. In the figure, the two circles intersect at two points C and D. A and B

are two points on the larger circle such that AC and BD are produced

to meet at a point E on the smaller circle. If BA // FE, determine

whether FE is the tangent to the smaller circle at E.

30. In the figure, the two circles intersect at two points B and C. A is a

point on the larger circle. AB produced intersects the smaller circle at

D. The tangent to the larger circle at C intersects the smaller circle at

another point E. Is AC is parallel to DE? Explain your answer.

31. In the figure, AC, CE and EG are the tangents to the circle at B, D and

F respectively. ∠ACE = 84° and

⌢

BD :⌢

DF = 6 : 7.

(a) Find ∠FBD.

(b) Find ∠DEF.

32. In the figure, the two circles intersect at two points C and D. The

tangents at C and D to the larger circle intersect the smaller circle at B

and A respectively. AC and BD are produced to meet at E on the larger

circle. If ∠AEB = 30°,

(a) find ∠CBA,

(b) find ∠EBC.

A

B

C

D

E

F

A

B

C

D

E

D

A

B

C E

F

G

84°

A

B C

D

E

30°

F

A

B

C D

E G

130° 83°

86

33. In the figure, AC touches the circle at C. ABD is a straight line.

AB = BC = 1 cm and AC = x cm.

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

(b) It is given that BD = DC.

(i) Express the length of AD in terms of x.

(ii) Find the value of x.


34. In the figure, CD touches the circle at D. ABC is a straight line. AB is

a diameter of the circle. BC = 2 cm, CD = 4 cm, BD = x cm and the

radius of the circle is r cm.

(a) Find the value of r.

(b) Find the value of x.


35. In the figure, DF is the tangent to the circle at E. AB // DF and

CA // EB. BCD is a straight line. ∠BDF = 42° and ∠CED = θ.

(a) Express ∠ABE in terms of θ.

(b) Hence, find θ.

36. In the figure, the two circles intersect at two points A and C. The

tangent to the larger circle at B intersects the smaller circle at D and E.

(a) Name an angle which is equal to ∠CBD.

(b) Name an angle which is equal to ∠CDB.

(c) If ∠BAE = 60°, find ∠BCD.

37. In the figure, the two circles intersect at two points A and E. BD

touches the two circles at B and D. AD intersects the larger circle at C.

DE produced intersects the larger circle at F. Let O be the centre of

the larger circle.

(a) Determine whether FC is parallel to BD.

(b) If C is the centre of the smaller circle and CD = 4 cm, find the

radius of the larger circle.

A

B

D C

E

D

A

B

C

2 cm

4 cm

x cm

C A

B

D

1 cm 1 cm

x cm

A

B

C

D

E

F

O

E

A B

C

D F 42° θ

87

Answers

Consolidation Exercise 11E

1. 25° 2. 70°

3. 75° 4. 63°

5. 58° 6. 36°

7. 40° 8. 48°

9. 29° 10. 140°

11. m = 70°, n = 75° 12. p = 25°, q = 25°

13. (a) 52° (b) 76°

14. yes 15. no

16. yes

17. (a) 60° (b) yes

18. (a) ∠AEC = x, ∠ADB = x

(b) yes

19. (a) yes (b) 6 cm

20. (a) △CED (b) 4 cm

21. 28° 22. 36°

23. (a) 50° (b) no

24. (a) 41° (b) 21°

25. yes

26. (a) △DCB (b) 2 cm

27. (a) 51° (b) yes

28. (a) 25° (b) 64°

29. yes 30. yes

31. (a) 56° (b) 68°

32. (a) 30° (b) 45°

33. (a) yes

(b) (i) (x + 1) cm

(ii) 2

51+

34. (a) 3 (b) 5

56

35. (a) 42° + θ (b) 32°

36. (a) ∠CAB (b) ∠CAE

(c) 120°

37. (a) yes (b) 6 cm

88

F4B: Chapter 11F

Date Task Progress

Book Example 20


(Video Teaching)

Book Example 21


(Video Teaching)

Book Example 22


(Video Teaching)

Book Example 23


(Video Teaching)



(Full Solution)


11F Level 1


Teacher’s

Signature

___________

( )


11F Level 2


Teacher’s

Signature

___________

( )


11F Multiple Choice


Signature

___________

( )


Self-Test


_________

89

O A

C

D

C1

A

B

C

O

T

A

O

O

O

B

O

C

O D

O

O

A

B

C

A B

Q C

P

D R

O



� Consolidation Exercise 11F


Level 1

1. In the figure, A is a point outside a circle C1 with centre O. The circle

with diameter OA cuts C1 at C and D. Prove that AC and AD are the

tangents to C1 at C and D respectively.

2. In the figure, AB is a chord of the circle. C is the foot of perpendicular

from B to the tangent to the circle at A. Prove that AB is the angle

bisector of ∠OBC.

3. In the figure, AB is a diameter of the circle. C is another point on the

circle. TA and TC touch the circle at A and C respectively. Prove that

OT // BC.

4. In the figure, AB is a diameter of the circle. QR is the tangent to the

circle at P. C and D are two points on QR such that AC ⊥ QR and

BD ⊥ QR.

(a) Prove that AC // OP // BD.

(b) Prove that CP = DP.

5. In the figure, A and B are two points on the circle. AD is the tangent to

the circle at A and ∠BOD = 90°. AB and OD intersect at C. Prove that

AD = CD.

90

A B

C

D

E

F

O

A

B

C

D

P Q

R

A

B

D E

O C

6. In the figure, �PQR is an isosceles triangle with PQ = PR and

∠QPR = 36°. QS is the angle bisector of ∠PQR, where SR // PQ.

Prove that P, Q, R and S are concyclic.

7. In the figure, AB is a diameter of the circle. C and D are another

points on the circle. AC produced and AD produced cut the tangent to

the circle at B at E and F respectively. Prove that CDFE is a cyclic

quadrilateral.

8. In the figure, two circles intersect at A and B. TC and TD are the

tangents to the smaller circle and the larger circle at C and D

respectively. CBD is a straight line. Prove that A, C, T and D are

concyclic.

9. In the figure, AB, BC and CD are the chords of the circle. P, Q and R

are the mid-points of AB, BC and CD respectively. Prove that

∠BPQ = ∠CRQ.

10. In the figure, the chords AE and BC are perpendicular to each other.

AD is a diameter of the circle. Prove that

⌢

CD =

⌢

BE .

