Module 8A - Circular Measure

7/31/2019 Module 8A - Circular Measure

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CIRCULAR MEASURE FORM 4

PROGRAM KECEMERLANGAN AKADEMIK (szk smkpn) Page 1

MODULE 8(A)ADDITIONAL MATHEMATICS

TOPIC : CIRCULAR MEASURE

1.

The diagram shows a circle with centre O. The reflex angle POQ is 1.2π rad and thelength of the minor arc PQ is 16π cm. Find the radius of the circle.

[3 marks][Answer : 20cm ]

2.

The diagram shows a circle PAQ with centre O, of radius 8 cm. SR is an arc of a circlewith centre O. The reflex angle POQ is 1.6π radians. Given that P and Q are midpointsof OS and OR respectively, find the area of shaded region, giving your answer in termsof π

[3 marks][Answer : 38.4π ]

3.

Diagram shows sectors OAB and OXY with centre O. It is given that OA = 8 cm, OX =4 cm, ∠ AOB = 1.2 radians and the length of arc XY = 3.2 cm. Calculatea) the value of θ, in radian,

b) the area, in cm², of the shaded region. [4 marks]

O

P

Q

1.2π 16π cm





[Answer :a) 0.8 b) 32 ]

4.

Diagram shows two sectors in which the arcs PQ and RS are arcs of two concentriccircles with centre O. Given that PR = QS = 4 cm, the length of arc PQ = 10 cm and

∠ POQ =1

2radian. Calculate

a) the length of arc RS, in cm,

b) the perimeter, in cm, of the shaded region.[4 marks]

[Answer :a) 17 b) 30 ]

5.

Diagram shows two sectors OBC and OAD with common centre O. The length of thearc BC is 6 cm and OA : AB = 3 : 1, finda) the length of OB,b) the perimeter of the shaded region.

[3 marks][Answer :a) 8 b) 14.5 ]

6.

Diagram shows sector of a circle with centre O and OAC is a right-angled triangle.Given that OA = 10 cm and BC = 4 cm, calculatea) the value of θ, in radian,

b) the area, in cm², of the shaded region. [4 marks]





[Answer :a) 0.9273 b) 22.37 ]

7.

Diagram shows a circle with centre O. It is given that the radius of the circle is 5 cmand the length of the chord AB is 7 cm. Find

a) ∠ AOB in radians,

b) the area of the shaded segment[4 marks]

[Answer :a) 1.5508 b) 6.888 ]

8.

Diagram shows a sector of a circle with centre A. Given that the ratio of the length ofarc PQ to radius is 2:3, find the value of θ in radians and the radius of the sector whenthe area of sector APQ is 19.45 cm².

[4 marks]

[Answer : 2

3 ]

9.

Diagram shows a circle with centre O. Given the length of the major arc AB is 54.32cm. Find the length, in cm of the radius. (use π=3.142)

[4 marks][Answer : 9.311 ]





10.

Diagram shows a semicircle with centre O. Given the diameter POR = 8 cm and thelength of arc PQ is 3.87 cm. Find

a) the value of θ, in radian,b) area of the sector OQR.

[4 marks][Answer :a) 0.9675 b) 17.4 ]

11.

Diagram shows sector PAQ with centre A. The length of radius is 5.8 cm and theperimeter of the sector is 18 cm. Find the value of θ, in radians.

[3 marks][Answer :1.103 ]

12.

Diagram shows a semicircle with centre O. Given the length of AC is 20.5 cm. Find thelength of the diameter AB correct to two decimal places.


13.

Diagram shows a circle with centre O. PR and QR are tangents to the circle at points Pand Q respectively. Given OR = 12 cm and ∠ PRQ = 0.8 radian . Find

(a) the radius, in cm , of the circle(b) the area , in cm 2 , of the shaded region

[ 4 marks ]

[Answer :a) 4.673 b) 25.57 ]

Q

P

O R





14.

Diagram shows a circle with centre O and radius 5 cm. Given that AB is a tangent tothe circle at A and the length of OB is 10 cm, find

a) the angle θ in radians,b) the area of the shaded region.

[3 marks]

[Answer :a)

3

π b) 8.56 ]

15.

Diagram shows two sectors OAD and OBC with centre O. Given that OD=5 cm, DC=3

cm and ∠ BOC=6

πradians, find the area of the shaded region in terms of π .

[3 marks]

[Answer : 13

4

π ]

16.

