Download - Part A: Key Points

F5 數學 1.1-1.4 Properties of Circles

P. 1

Part A: Key Points

1. (a) If ABON ⊥ , then BNAN = . (b) If BNAN = , then ABON ⊥ .

(line from centre ⊥ chord bisects chord) (line joining centre to mid-pt. of chord ⊥ chord)

(圓心至弦的垂線平分弦) (圓心至弦中點的連線垂直弦)

Note: The perpendicular bisector of a chord passes through the centre of the circle.

(⊥ bisector of chord passes through centre)

2. (a) If CDAB = , ABOM ⊥ and CDON ⊥ , (b) If ONOM = , ABOM ⊥ and CDON ⊥ ,

then ONOM = . then CDAB = .

(equal chords, equidistant from centre) (chords equidistant from centre are equal)

(等弦與圓心等距) (與圓心等距的弦等長)

QUICK CHECK


P. 2

Part B: Key Points 1. (a) q = 2p ( at centre twice at ☉ce) (圓心角兩倍於圓周角)

(b) If AB is a diameter, then = 90APB . (c) If = 90APB , then AB is a diameter.

( in semi-circle) (半圓上的圓周角) (converse of in semi-circle) (半圓上的圓周角的逆定理)

2. x = y (s in the same segment) (同弓形內的圓周角)

QUICK CHECK


P. 3

Part C: Key Points 1. (a)

(b)

If CODAOB = , then If 𝐴𝐵⌢

= 𝐶𝐷⌢

, then

(i) 𝐴𝐵⌢

= 𝐶𝐷⌢

(equal s, equal arcs) (i) CODAOB = (equal arcs, equal s)

(等角對等弧) (等弧對等角)

(ii) CDAB = (equal s, equal chords) (ii) CDAB = (equal arcs, equal chords)

(等角對等弦) (等弧對等弦)

(c)

If CDAB = , then

(i) CODAOB = (equal chords, equal s) (等弦對等角)

(ii) 𝐴𝐵⌢

= 𝐶𝐷⌢

(equal chords, equal arcs) (等弦對等弧)

2. (a)

(b)

𝐴𝐵⌢

: 𝐶𝐷⌢

= 𝑥: 𝑦 (arcs prop. to s at centre) 𝐴𝐵⌢

: 𝐶𝐷⌢

= 𝑚: 𝑛 (arcs prop. to s at ☉ce)

(弧與圓心角成比例) (弧與圓周角成比例)

QUICK CHECK

1. 2. 3.

Find

︵︵BCAB : .


P. 4

4. 5. 6.

Find BC. Find A, B and C.

5. 8. 9.

Find ︵AB .

10.

(a) Find ∠CAD.

(b) Hence, find

︵︵CDAC : .


P. 5

Part D: Key Points (a) ∠A + ∠C = 180° and ∠B + ∠D = 180° (b) ∠DCE = ∠A

(opp. s, cyclic quad.) (ext. , cyclic quad.)

(圓內接四邊形對角) (圓內接四邊形外角)

QUICK CHECK

1. 2. 3.

Find ∠BAC.

4. 5. 6.

Find ∠ACD. Find ∠CFD.

7.

Find CED.


P. 6

Part E: Key Points (a) If qp = , then A, B, Q and P are concyclic. (b) If =+ 180CA or =+ 180DB , then

(converse of s in the same segment) A, B, C and D are concyclic.

(同弓形內的圓周角的逆定理) (opp. s supp.) (對角互補)

(c) If pq = , then A, B, C and D are concyclic.

(ext. = int. opp. ) (外角 = 內對角)

Note: Any three non-collinear points are concyclic, and there is one and only one circle that can be drawn

passing through them.

QUICK CHECK

Prove concyclic.

1. 2. 3.

4. 5. 6.


P. 7

1. In the figure, AOM is a straight line and M is the mid-point of

chord BC.

If BC = 6 cm and the radius of the circle is 5 cm, find the

lengths of

(a) OM, (b) AC.

(Leave your answers in surd form if necessary.)

a) 4

b) 3√10

2. In the figure, M and N are points on chords AB and CD

respectively such that OM ⊥ AB and ON ⊥ CD. AON is a

straight line. OM = ON = 6 cm and CD = 9 cm. Find the

length of AN.