P

O

Q

O

36° S

O

R

O

A B

C

D

T

91

A

C

B

D E

A

B

C D E

A

B

C

D

E

F G

P

Q

A B

C

D

E

F

O

A

B C D

E

A B

C

D E

F

11. In the figure, two circles touch each other internally at A. The tangent

to the smaller circle at D cuts the larger circle at B and C. AEB is a

diameter of the larger circle, where E is a point on the smaller circle.

Prove that AD is the angle bisector of ∠BAC.

12. In the figure, A, B and D are three points on the circle. The tangent to

the circle at B and AD produced intersect at C. E is a point on AD

such that BE bisects ∠ABD. Prove that CB = CE.

13. In the figure, two circles touch each other internally at A. PQ is the

tangent to the circles at A. B, C, D and G are points on the larger

circle. E and F are points on the smaller circle. AEB, AFC and DEFG

are straight lines.

(a) Prove that BC // DG.

(b) Prove that ∠BAD = ∠CAG.

14. In the figure, AB and CD are two equal chords of the circle. BA and

DC are produced to E and F respectively such that AE = CF.

(a) Prove that OE = OF.

(b) Prove that the perpendicular bisector of EF passes through the

centre O.

Level 2

15. In the figure, AB is the tangent to the circle at A and AB = AC. The

line joining B and C cuts the circle at D. E is the mid-point of the arc

AC. Prove that ABDE is a parallelogram.

16. In the figure, two equal circles intersect at B and D. ABC, AFD and

BFE are straight lines.

(a) Prove that AD = CD.

(b) Prove that �DEF is an isosceles triangle.

92

A

B

C

D

E

F P

A B

C

D

E

O

H

A

B

C

D

D

E

O

A

B

C D

E

H K

A

B

C

D E

O

A

B C

D

S

R

P

Q

T

17. In the figure, D and E are points on AB and BC respectively such that

PD ⊥ AB and PE ⊥ BC. BF is an altitude of �BDE.

(a) Prove that B, D, P and E are concyclic.

(b) Prove that ∠ABF = ∠PBE.

18. In the figure, A, B and C are three points on the circle. AD is a

diameter of the circle. E is a point on AB such that OE ⊥ AB. H is the

orthocentre of �ABC.

(a) Prove that BDCH is a parallelogram.

(b) Prove that OE =2

1CH.

19. In the figure, O is the centre of the larger circle. The centre C of the

smaller circle lies on the larger circle. The tangent to the smaller

circle at D cuts the larger circle at A and E. AB is a diameter of the

larger circle.

(a) Prove that �ABC ~ �CED.

(b) Let R and r be the radii of the larger and the smaller circles

respectively. Prove that CA � CE = 2Rr.

20. In the figure, ABCD is a cyclic quadrilateral. AC intersects BD at E

and AC ⊥ BD. H and K are the orthocentres of �ABD and �ABC

respectively. Prove that CDHK is a rhombus.

21. In the figure, ABCD is a square. ABD is a sector with centre A and

radius AB. T is a point on the arc BD. The tangent at T cuts BC and

CD at P and Q respectively. The diagonal BD cuts AP and AQ at S

and R respectively. Prove that P, Q, R and S are concyclic.

22. In the figure, A, B and C are three points on the circle. BD and CE are

two altitudes of �ABC. Prove that OA ⊥ ED.

93

A B

C

D

E

F O

M

A

B

C

D

E

C1

C2

F

A

D

C

B

E

F G

O

A

B

C

D

E

P

23. In the figure, AB is a diameter of the circle. E is another point on the

circle. AC, BD and CD are the tangents to the circle at A, B and E

respectively. AD and BC intersect at M. EM produced cuts AB at F.

(a) Prove that �ACM ~ �DBM.

Hence, prove that EF // CA.

(b) Prove that EM = MF.

24. In the figure, C1 and C2 are two circles. CA and CB are the tangents to

C2 at A and B respectively. AB is the tangent to C1 at B. C1 and C2

intersect at B and D. AD produced cuts C1 at E. AE and BC intersect at

F. Prove that BF = CF.

25. In the figure, the tangents to the circle at A and B intersect at D.

DEFG, AFC and AEB are straight lines. It is given that CB // GD.

(a) Prove that O, A, D and B are concyclic.

(b) Prove that A, F, B and D are concyclic.

(c) Prove that OF ⊥ DG.

26. In the figure, two circles touch each other externally at P. E is a point

on the common tangent to the circles at P. A straight line from E cuts

the larger circle at A and B. Another straight line from E cuts the

smaller circle at C and D.

(a) (i) By considering �AEP and �PEB, prove that

EP2 = EA × EB.

(ii) By considering �DEP and �PEC, prove that

EP2 = ED × EC.

(b) Using the results of (a), prove that �EAD ~ �ECB.

(c) Hence, prove that A, B, C and D are concyclic.

94

F4B: 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)


12A Level 1


Teacher’s

Signature

___________

( )


12A Level 2


Teacher’s

Signature

___________

( )


12A Multiple Choice


Signature

___________

( )


Self-Test


_________

95


Objective: To review Pythagoras’ theorem, trigonometric ratios, trigonometric ratios of special angles,

trigonometric identities and transformation (reflection and rotation).

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

Pythagoras’ Theorem

In each of the following, find the value of x. [Nos. 1–2] � Review Ex: 1

1. x2 = ( )2 + ( )2 2.

x = )()( +

= )(

=

Trigonometric Ratios

In each of the following, find the values of sin θ, cos θ and tan θ. [Nos. 3–4] � Review Ex: 2

3. sin θ =)(

)( 4.

cos θ =)(

)(

tan θ =)(

)(

Trigonometric Ratios of Special Angles

Find the values of the following expressions without using a calculator. [Nos. 5–7] � Review Ex: 3

5. tan 45° – cos 60° 6. sin 60° tan 30° 7. cos2 45° + sin 30°

= = =

Trigonometric Identities

Simplify the following expressions. [Nos. 8–11] � Review Ex: 4

8. sin θ cos (90° – θ) 9. θ

θ

cos

)90(tan −°

= =

7 6

x

x

12

5

12

37 35 θ

8

θ 15 7

tan (90° – θ) =θtan

1

tan θ =θ

θ

cos

sin

cos (90° – θ) = sin θ

96

10. )90(sin

sin1 2

θ

θ

−°

−= 11. 1 –

)90(tan

cos2

2

θ

θ

−°=

Transformation in the Coordinate Axis: Reflection

12. If P(4 , 2) is reflected in the x-axis to Q, 13. If A(–1 , 5) is reflected in the y-axis to B,

find the coordinates of Q. find the coordinates of B. � Review Ex: 5

The coordinates of Q

=

Transformation about the Origin: Rotation

14. If A(2 , 6) is rotated anticlockwise about the 15. If M(7 , –3) is rotated anticlockwise about the

origin O through 90° to B, find the coordinates origin O through 270° to N, find the

of B. coordinates of N. � Review Ex: 6

The coordinates of B

=

16. If S(–6 , 1) is rotated anticlockwise about the origin O through 180° to T, find the coordinates of T.


17. (a) Simplify)90(tan

sin1 2

θ

θ

−°

−.