Diagram shows two sectors with centre O. Given that OC=5cm, CB=x cm,

∠ AOB= ∠ COD and AB:CD = 3:1, find the value of x.[3 marks]

θ 5 cm

O A

C

B





[Answer :10 ] 17.

Diagram shows the semicircle ABCD with centre O and radius 7 cm. If the length ofarcs AB, BC and CD are the same, find the perimeter of the shaded region.


18.

Diagram shows a circle with centre O and radius 8.5 cm. Given that the area of theminor sector OPQ is 64 cm², find the length of the major arc PQ.


19.

Diagram shows a sector LOM with centre O. The length of arc LM is 14.25 cm and theperimeter of the sector LOM is 34.25 cm. Calculate the value of θ, in radian.

[3 marks][Answer :1.425 ]

20.

Diagram shows a circle with centre O. Given that the length of the major arc is 48.24cm, find the radius, in cm. (use π = 3.142)

[3 marks]

[Answer : 8.249 ] END OF MODULE

25˚

P

O

Q





ANSWERSMODULE 8(A)

ADDITIONAL MATHEMATICSTOPIC : CIRCULAR MEASURE

1. 2 1.2 0.8

16 (0.8 )

20

r

r cm

π π π

π

− =

=

=

2.

2 2

2 1.6 0.4

1 116 0.4 8 0.4

2 2

38.4

π π π

π π

π

− =

= × × − × ×

=

3. a) 3.2 4

3.2

4

0.8

θ

θ

=

=

=

b)2 21 1

8 1.2 4 0.82 2

32

= × × − × ×

=

4. a)

( )

1020

12

120 14

2

17

OP

RS

= =

= + ×

=

b) =10 + 4 + 12 + 4=30

5. a) 6 0.75

6

0.75

8

r

r

=

=

=

b) =2 + 6 + 2 + 6(0.75)=14.5

6. a) 6cos

10

0.9273

θ =

=





b)21 1

10 0.9273 6 82 2

22.37

= × × − × ×

=

7. a) 3.5sin

5

0.7754

2(07754) 1.5508

COB

COB

AOB

∠ =

∠ =

∠ = =

b)

2

OC 5 cos0.7754 3.57071

1 15 1.5508 7 3.570712 2

6.888

= × =

= × × − × ×

=

8. 2(19.45)

23

7.639

2

3

2

3

r

r

r

θ

θ

=

=

=

=

9. 2 0.45

5.834

s r

r(5.834) 54.32

r 9.311

θ = π −

=

= θ

=

=

10. a)

3.87 4

0.9675

s r

rad

θ

θ

θ

=

=

=

b)( ) ( )

214 2.175

2

17.4

A =

=

A

B

C

O





11. ( )PQ 18 2 5.8

6.4

s

j6.4

5.8

1.103

= −

=

θ =

=

=

12. AOC 0.85

2.29rad

20.5 j

2.29

8.95

AOB 2(8.95)

17.90

∠ = π −

=

=

=

=

=

13. a) OP = Sin 22.92º (12)= 4.673 cm

b)A = 21

(4.673) (2.342)2

= 25.57 cm²

14. a) 5 1Cos10 2

rad3

θ = =

πθ =

b)

( )

2

2

2

2

1Areaof OAB 5 10 sin

2 3

21.65cm

1Areaof sector OAC 5

2 3

13.09cm

Areaof shadedregion 21.65 13.09

8.56cm

π= × × ×

=

π =

=

= −

=

15.

2

1 1A 8 8 5 5

2 6 2 6

13units

4

π π = × × × − × × ×

π=





16.

( )

AB :CD 3 :1

5 x 3

5 1

x 10cm

=

+ θ=

θ

=

17. 7ArcBC 7 cm

3 3

CQsin

3 7

7 3CQ cm

2

OQCos

3 7

7OQ cm

2

Perimeter of shadedregion

7 7 3 72 2

3 2 2

26.45cm

π π= × =

π=

=

π=

=

∴

π = + +

=

18.

( )

21 8.5 642

1.772rad

PQ 8.5 2 1.772

38.35cm

× × θ =

θ =

= π −

=

19. 34.25 14.25r OL OM 10cm

2

14.251.425rad

10

−= = = =

θ = =

20. Angle subtend by major arc PQ=360˚ - 25˚ = 335˚

335180

5.848rad

s r

sr

48.248.249cm

5.848

π= ° ×

°

=

= θ

=θ

= =





END OF MODULE

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Module 8A - Circular Measure