chapter 14 i circles enhance

7/28/2019 Chapter 14 I Circles ENHANCE

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O

CHAPTER 11: CIRCLES III

11.1 PROPERTIES OF ANGLES IN CIRCLES

1. Angle subtended by an arc at the centre is twice the angle subtended at the circumference

by the same arc.

x = 2y

Example

a) Find the value of x.

x = 110 2 = 55

Exercises

i). Find the value of x.

ii)

iii)

2. Angle subtended at the circumference in a semicircle is 90 .

x = y = 90

Example

b) Find the value of x.

x = 90 60

= 30

Exercises

Find the value of x.

i)

x =

Circles III

O

35x

42

O

x

1

x =

x =

ox 54

x

O

110

x =

y

x

O

xy

60

x

O


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ii)

x =

ii)

x =

3. a) Angles subtended at the circumference by the same arc are equal. b

r = s

b) Angles subtended at the circumference by arcs of the same length are the same.

x = y

Example

Find the value of

y = 47o

Exercises

i) Find the value of xand y.

x = y =

Circles III

x 47

y

38

2

47o y

35 xo x

o

rs

x

y


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ii) Find the value of y.

y =

iii) Find the value of x and y.

x y+ =

4 a) The sum of interior opposite angles of a cyclic quadrilateral is 180o.

p + r = 180o

q + s = 180o

b) The exterior angle and the corresponding interior opposite angle of cyclic quadrilateral are

equal

x = y

Example

Find the value of xand y.

x = 180 - 85 y = 180 - 95

= 95 = 85

Exercises

Find the value of x and y.

i)

x = y =

ii)

iii)

Circles III

27y

x

80y

115

A

x

B O50 110

yD

C

3

P Q95 x

y

85 R

S

s

p

q r

60o

17oxy

x

y


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x =

y =

x =

y =

11.2. PROPERTIES OF TANGENTS TO CIRCLES.

11.2.1 Tangent to a circle

11.2.2

i) Tangents drawn from an external point are equal, AB = CB

ii) The tangents subtend equal angles at the centre, AOB = COB

iii) The line joining the external point to the centre of the circle bisects the angle between

the tangents, ABO = CBO

p + q = 180

Example


Exercise

a i)

Circles III

A

x

B O50 110

yD

C

P

xS y 100

110R Q

O

4

O

Tangent

A tangent to a circle is

perpendicular to the

drawn to the point of

contact.

B

C

A

P

O pqo


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x = 90 - 65

= 25

x =

Example

x = 180 - 50

= 130

Exercises

b i) Find the value of x.

x =

ii) Find the value of x and y

x = y =

iii) Find the value of x and y.

x = y =

11.2.3 ANGLES BETWEEN TANGENTS AND CHORDS

Circles III

65

x

O

P

O Q

70 x

R

P

O Qx

50

P

O Qx

15

R

y54

x

Q

R

O

13y

x

N

O

P

5

P


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47

x

35

y

P

Q

R

Example 1

Given STU is a tangent to the circle with centre O.

Find the value of x and y.

x = 47 y = 63

1a) Given PQR is a tangent to the circle with

centre O. Find the value of x and y.

x = y =

Example 2

x = 58 y = 180- 116

= 64

2a)

x = y =

Circles III

a

by

x

S

T

U

63

47x

y

x y

75

60

P Q R

y

x58

T

U

6

S

Angle between a tangent and a

chord through the point of contact

is equal to the angle in the

alternate segment.

a = b

x = y


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x48

Q

O

N

M

R

P

b)

x =

y =

c)

x =

y =

Questions based on examination format (Paper 1)

1. In Diagram 1, the tangent PQR touches the circle at Q and MNR is a straight line. Find the

value of x.

A 38 C 109

B 52 D 142

2. In diagram 2, PQR is the tangent to the circle with centre O. Given that the length of arc MN is

equal to length of arc NQ. Find the value of x.