(b) When θ = 60°, find the value of)90(tan

sin1 2

θ

θ

−°

−without using a calculator.

sin2 θ + cos2 θ = 1

sin (90° – θ) = cos θ

(m , n)

(m , –n)

x

y (2 , 6)

(___ , ___)

x

y

90°

O

(___ , ___) (7 , –3)

x

270°

O

y

(___ , ___)

(–6 , 1)

x

180° O

(m , n) (–m , n)

y

97

4B Lesson Worksheet 12.1A & B (Refer to Book 4B P.12.6)

Objective: To understand angles of rotation, quadrants and the meaning of trigonometric ratios of any angle.

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

Angles of Rotation Quadrants

In the figure, θ is called the angle of rotation.

The starting position OA is called the initial

side of θ. The final position OP is called the

terminal side of θ.

1. In each of the following, sketch the angle and write down the quadrant in which the angle lies.

(a) 150° (b) 300° (c) 400° (d) –30°

Quadrant_______ Quadrant_______ Quadrant_______ Quadrant_______

2. Determine the quadrant in which each of the following angles lies. �Ex 12A: 2

Angle 40° 130° 264° 392° –107° 294° –76°

Quadrant

Definitions of Trigonometric Ratios of Any Angle

sin θ =OP

Py

oflength

of coordinate-=

r

y,

cos θ =OP

Px

oflength

of coordinate-=

r

x,

tan θ =Px

Py

of coordinate-

of coordinate-=

x

y,

where r = 22 yx + .

In each of the following, find the values of sin θ, cos θ and tan θ. [Nos. 3–4] �Ex 12A: 6, 7

3. sin θ = 4.

cos θ =

tan θ =

x

y

O x

y

O x

y

O x

y

O

x

y

O

θ 5

P(–4 , 3)

x

y

O

P(–3, 7− )

θ

4

98


In the figure, the coordinates of P are (–7 , 24).

Find the values of sin θ, cos θ and tan θ.

OP = 22 24)7( +− = 25 ∴ sin θ =25

24

cos θ =25

7−

tan θ =7

24−

In the figure, the coordinates of P are (12 , –5).


OP = 22 )()( + = ∴ sin θ =

cos θ =

tan θ =

In each of the following, find the values of sin θ, cos θ and tan θ. [Nos. 5–8] �Ex 12A: 8

5. OP = 22 )()( + 6.

=

∴ sin θ =

cos θ =

tan θ =

7. 8.


9. (a) In the figure, find the value of x.

(b) Find the values of sin θ, cos θ and tan θ.

x

y

O

θ

P(–7 , 24)

x

y

O

θ

P(–20 , –21)

x

y

O

P(2 , 5− )

θ x

y

O

θ

P( 15− , –7)

x

y

O

θ

P(12 , –5)

x

y

O

P(–9 , 40) θ

x

y

O

P(x , 21− )

θ

5

Since P lies on the ( left /

right ) of the y-axis, x is

( positive / negative ).

99

4B Lesson Worksheet 12.1C, D & E (Refer to Book 4B P.12.11)

Objective: To understand the signs and the patterns of the values of trigonometric ratios, and to find the

trigonometric ratios of any angle using a calculator.

Signs of Trigonometric Ratios

The sign of a trigonometric ratio of an angle is determined by the

quadrant in which that angle lies.

A: The values of all trigonometric ratios are positive.

S: Only the values of sine ratios are positive.

T: Only the values of tangent ratios are positive.

C: Only the values of cosine ratios are positive.


Given that cos θ =17

8and θ lies in quadrant IV,

find the values of sin θ and tan θ.

Let P(x , y) be a point on the terminal side of θ and

OP = r.

Let x = 8 and r = 17. ◀ cos θ =r

x

17 = 228 y+ ◀ r = 22 yx +

172 = 64 + y2

y2 = 225

y = 15 (rejected) or –15 ◀ Since P lies in quadrant IV, ∴ sin θ =17

15− ◀ sin θ =

r

y

tan θ =8

15− ◀ tan θ =

x

y

Given that sin θ =13

12and θ lies in quadrant II, find

the values of cos θ and tan θ.

Let P(x , y) be a point on the terminal side of θ and

OP = r.

Let ( ) = ( ) and ( ) = ( ).

( ) = 22 )()( +

( )2 = ( ) + ( )

=

1. Given that tan θ =3

4and θ lies in quadrant III, 2. Given that sin θ =

25

7− , where 270° < θ < 360°,

find the values of sin θ and cos θ. find the values of cos θ and tan θ. �Ex 12A: 9–12

Let P(x , y) be a point on the terminal side of θ

and OP = r.

Let ( ) = ( ) and ( ) = ( ).

( ) = 22 )()( +

=

y is negative.

x

y

O

y

x O

θ

P(x , y)

r

y

x O

θ

P(x , y)

r

x

y

O

100

Patterns of the Values of Trigonometric Ratios

θθθθ 0°°°° 90°°°° 180°°°° 270°°°° 360°°°°

sin θθθθ 0 1 0 –1 0

cos θθθθ 1 0 –1 0 1

tan θθθθ 0 meaningless 0 meaningless 0

Find the values of the following expressions without using a calculator. [Nos. 3–8]

3. sin 90° + cos 180° 4. cos 270° – tan 360° 5. sin 270° × tan 180° �Ex 12A: 13

= ( ) + ( ) = =

=

6. cos 0° + sin 270° – tan 0° 7. °

°+°

360 cos

180 tan0 sin 8. (sin 360° – cos 180°)2

= = =

Finding Trigonometric Ratios of Any Angle Using a Calculator

Find the values of the following trigonometric ratios using a calculator. [Nos. 9–12]

(Give the answers correct to 3 decimal places.) �Ex 12A: 14, 15

9. cos 36° 10. tan (–194°)

= =

11. sin 145.6° 12. cos (–87.4°)

= =


13. It is given that cos θ =5

4− , where 0° < θ < 180°.