13.5


P. 8

3. In the figure, two parallel chords AMB and CND are 6

cm apart.

MON is a straight line. OM ⊥ AB, AB = 10 cm

and CD = 12 cm.

Let OM = x cm and r cm be the radius of the

circle.

(a) By considering △OAM, express r2 in terms

of x.

(b) Hence, find x and r.

(Give your answers correct to 3 significant

figures if necessary.)

a) 𝑟2 = 𝑥2 + 25

b) 𝑥 = 3.92 𝑟 = 6.35

4. B1. In the figure, AO // CB and ∠OAB = 32°. Find x and y.

𝑥 = 64° 𝑦 = 26°


P. 9

5. In the figure, chords AB and ED are produced to C. AD and BE intersect at K. ∠AOE = 66°

and ∠BAD = 13°. Find

在圖中，弦 AB 的延線與 ED 的延線相交於 C。AD 與 BE 相交於 K。

求 (a) ∠AKE, (b) ∠BCE.

a) 46° b) 20°

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

= 22 cm and

ACB = 55. Find 𝐵𝐶⌢

.

14

7. In the figure, AB is a diameter of the circle. ABD = 30 and

DOC = 80. Find 𝐴𝐷⌢

: 𝐷𝐶⌢

: 𝐶𝐵⌢

.

3: 4: 2

8. In the figure, PDA, PCB, QBA and QCD are straight

lines. AC and BD intersect at R. APB = 52,

CAB = 35 and ARB = 84. Find x and y.

𝑥 = 16° 𝑦 = 51°

y


P. 10

9. In the figure, ABCD is a kite, where AB = AD and BC = DC. It is

given that BAD = 110° and BDC = 55°.

(a) Prove that ABCD is a cyclic quadrilateral.

(b) Let M be the mid-point of AC. Using the result of (a),

find BMC.

a)

b) 𝟏𝟏𝟎°

10. E7. In the figure, APC, AQD and BPQE are straight lines

and 𝐵𝐴⌢

= 𝐴𝐸⌢

.

(a) Prove that =++ 180zyx .

(b) Prove that C, D, Q and P are concyclic.

11. 6. In the figure, O is the centre of the circle ANB. The chord AB

meets the radius ON at M such that AM = MB = 6 cm and

MN = 3 cm.

(a) Find the radius of the circle.

(b) Find𝐵𝑁⌢

a) 𝟕. 𝟓

b) 𝟔. 𝟗𝟓

110°

55°

A

B

C

D


P. 11

12. 4. In the figure, 𝐴𝐵⌢

= 𝐵𝐶⌢

= 𝐶𝐷⌢

= 𝐷𝐸⌢

= 𝐸𝐹⌢

. If AOF = 120°, find x.

𝟒𝟖°

13. 7. In the figure, ABCD is a quadrilateral and its diagonals intersect at K. KAD = 20, KAB

= 40 and ABC : BCD : CDA = 5 : 6 : 4. (a) Prove that ABCD is a cyclic quadrilateral.

(b) Find BKC.

a)

b) 𝟏𝟐𝟎°

14. 8. In the figure, O is the centre of the circle. ADC is a straight line and DA = DB.

(a) Prove that BDC = 2BAD. (b) Show that O, D, B and C are concyclic.


P. 12

15. In the figure, ABCD is a quadrilateral and its diagonals intersect at K. ADK = 37, AKB

= 75, DCK = 33 and BCD = 70. (a) Prove that ABCD is a cyclic quadrilateral.

(b) Find ABC.

1°

a)

b) 𝟕𝟏°

16. 4. In the figure, BCD = 57°, CB and DF are produced to meet at A, BF and CD are

produced to meet at E such that DEF = 31°. Find x and y.

𝒙 = 𝟗𝟐°

𝒚 = 𝟑𝟓°


P. 13

Part F: Key Points

(a) If PQ is the tangent to the circle at T, (b) If ,OTPQ ⊥ then PQ is the tangent to

then OTPQ ⊥ . the circle at T.