A 21 C 69B 48 D 100

Circles III 7

88o

15o

x yx

y81o

31o

DIAGRAM 1

DIAGRAM 2

38

x

N

Q

P

R

M


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P

R45

110

S

xO

Q

48

xO

RP

3. In diagram 3, O is the centre of a circle and PQR is a tangent . Find the value of x.

A 45 C 115

B 90 D 220

4. In Diagram 4, PQR is the tangent to the circle with centre O. Find the value of x.

A 21 C 42

B 24 D 60

5. In Diagram 5, ABC is the tangent to the circle BDEF. The value of x is

A 48 C 56

B 54 D 59

Circles III 8

A CB

E

65O

56O

D

F

xO

DIAGRAM 3

DIAGRAM 4

Q

DIAGRAM 5


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R

Q

P

65

y

x

38

D

80 36

yEF

BA

x

N

MLO

38

K

6. In Diagram 6, KOLM is a straight line and MN is the tangent to the circle with centre

O. Find the value of x..

A 14 C 38

B 19 D 142

7. In diagram 7, PQR is the tangent to the circle .Find the value of x + y.

A 103 C 141

B 115 D 168

8. In diagram 8, ABC is the tangent to the circle at B. Given that EF = ED and CDF is

the straight line. Find the value of y .

A 28 C 60

B 44 D 62

Circles III 9

DIAGRAM 7

DIAGRAM 8


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70y

45

O

P

Q

SRT

x

5342

y

D

C

B

E

A

y

100

70

xQ

P

T

S

R

9. In the diagram, TRS is the tangent to the circle with centre O at R.

The value of y is

A 80 C 110

B 100 D 115

10. In diagram 10, tangent AE touches the circle ABCD at A. Given BC is parallel to AD.

Find the value of x + y.

A 36 C 53

B 42 D 95

11. In Diagram 11, PT is the tangent to the circle PQRS and RST is the straight line.

The value of x + y is

A 110 C 150

B 130 D 170

Circles III 10

DIAGRAM 11


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D

C

O

A

y

42

T

SP

y

O

64

RQ

x

QP

O

R

148

B

A

12. In Diagram 12, tangent CD touches circle ABC with centre O at C.

The value of y is

A 25 C 48

B 32 D 53

13. In Diagram 13, PST is tangent to the circle with centre O, at S and POR is straight line.

The value of y is

A 28 C 46

B 38 D 62

14. In the diagram below, ABC and APQ are tangents to the circle with centre O.

The value of x is

A 48 C 74

B 58 D 84

Circles III 11

DIAGRAM 12

DIAGRAM 13

DIAGRAM 14


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P

x

30O

58

NMK

x

24

50

OP

N

15. Diagram 15 shows the tangent KMN to the circle with centre O at M.

The value of x is

A 52 C 58

B 56 D 62

16. In Diagram 16, KLM is a tangent to the circle. The value of x is

A 76 C 80

B 78 D 82

17. In Diagram 17, MNP is a tangent to the circle QRN at N with centre O. The value of x

is

A 40 C 74

B 64 D 124

18. In Diagram 18, APB is a tangent to the circle QRN at P. Given QPS = SPB.

Circles III 12

KL

M

P

K

Q

R35o

65o xo

DIAGRAM 15

DIAGRAM 16

DIAGRAM 17

Q


13/20

40

x

S

R

Q

B

P

A

xO

R

40

QP

S

6548x

T

R

S

QP


A 70 C 140

B 110 D 160

19. In Diagram 19, PQ is a tangent to the circle PRS with centre O. SRQ is a straight line.


A 25 C 50

B 40 D 100

20. In diagram 20, PQR is a tangent to the circle QST. Given that PTS is a straight line.


A 17 C 67

B 42 D 113

Past Year SPM Questions

Circles III 13

DIAGRAM 18

DIAGRAM 20

DIAGRAM 19


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T

S

55O

100

yR

November 2003

In Diagram 3, RST is a tangent to the circle with centre O, at point S.