(a) Determine the quadrant in which θ lies.

(b) Find the values of sin θ and tan θ.

36 194

145.6

101


12 Basic Trigonometry


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

Level 1

Find θ in each of the following figures. [Nos. 1–4]

1. 2.

3. 4.

5. Write down the quadrant in which each of the following angles lies.

(a) 49° (b) –54°

(c) 190° (d) –199°

(e) 299° (f) 400°

6. In each of the following, determine the quadrant in which θ lies and the signs (+/–) of the

trigonometric ratios.

θ Quadrant Sign of sin θ Sign of cos θ Sign of tan θ

(a) 204°

(b) –192°

(c) 371°

(d) –47.2°

x

y

65°

O

θ

x

y

θ

O

–77°

x

y

133°

O

θ

x

y

θ

O

–148°

102

Write down the quadrant(s) in which θ lies in each of the following conditions. [Nos. 7–10]

7. (a) sin θ > 0 (b) cos θ < 0

8. (a) cos θ = 0.4 (b) tan θ > 1

9. (a) cos θ < 0 and tan θ > 0 (b) sin θ tan θ > 0

10. (a) tan θ = –2 and sin θ > 0 (b) cos θ = 0.7 and tan θ < 0

In each of the following figures, find the values of sin θ, cos θ and tan θ. [Nos. 11–16]

11. 12.

13. 14.

15. 16.


4− and θ lies in quadrant II, find the values of sin θ and cos θ.

18. Given that cos θ =2

1 and θ lies in quadrant IV, find the values of sin θ and tan θ.

19. Given that sin θ =13

5− and θ lies in quadrant III, find the values of cos θ and tan θ.

x

y

O

θ

P(– 3 , 6 )

x

y

θ

O

P(–6 , –2.5)

x

y

O

θ

P(2 , 21 )

x

y

θ

O

P(15 , –8)

17

x

y

O

θ

P(–9.6 , –2.8)10

x

y

θ

O

P(–3 , 4)

5

103


24− and 180° < θ < 270°, find the values of sin θ and tan θ.


3 and 90° < θ < 180°, find the values of cos θ and tan θ.


15− and 270° < θ < 360°, find the values of sin θ and cos θ.

23. Find the values of the following expressions without using a calculator.

(a) sin 180° + cos 180° (b) tan 360° – cos 270°

(c) sin 270° × cos 90° (d) °

°

90sin

0tan

24. Find the values of the following trigonometric ratios using a calculator.

(Give the answers correct to 3 significant figures.)

(a) sin 152° (b) tan 200°

(c) cos 420.5° (d) sin (–49.7°)

Level 2

Given that θ is an acute angle in quadrant I, find the quadrant in which each of the following angles lies.

[Nos. 25–28]

25. 180° – θ 26. 90° – θ

27. 270° + θ 28. 360° + θ

In each of the following figures, find the values of sin θ, cos θ and tan θ. [Nos. 29–34]

29. 30.

31. 32.

x

y

O

θ

P(–10 , y)

26

x

y

θ

O

P(x , 6)

10

x

y

O θ

P(x , –1.5)

1.7

x

y

O

θ

P(x , 2)

3

104

33. 34.


24− , where 90° < θ < 270°, find the values of sin θ and cos θ.


1, where 90° < θ < 360°, find the values of sin θ and tan θ.


8− and cos θ < 0, find the values of cos θ and tan θ.

38. Given that tan θ = 22− and sin θ > 0, find the values of sin θ and cos θ.


24, find the values of cos θ and tan θ.


3− , find the values of sin θ and cos θ.


1, find the value of sin2 θ + tan2 θ.


5− , find the value of sin θ cos θ.

43. Find the values of the following expressions without using a calculator.

(a) cos 0° + sin 90° – tan 180° (b) 2 sin 180° – cos2 270° × tan 110°

(c) °

°

50tan

0sin3– 4 cos 360° (d)

°−°

°−°

180cos360sin

270sin360tan

44. (a) If the polar coordinates of R are (4 , 60°), find the rectangular coordinates of R.

(b) P is a point lying on the terminal side of an angle θ. If the rectangular coordinates of P are

( 36 , –6), find the polar coordinates of P.

45. It is given that

i

i

25

267

−

−= a + bi, where a and b are real numbers with i = 1− .

� (a) Find the values of a and b.

(b) P(a , b) is a point on a rectangular coordinate plane and OP is the terminal side of an angle θ.


x

y

θ

O

P(– 32 , y)

4 x

y

θ

O

P(5 , y)

6

105

Answers


1. 212° 2. –227°

3. 283° 4. –295°

5. (a) quadrant I (b) quadrant IV

(c) quadrant III (d) quadrant II

(e) quadrant IV (f) quadrant I

6. (a) III, –, –, + (b) II, +, –, –

(c) I, +, +, + (d) IV, –, +, –

7. (a) quadrant I or II (b) quadrant II or III

8. (a) quadrant I or IV(b) quadrant I or III

9. (a) quadrant III (b) quadrant I or IV

10. (a) quadrant II

(b) quadrant IV

11. sin θ =5

4, cos θ =

5

3− , tan θ =

3

4−

12. sin θ =25

7− , cos θ =

25

24− , tan θ =

24

7

13. sin θ =17

8− , cos θ =

17

15, tan θ =

15

8−

14. sin θ =5

21, cos θ =

5

2, tan θ =

2

21

15. sin θ =13

5− , cos θ =

13

12− , tan θ =

12

5

16. sin θ =3

6, cos θ =

3

3− ,

tan θ =3

6− (or – 2 )

17. sin θ =5

4, cos θ =

5

3−

18. sin θ =2

3− , tan θ = 3−

19. cos θ =13

12− , tan θ =

12

5

20. sin θ =25

7− , tan θ =

24

7

21. cos θ =2

1− , tan θ = 3−

22. sin θ =17

15− , cos θ =

17

8

23. (a) –1 (b) 0

(c) 0 (d) 0

24. (a) 0.469 (b) 0.364

(c) 0.492 (d) –0.763

25. quadrant II

26. quadrant I

27. quadrant IV

28. quadrant I

29. sin θ =5

3, cos θ =

5

4− , tan θ =

4

3−

30. sin θ =3

2, cos θ =

3

5, tan θ =

5

2

31. sin θ =17

15− , cos θ =

17

8, tan θ =

8

15−

32. sin θ =13

12, cos θ =

13

5− , tan θ =

5

12−

33. sin θ =6

11− , cos θ =

6

5, tan θ =

5

11−

34. sin θ =2

1− , cos θ =

2

3− , tan θ =

3

1

35. sin θ =25

24, cos θ =

25

7−

36. sin θ =4

15− , tan θ = – 15

37. cos θ =17

15− , tan θ =

15

8

38. sin θ =3

22, cos θ =

3

1−

39. quadrant I: cos θ =25

7, tan θ =

7

24;

quadrant II: cos θ =25

7− , tan θ =

7

24−

40. quadrant II: sin θ =5

3, cos θ =

5

4− ;

quadrant IV: sin θ =5

3− , cos θ =

5

4

41. 9

80 42.