(tangent ⊥ radius) (切線 ⊥ 半徑) (converse of tangent ⊥ radius) (切線 ⊥ 半徑的逆定理)

Note: The perpendicular to a tangent PQ at its point of contact T

passes through the centre O of the circle.

(⊥ to tangent at its point of contact passes through centre)

(垂直切線且通過切點的直線通過圓心)

Part G: Key Points

If two tangents, TP and TQ, are drawn to a circle from an external

point T and touch the circle at P and Q respectively, then

(i) TQTP = ,

(ii) QOTPOT = ,

(iii) QTOPTO = .

(tangent properties) (切線性質)

Part H: Key Points

(a) A tangent-chord angle of a circle is equal to an angle in the alternate segment.

(i)

(ii)

ABTATQ = ACTATP =

(∠ in alt. segment) (交錯弓形的圓周角)


P. 14

(b) If TP is a straight line such that ABTATP = ,

then TP is the tangent to the circle at T.

(converse of ∠ in alt. segment)

(交錯弓形的圓周角的逆定理)

QUICK CHECK

F1.

BOA and BCT are straight lines.

F2.

AOB and BCT are straight lines.

F3.

TB is the tangent to the circle at B.

F4.

CD is a diameter of the circle.

F5.

CBT is a straight line.

F6.

COT is a straight line.

QUICK CHECK

G1.

G2.

TCOD is a straight line.

63°

94° O

x

y

B

A T

C

50°

66°

O

T A

C

x

y

44° x y

68°

A

B

T

C

A

T

B D

O

C 16°

x y


P. 15

QUICK CHECK

In each of the following figures, AB is the tangent to the circle at T. Find the unknown(s). (H1 −H4)

H1.

H2.

CR is the tangent to the circle at C.

H3. H4.

CD is the tangent to the circle at Q.

17. F7. In the figure, AB is the tangent to the circle at C and

BO⊥OA.

(a) Show that △BOA ~ △BCO.

(b) If OA = 8 cm and OB = 6 cm, find the radius of

the circle.

a) b) 4.8

D

A T

B

C

E x y

25°

48°

37°

R

E

x

T

D

y

73°

x

24° A B

T

C

D Q

P S


P. 16

18. F8. In the figure, CP is a diameter of the circle and

intersects AB at M.

OB is produced to Q. It is given that CP⊥AB,

AB = 6 cm and

CM = 9 cm.

(a) Find the radius of the circle.

(b) Furthermore, if BQ = 8 cm and PQ = 12

cm, is PQ the tangent to the circle at P?

Explain your answer.

a) 5 b) 𝑁𝑜

19. F9. In the figure, TA is the tangent to the circle at

A. M is the mid-point

of AB. MT cuts the circle at D and intersects

AC at E. It is given

that TABA⊥ , ∠ATM = 50° and ∠ABC =

58°.

(a) Is AB a diameter of the circle? Explain

your answer.

(b) Hence, find x and y.

a) 𝑌𝑒𝑠 b) 𝑥 = 20°

𝑦 = 52°

P

B

O

Q

A M

C


P. 17

20. E1. In the figure, AFB is a diameter of the circle. AD

intersects the circle at C. CB and DF intersect at E and

DF ⊥ AB.

(a) Prove that A, F, E and C are concyclic.

(b) Prove that AE is a diameter of the circumscribed

circle of quadrilateral AFEC. 證明 AE 是四邊形

AFEC 的外接圓的一條直徑。

21. G3. In the figure, AB and BE are tangents to

the circle at A and C respectively. AOD is a

straight line and ∠ABO = 30°.

(a) Find ∠COD.

(b) Show that BO // CD.

a) 60°

22. G4. In the figure, a circle is inscribed in △ABC and

∠AOC = 115°.

(a) Find the value of ∠BAC + ∠BCA.

(b) Hence, find ∠OBC.

a) 130° b) 25°


P. 18

23. G5. The figure shows the inscribed circle of

△ABC. APO, BQO and CRO are straight lines.

ABO = 25 and BAO = 35.

Find (a) ACR.