Diagram 3

The value of y is

A 45 C 70

B 60 D 125

June 2004

In the Diagram 3, O is the centre of the circle and JKL is a tangent to the circle at point K.

DIAGRAM 3

The value of n is

A 25 C 50

B 40 D 65

November 2004

In Diagram 2, JK is a tangent to the circle KLM at K and JML is a straight line

Circles III 14

J

L

K

M

O

15

55

n


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Q

Ox

40

DIAGRAM 2

The value of x is

A 15 C 60

B 45 D 75

June 2005

In Diagram 2, O is the centre of circle QST and PQR is the tangent to the circle at Q.

DIAGRAM 2


A 20 C 50

B 25 D 80

Circles III 15

J

M

K

L

30 x


16/20

November 2005

In Diagram 3, TPS is a tangent to the circle PQR at P.

DIAGRAM 3

The length of arc PR is equal to the length of arc QR. Find the value of x.

A. 22 C. 44

B. 34 D. 68

June 2006

In Diagram 2, SPT is a tangent to the circle with centre O, at P.

DIAGRAM 2

Find POQ

A 60 C 100

B 80 D 140

November 2006

Circles III 16

50

Q

O

S

P

T

T

P

S

68

Q

x

R


17/20

In Diagram 2, QRT is a tangent to the circle centre O, at R. PUS, OPQ and OUR are straight lines.

DIAGRAM 2


A 40 C 50

B 45 D 65

June 2007

In Diagram 3, TQU is a tangent to the circle centre O, at point Q. SPQ is straight line.


A 114 C 120

B 117 D 126

November 2007

Circles III 17

25

X

U

P

O

R

Q

S

T

66o

27o

xo

T

U

QO

S

R

P

Diagram 3


18/20

Diagram 3 shows a circle, PFQ, centre O. EFG is a tangent to the circle at F.


A 32 C 61

B 48 D 74

June 2008

1. In Diagram 4, URT is the tangent to the circle PQRS at point R. PQR is an equilateral triangle and

URS = 20o .


A 30 C 40

B 35 D 45

2. In Diagram 5, O is the centre of a circle. TQU is the tangent to the circle at point Q. PSR is a straight

line and OPS = 10o .

Circles III 18

16

O

FGE

58x

QP

Diagram 3

RTU

20

xQP

S

Diagram 4


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A 18 C 28

B 22 D 33

November 2008

In Diagram 3, JKL is the tangent to the circle centre O at K and JNM is a straight line .


A 45 C 15

B 25 D 5

ANSWERS

Circles III 19

10O

Q UT

43

x

P

32S

Diagram 5

50

O

K

LJ35

xN

M

Diagram 5


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Exercise 11.1

1 i) 70 ii) 110 iii) 84

2. i) 36 ii) 55 iii) 30

3 i) x = 47 y = 38

ii) y = 27

iii) x = 17 y = 60

4 i) x = 115 y = 100

ii) x = 55 y = 125

iii) x = 30 y = 50

Exercise 11.2

a i) x = 20

b) i) x = 150

ii) x = 90 y = 126

iii) x = 154 y = 13

Exercise 11.2.3

1 a) x = 60 y = 45o

2 a) x = 47 y = 35

b) x = 61 y = 44

c) x = 99 y = 68

1. C 11. A

2. C 12. C

3. C 13. B

4. A 14. C

5. D 15. D

6. A 16. C

7. C 17. B

8. A 18. B

9. B 19. A

10. D 20. A

2003 NOV B2004 JUNE A 2004 NOV B

2005 JUNE B 2005 NOV C

2006 JUNE C 2006 NOV A

2007 JUNE B 2007 NOV B

2008 JUNE C 2008 NOV A

Circles III 20

chapter 14 i circles enhance

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