169

60−

43. (a) 2 (b) 0

(c) –4 (d) 1

44. (a) (2 , 32 ) (b) (12 , 330°) 45. (a) a = 3, b = –4

(b) sin θ =5

4− , cos θ =

5

3, tan θ =

3

4−

106

F4B: 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)


12B Level 1


Teacher’s

Signature

___________

( )


12B Level 2


Teacher’s

Signature

___________

( )

107


12B Multiple Choice


Signature

___________

( )


Self-Test


_________

108


Objective: To use trigonometric identities to simplify expressions.

Relations among Trigonometric Ratios of the Same Angle

For any angle θ,

(a) sin2 θ + cos2 θ = 1 (b) tan θ =θ

θ

cos

sin


Simplifyθ

θ

2cos1

sin

−.

θ

θ

2cos1

sin

−=

θ

θ

2sin

sin

=θsin

1

Simplify1cos

tan2

−θ

θ.

1cos

tan2

−θ

θ=

)(

)(

sinθ

=

Simplify the following expressions. [Nos. 1–6] �Ex 12B: 1–4

1. tan θ (1 – sin2 θ) = 2. 1 + tan2 θ =

3. 1tan

cossin

−

−

θ

θθ= 4.

θθθ

θθ

sincoscos

sincos1 2

−

−−=

5. (sin θ + 1)(sin θ – 1) = 6. θ

θ

θ

2sin

tan

sin1−

=

� sin2 θ + cos2 θ = 1

1 – cos2 θ = sin2 θ sin2 θ + cos2 θ = 1

cos2 θ – 1 =

(a + b)(a – b) ≡ a2 – b2

109


Given that tan θ =5

3, find the value of

θ

θθ

sin

cos3sin2 +.

θ

θθ

sin

cos3sin2 +=

θ

θ

θ

θθ

cos

sincos

cos3sin2 +

=θ

θ

tan

3tan2 +

=

5

3

35

32 +

= 7

Given that tan θ =4


θθ

θθ

cossin2

cos2sin3

+

−.

θθ

θθ

cossin2

cos2sin3

+

−=

)(

cossin2

)(

cos2sin3

θθ

θθ

+

−

=)()(

)()(

+

−

=


2, find the value of 8. Given that cos θ =

6


θ

θ

θ tan

cos

sin

1− .

θ

θ

2

2

cos1

tan

−. �Ex 12B: 5–8



3− , find the value of

)sin1)(sin1(

1

θθ +−.

� Divide the

numerator and

the denominator

by cos θ.

Express

θ

θ

θ tan

cos

sin

1−

in terms of

sin θ only.

Express

θ

θ

2

2

cos1

tan

−

in

terms of cos θ

only.

110


Objective: To simplify trigonometric ratios of n ⋅ 90° ± θ to those of θ.

Review: Trigonometric Ratios of Special Angles

Simplify the following expressions. [Nos. 1–4]

(Leave the radical sign ‘√’ in the answers if necessary.)

1. sin 30° – cos 45° = ( ) – ( ) 2. tan 45° + cos 60° = ( ) + ( )

= =

3. tan 30° × sin 60° = 4. °

°

30cos

60tan=

Simplifying Trigonometric Ratios of n ⋅⋅⋅⋅ 90°°°° ±±±± θθθθ to Those of θθθθ

(a) –θ, 180° ± θ, 360° ± θ (b) 90° ± θ, 270° ± θ


Simplify sin (270° + θ) + cos (–θ).

sin (270° + θ) + cos (–θ)

= –cos θ + cos θ

= 0

Simplify cos (360° + θ) – sin (90° – θ).

cos (360° + θ) – sin (90° – θ)

= ( ) – ( )

=


5. )180(cos

)270(sin

θ

θ

−°

−° 6. sin (90° + θ) + cos (360° – θ) �Ex 12B: 9–18

= )(

)(

=

=

7. tan (180° + θ) ⋅ tan (90° – θ) 8. tan (360° – θ) ⋅ cos (–θ)

= =

111

9. )(tan

)180(sin

θ

θ

−

−° 10.

)180(cos

)270(tan

θ

θ

+°

+°

= =


Without using a calculator, find the value of

sin 240°.

sin 240° = sin (180° + 60°)

= –sin 60°

=2

3−

Without using a calculator, find the value of

cos 135°.

cos 135° = cos [180° – ( )]

=

Without using a calculator, find the values of the following trigonometric ratios. [Nos. 11–14]


11. tan 120° = tan [180° – ( )] 12. sin 300° = �Ex 12B: 19

=

13. cos 225° = 14. tan 210° =


15. Simplify)270(tan)180(sin

)270(cos22 2

θθ

θ

−°+°

−°−.

112




Level 1


1. θ

θθ

2sin1

cossin

− 2.

θθ22 tan

1

sin

1−

3. θ

θθ

tan

cossin + 4. θ

θθtan

cossin

1−

5. )sin1(tancos

1θθ

θ−

+ 6.

θθ

θ

sintan

cos1

−

−


3, find the value of 2 sin θ + 3 cos2 θ.

8. Given that cos θ = –0.4, find the value of

θθ

θθ

tan3sin

tansin2

+

−.

9. Given that tan θ = –2, find the value of

θθ

θθ

22 cossin3

cossin2

−.

10. Complete the following table.

α sin α cos α tan α

(a) 180° + θ –cos θ

(b) –θ –tan θ

(c) 90° + θ cos θ

(d) 270° – θ


11. tan (180° + θ) – tan θ 12. cos (180° + θ) + cos θ

13. sin (90° + θ) – cos (–θ) 14. sin (–θ) + cos (90° – θ)

15. )270( cos

1

)90( cos

1

θθ +°+

+° 16.