(b) 𝑃𝑄⌢

: 𝑄𝑅⌢

: 𝑅𝑃⌢

.

a) 30° b) 24: 25: 23

24. G6. In the figure, AB, BC and CA are

tangents to the circle at D, E and F

respectively. AB = 15 cm, BC = 18 cm and AC

= 10 cm. Find BD.

11.5

25. G7. In the figure, OPQR is a square and OPTR is a

quarter of the circle.

A and B lie on PQ and QR respectively. PQ, AB and

QR are tangents

to the circle at P, T and R respectively. It is given

that OP = 6 cm and

AP = 2 cm. Find

(a) the length of BR,

(b) the area of the shaded region.

(Give your answer in terms of . )

a) 3 b) 30 − 9𝜋


P. 19

26. H5. In the figure, UR is a diameter of the circle,

while PQ is the tangent to the circle at T.

∠SUR = 50° and US // PQ. Find x.

𝑥 = 20°

27. H6. In the figure, AC is the tangent to the

circle at A and BC cuts AD at E. If ACE = 25,

DAB = 30 and BDE = 54, find CBD

67°

28. H7. In the figure, n𝐶𝐷⌢

= 𝐷𝑇⌢

, CB // AT and ∠CBT =

60°. If BD is a diameter of the circle, is AT the tangent

to the circle at T? Explain your answer.

𝑌𝑒𝑠

29. H8. In the figure, AB and BD are tangents to

the circle at A and C respectively. AOED is a

straight line. It is given that cm 5=AB and

cm 13=BD .

(a) Find the length of CD.

(b) Find the radius of the circle.

a) 8

b) 10

3


P. 20

30. H9. In the figure, ΔABC is an

isosceles triangle, AB = BC = 20 cm and

tan ∠BAC = 4/3.

If the length of the radius of the inscribed

circle of ΔABC is r cm,

find the value of r.

𝑟 = 6

31. 7. In the figure, AB is a chord on the circle and TA is the tangent to the circle at A. OT bisects AB at C. (a) Prove that A, O, B and T are concyclic. (b) Join BT. Is BT the tangent to the circle at B? Explain your answer.

a) b) Yes

32.

7.06


P. 21

33. E5. In the figure, ABC, AFG, ADE and CGE are straight lines and DEG =

70.

(a) Find ABD.

(b) Prove that DEGF is a cyclic quadrilateral.

a) 𝟕𝟎°

34. D7. In the figure, O1 and O2 are the centres of the

two circles ABD and BCD respectively, where O1 lies

on circle BCD and O2 lies on circle ABD.

Let ∠BAD = x.

(a) Express ∠BCD in terms of x.

(b) Find x.

a) 90° =𝑥

2

b) 𝑥 = 60°

35. In the figure, CDA and CEB are straight lines, DA = EB = k, where k is a constant.

In the smaller circle, 𝐷𝐸⌢

:𝐶𝐷⌢

= 1 : 2.

(a) Prove that △CAB △CED.

(b) Prove that CD = CE.

(c) Find x and y.

(d) Prove that AB // DE.

a)

b)

c) 𝑥 = 36°

𝑦 = 72°

d)

O1 O2 A

B

C

D


P. 22

36.

a)

b)𝑌𝑒𝑠

37.

15°

38.

5 𝑐𝑚


P. 23


P. 24


P. 25


P. 26


P. 27


P. 28


P. 29


P. 30


P. 31


P. 32


P. 33


P. 34

61. In the figure, a circle is inscribed in quadrilateral ABCD. P, Q, R

and S are the points of contact.

If BC = 9 cm, CD = 10 cm and AD = 7 cm, find the length of AB.

Revision Exercise 2

Revision Exercise 1

1. C 2. B 3. A 4. C 5. B 6. B 7. D

8. B 9. A 10. C 11. B 12. C 13. B 14. D

15. D 16. C 17. B 18. A 19. B 20. D 21. C

22. D 23. B 24. A 25. B 26. C 27. A 28. A

29. C 30. D 31. C 32. B 33. D 34. C 35. A

36. B 37. D 38. A 39. C 40. C 41. B 42. A

43. B 44. C 45. B 46. C 47. C 48. B 49. A

50. B 51. D 52. B 53. C 54. D 55. A 56. B

57. B 58. D 59. A 60. B 61. 6 cm