)360( sin

)180( sin

θ

θ

−°

−°

17. )90( tan

)360( tan

θ

θ

−°

−° 18.

)270( cos

)360( cos

θ

θ

−°

+°

113

19. tan (90° + θ) ⋅ sin (180° + θ) 20. cos (360° – θ) ⋅ tan (–θ)

21. )270( tan

)90( sin

θ

θ

−°

−° 22.

)270( sin

)270( cos

θ

θ

+°

−°



23. cos 120° 24. sin 240°

25. tan 330° 26. cos 300°

27. sin 315° 28. tan (–135°)

Level 2

29. Given that cos (270° + θ) = –0.6 and cos θ > 0, find the values of sin θ, cos θ and tan θ.

30. Given that tan (180° – θ) = 3, find the values of sin θ, cos θ and tan θ.



31. θ

θ

θ cos1

sin

tan

1

++ 32.

−

− 1

cos

11

sin

122θθ

33. θ

θ

θ

θ

cos

sin1

sin1

cos −+

− 34.

1cossin

1

1cossin

1

++−

−+ θθθθ

35. 1tan

cos

1tan

1

sin

++

+θ

θ

θ

θ

36. )360( tan

)270( cos

)180( cosθ

θ

θ−°⋅

+°

−°

37. )( tan)360( sin)360( cos

1θθ

θ−⋅+°+

−° 38. 1 + sin (270° – θ) ⋅ sin (90° + θ)

39. tan (180° – θ) ⋅ sin (180° + θ) + cos (360° – θ) 40. )180( sin

)( cos)270( tan

θ

θθ

+°

−⋅−°– 1

41. )]( sin1)][90( cos1[

)180( cos)270( sin1

θθ

θθ

−++°−

−°⋅−°− 42.

)180( sin

)90( sin1

)( cos1

)270( cos

θ

θ

θ

θ

−°

+°++

−+

+°

43. sin4 (180° + θ) – sin4 (270° + θ) + cos2 θ 44. )270( sin22

300tan

)90( sin1

120sin

θθ +°+

°−

+°+

°

45. sin (90° – θ) ⋅ cos (180° + θ) – cos (90° + θ) ⋅ sin (–θ)

46. tan 240° ⋅ cos (270° + θ) – 2 cos 150° ⋅ sin (180° + θ)

114


3− , find the value of tan (90° + θ) ⋅ tan (360° – θ) – cos (180° – θ) ⋅ sin (270° + θ).

48. Given that tan (270° – θ) = –2, find the value of

θθ

θθ

22 sin3cos4

cossin

−.

49. Given that cos (360° – θ) =5


)180(tan1

)( cos210sin42

θ

θ

+°+

−⋅°.


50. sin 330° – tan 135° 51. cos 150° ⋅ tan 210°

52. tan 300° ÷ cos 390° 53. sin 135° + cos (–225°)

54. In △ABC, ∠A = 45°. Find the values of the following without using a calculator.

(a) tan (∠B + ∠C)

(b) sin (∠B + ∠C – ∠A)

(c) cos (3∠A + ∠B + ∠C)

55. In a quadrilateral PQRS, ∠P = 120° and ∠Q = 60°. Find the values of the following without using a

calculator.

(a) sin (∠R + ∠S)

(b) cos

∠−

∠+∠+∠P

SRQ

4

(c) tan

∠+∠+

∠+∠

4

2

2

2 RQSP

56. α, β and θ are the interior angles of a triangle with α + β = 90°. Prove the following identities.

(a) sin α – cos β = cos θ

(b) tan α tan β – tan2

θ= 0

(c) sin α = sin β tan (θ – β)

(d) cos2 α + cos2 β + cos 2θ = 0

57. Without using a calculator, find the value of cos 1° + cos 2° + cos 3° + … + cos 179°.

58. Without using a calculator, find the value of

tan 91° ⋅ tan 179° + tan 92° ⋅ tan 178° + tan 93° ⋅ tan 177° + … + tan 179° ⋅ tan 91°.

115

Answers


1. tan θ 2. 1

3. θsin

1 4.

θtan

1

5. cos θ 6. θtan

1

7. 16

45 8.

13

9−

9. 11

4−

10. α sin α cos α tan α

(a) 180° + θ –sin θ –cos θ tan θ

(b) –θ –sin θ cos θ –tan θ

(c) 90° + θ cos θ –sin θ

θtan

1−

(d) 270° – θ –cos θ –sin θ

θtan

1

11. 0 12. 0

13. 0 14. 0

15. 0 16. –1

17. –tan2 θ 18. θtan

1−

19. cos θ 20. –sin θ

21. sin θ 22. tan θ

23. 2

1− 24.

2

3−

25. 3

1− 26.

2

1

27. 2

1− 28. 1

29. sin θ = –0.6, cos θ = 0.8, tan θ = –0.75

30. quadrant II: sin θ =10

3, cos θ =

10

1− ,

tan θ = –3;

quadrant IV: sin θ =10

3− , cos θ =

10

1,

tan θ = –3

31. θsin

1 32. 1

33. θcos

2 34.

θθ cossin

1

35. θθ cossin

1

+ 36. 1

37. cos θ 38. sin2 θ

39. θcos

1 40.

θ2sin

1−

41. tan2 θ 42. θsin

2

43. sin2 θ 44. θ

2sin

3

45. –1 46. 0

47. 16

9 48.

13

2−

49. 125

16− 50.

2

1

51. 2

1− 52. –2

53. 0

54. (a) –1 (b) 1

(c) 0

55. (a) 0 (b) 2

1

(c) 1

57. 0 58. 89

116

F4B: Chapter 12C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)



(Full Solution)


12C Level 1


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12C Level 2


Teacher’s

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___________

( )


12C Multiple Choice


Signature

___________

( )


Self-Test


_________

117

4B Lesson Worksheet 12.3A, B & C (Refer to Book 4B P.12.30)

Objective: To understand the graphs of sine, cosine and tangent functions and their properties.

Graphs of Trigonometric Functions

(a) (b) (c)

Trigonometric function y = sin x y = cos x y = tan x

Maximum value 1 1 not exist

Minimum value –1 –1 not exist

Period 360° 360° 180°

Find the maximum and minimum values of the following functions. [Nos. 1–2]

1. y = 3 sin x 2. y = –2 cos x �Ex 12C: 1, 2

( ) ≤ sin x ≤ ( ) ( ) ≤ cos x ≤ ( )

Maximum value of y = 3( ) =

Minimum value of y = ( )( ) =


Find the maximum and minimum values of

y = 2 sin x – 1.

–1 ≤ sin x ≤ 1

Maximum value of y = 2(1) – 1 = 1

Minimum value of y = 2(–1) – 1 = –3

Find the maximum and minimum values of

y = 3 cos x + 2.

( ) ≤ cos x ≤ ( )

Maximum value of y = 3( ) + 2 =

Minimum value of y =


3. y = –4 + 2 sin x 4. y = 2 cos x – 5 �Ex 12C: 3–9

5. y = 6 – 3 cos x 6. y = 1 – 4 sin x

118

7. (a) y = 2 + sin x 8. (a) y = 6 – cos x

(b) y =xsin2

1

+ (b) y =

xcos6

1

−

Maximum value of y

=)sin2( of valueminimum

1

x+

=

Minimum value of y

=)sin2( of valuemaximum

1

x+

=

9. (a) y = 8 – 3 sin x 10. (a) y = 4 cos x + 5

(b) y =xsin38

2

− (b) y =

5cos4

4

+x


11. (a) Find the maximum and minimum values of y = sin2 x.

(b) Hence, find the maximum and minimum values of z = 3 sin2 x + 4.

For –1 ≤ sin x ≤ 1,

(–1)2 = ____

02 = ____

12 = ____

119




Level 1


1. y = 4 cos x 2. y = –2 sin x 3. y = 6 + cos x

4. y = 1 – 3 sin x 5. y = –5 cos x – 2 6. y = 4 sin x – 4

7. y = 2(cos x + 1) – 3 8. y =xsin2

1

+ 9. y =

xcos25

1

−

10. y =xsin3

4

− 11. y =

xcos34

2

+ 12. y =

)1(sin23

1

−− x

Level 2


13. y = 4 + sin2 x 14. y = 4 cos2 x – 1 15. y = 2 – 3 cos2 x

16. y = –5 – 2 sin2 x 17. y =x

2cos5

1

+ 18. y =

x2sin3

2

−

19. y =3sin2

102

−x 20. y =

x2cos25

3

−− 21. y =

x2

sin2

3

+

−

22. y =x

2cos53

4

+

− 23. y = (4 + 2 sin x)2 24. y = (3 cos x – 5)2

25. y = (4 cos x + 3)2 26. y = (–1 + 2 sin x)2 27. y = 2 sin x + cos (90° – x)

28. y = cos x – 3 sin (270° – x) + 4 29. y = 2 sin2 x – sin2 (180° – x) – 1 30. y = sin2 x + 3 sin2 (90° + x)

31. y =xx

22 cos3sin4

24

+ 32. y =

)90(cos5cos2

2022

xx −°+ 33. y =

x

x

sin5

sin3

+

−

� 34. (a) Using the method of completing the square, express u2 + 4u – 6 in the form (u + a)2 + b, where a

and b are real numbers.

(b) Wilson claims that the minimum value of y = sin2 x + 4 sin x – 6 is the value of b found in (a). Do

you agree? Explain you answer.

120

Answers


1. maximum: 4, minimum: –4


3. maximum: 7, minimum: 5





8. maximum: 1, minimum:

3

1

9. maximum:

3

1, minimum:

7

1



7

2

12. maximum:

3

1, minimum:

7

1




16. maximum: –5, minimum: –7

17. maximum:

5

1, minimum:

6

1


3

2

19. maximum:

3

10− , minimum: –10

20. maximum:

7

3− , minimum:

5

3−

21. maximum: –1, minimum:

2

3−

22. maximum:

2

1− , minimum:

3

4−












3

1

34. (a) (u + 2)2 – 10

(b) no

121

F4B: Chapter 12D

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 13


(Video Teaching)

Book Example 14


(Video Teaching)

Book Example 15


(Video Teaching)

Book Example 16


(Video Teaching)

Book Example 17


(Video Teaching)

Book Example 18


(Video Teaching)

Book Example 19


(Video Teaching)



(Full Solution)

122


12D Level 1


Teacher’s

Signature

___________

( )


12D Level 2


Teacher’s

Signature

___________

( )


12D Multiple Choice


Signature

___________

( )


Self-Test


_________

123

4B Lesson Worksheet 12.4A & B (Refer to Book 4B P.12.37)

Objective: To find the measure of the angle from the given values of trigonometric ratios and to solve

simple trigonometric equations.

[In this worksheet, give the answers correct to the nearest 0.1° if necessary.]

Trigonometric Equations

If f(x) = k (or f(x) = –k), where f(x) represents sin x, cos x or tan x, k > 0 and 0° ≤ x ≤ 360°, x can be found

as follows:

Step 1: Find an acute angle α such that f(α) = k. (α is called a reference angle.)

Step 2: According to the sign of f(x), determine the quadrant(s) in which x lies.

Step 3: If x lies in quadrant I, x = α. If x lies in quadrant II, x = 180° – α.

If x lies in quadrant III, x = 180° + α. If x lies in quadrant IV, x = 360° – α.


If sin x = 0.6, find x, where 0° ≤ x ≤ 360°.

Reference angle α

= 36.870°, cor. to the nearest 0.001° ∵ sin x > 0 ∴ x lies in quadrant I or quadrant II.

x = 36.870° or 180° – 36.870° ∴ x = 36.9° or 143.1°, cor. to the nearest 0.1°

If cos x = 0.2, find x, where 0° ≤ x ≤ 360°.

Reference angle α

= ( ), cor. to the nearest 0.001° ∵ cos x ( < / > ) 0 ∴ x lies in quadrant ( ) or quadrant ( ).

x = ( ) or ( ) – ( ) ∴ x = or ,

cor. to the nearest 0.1°

In each of the following, find x, where 0° ≤ x ≤ 360°. [Nos. 1–6]

1. tan x = 3 2. cos x =2

1− �Ex 12D: 1–6

Reference angle α = ( )

∵ tan x ( < / > ) 0

∴ x lies in quadrant ( ) or quadrant ( ).

x = ( ) or ( )

∴ x = or

3. sin x = 0.1 4. tan x = –1.7

0.6 0.2

124

5. sin x = 5 6. cos x = –1.4

For any angle x, ( ) ≤ sin x ≤ ( ).

∴ x has ( ) solutions.


Solve 4 cos x = 1, where 0° ≤ x ≤ 360°.

4 cos x = 1

cos x =4

1

x = 75.522° or 360° – 75.522° ∴ x = 75.5° or 284.5°, cor. to the nearest 0.1°

Solve 5 sin x = –3, where 0° ≤ x ≤ 360°.

5 sin x = –3

sin x =)(

)(−

x = ( ) or ( ) ∴ x = or ,


7. Solve 3 tan x = 5, where 0° ≤ x ≤ 360°. 8. Solve 7 cos x = –2, where 0° < x < 360°. �Ex 12D: 10–12

tan x =)(

)(

x = ( ) or ( ) ∴ x = or ,


9. Solve 4 sin x = –3, where 90° ≤ x ≤ 270°. 10. Solve 5 tan x = –6, where 0° < x < 180°.


11. Solve the following equations, where 0° < x < 180°.

(a) xcos3 = –1 (b) xtan2 = 7

Reference angle α

= 75.522°, cor. to the nearest 0.001° ∵ cos x > 0 ∴ x lies in quadrant I or quadrant IV.

Reference angle α = ( ),

cor. to the nearest 0.001° ∵ sin x ( < / > ) 0 ∴ x lies in quadrant ( ) or

quadrant ( ).

125




[In this exercise, give the answers correct to the nearest 0.1° if necessary.]

Level 1

In each of the following, find θ, where 0° ≤ θ ≤ 360°. [Nos. 1–8]

1. cos θ =2

1 2. tan θ = –2

3. sin θ = –1 4. cos θ =2

3−

5. tan θ = 3 6. sin θ =2

3

7. cos θ = –1.2 8. sin θ =4

1

Solve the following equations, where 0° ≤ θ < 360°. [Nos. 9–14]

9. 3 cos θ = 2 10. 4 sin θ = 2

11. 2 tan θ = –5 12. 6 sin θ = –4

13. –3 tan θ = 7 14. –5 cos θ = –3

Solve the following equations, where 0° < θ ≤ 180°. [Nos. 15–20]

15. 4 tan θ = 7 16. –4 sin θ = –3

17. 2 cos θ = –1 18. –5 sin θ = 5

19. 7 cos θ = –2 20. –6 tan θ = –3

Level 2

Without using a calculator, find θ in each of the following, where 0° < θ < 360°. [Nos. 21–24]

21. tan θ = tan 42° 22. cos θ = –cos 87°

23. sin θ = cos 1° 24. cos θ = –sin 63°

126

Solve the following equations. [Nos. 25–34]

25. 3 sin θ = 1, where 0° ≤ θ < 360°. 26. – 5 tan θ = 5, where 0° < θ < 180°.

27. 6 cos θ = – 2 , where 0° ≤ θ ≤ 360°. 28. – 2 tan θ = 2 – 2 , where 0° ≤ θ ≤ 360°.

� 29. 4 cos θ – 2 = 0, where 0° ≤ θ < 360°. � 30. 3 sin θ + 2 = 0, where 0° < θ < 270°.

� 31. 3 – 4 tan θ = 0, where 0° ≤ θ < 180°. � 32. –5 sin θ – 1 = 0, where 0° ≤ θ < 180°.

� 33. 5sin

3−=

θ, where 0° < θ < 360°. � 34.

θtan

2+ 6 = 0, where 0° ≤ θ < 270°.

Solve the following equations, where 0° ≤ θ ≤ 360°. [Nos. 35–44]

� 35. sin θ = 2 cos θ � 36. 4 sin θ – 3 cos θ = 0

� 37. 6 cos θ = –5 sin θ � 38. 2 sin θ + cos θ = 0

� 39. cos θ (sin θ + 1) = 0 � 40. (2 sin θ – 1)(tan θ + 2) = 0

� 41. 4 sin θ tan θ – 5 sin θ = 0 � 42. sin2 θ + 3 sin θ cos θ = 0

� 43. 3 sin (–θ) = –2 tan θ � 44. 5 cos (180° – θ) + 3 tan (90° – θ) = 0

� 45. For 0° ≤ θ ≤ 360°, how many distinct roots does the equation tan θ (2 sin θ – 1) = 0 have?

� 46. (a) Solve the equation 2 cos θ + 1 = 0, where 0° ≤ θ ≤ 360°.

(b) Does the equation (2 cos θ + 1)( 3 tan θ – 3) = 0 have 4 distinct real roots, where 0° ≤ θ ≤ 360°?

Explain your answer.

� 47. The equation (sin θ – k)(3 – 4 tan θ) = 0 has n distinct real roots, where 0° ≤ θ ≤ 360°. Find the range

of values of k for each of the following values of n.

(a) n = 2 (b) n = 5

� 48. (a) Solve 2 tan α – 3 = 0, where 0° ≤ α < 360°.

(b) Using the results of (a), solve 2 tan 2θ – 3 = 0, where 0° ≤ θ < 180°.

� 49. (a) Solve 2 sin β – 2 = 0, where 0° < β < 90°.

(b) Using the result of (a), solve 2 sin 24

−θ

= 0, where 0° < θ < 360°.

127

Answers


1. 60°, 300°

2. 116.6°, 296.6°

3. 270°

4. 150°, 210°

5. 60°, 240°

6. no solutions

7. no solutions

8. 14.5°, 165.5°

9. 48.2°, 311.8°

10. 30°, 150°

11. 111.8°, 291.8°

12. 221.8°, 318.2°

13. 113.2°, 293.2°

14. 53.1°, 306.9°

15. 60.3°

16. 48.6°, 131.4°

17. 120°

18. no solutions

19. 106.6°

20. 26.6°

21. 42°, 222°

22. 93°, 267°

23. 89°, 91°

24. 153°, 207°

25. 35.3°, 144.7°

26. 114.1°

27. 125.3°, 234.7°

28. 157.5°, 337.5°

29. 60°, 300°

30. 221.8°

31. 36.9°

32. no solutions

33. 216.9°, 323.1°

34. 161.6°

35. 63.4°, 243.4°

36. 36.9°, 216.9°

37. 129.8°, 309.8°

38. 144.7°, 324.7°

39. 90°, 270°

40. 30°, 116.6°, 150°, 296.6°

41. 0°, 51.3°, 180°, 231.3°, 360°

42. 0°, 108.4°, 180°, 288.4°, 360°

43. 0°, 48.2°, 180°, 311.8°, 360°

44. 36.9°, 90°, 143.1°, 270°

45. 5

46. (a) 120°, 240°

(b) no

47. (a) k < –1 or k > 1

(b) k = 0

48. (a) 56.3°, 236.3°

(b) 28.2°, 118.2°

49. (a) 45°

(b) 180°

Chapter 10 Basic Properties of Circles 10A p.2 10B p

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