Short Questions (Question 7 & 8)

Question 7:
Diagram below shows a circle with centre O and radius 12 cm.

Given that A, B and C are points such that OA = AB and  ∠OAC = 90°, find
(b)  the area, in cm2, of the shaded region.

Solution:
(a) For triangle OAC,
cos  ∠AOC = 6/12

(b)
= Area of BOC – Area of triangle AOC
½ (12)2 (1.047) – ½ (6) (12) sin 1.047 (change calculator to Rad mode)
= 75.38 – 31.17
= 44.21 cm2

Question 8:
Diagram below shows a sector QOR of a circle with centre O.

It is given that PS = 8 cm and QP = PO= OS = SR = 5 cm.
Find
(a) the length, in cm, of the arc QR,
(b) the area, in cm2, of the shaded region.

Solution:
(a) Length of arc QR = θ = 10 (1.75) = 17.5 cm

(b)
= Area of sector QOR – Area of triangle POS
½ (10)2 (1.75) – ½ (5) (5) sin 1.75 (change calculator to Rad mode)
= 87.5 – 12.30
= 75.2 cm2

Short Questions (Question 5 & 6)

Question 5 (4 marks):
Diagram shows a circle with centre O.

Diagram

PR
and QR are tangents to the circle at points P and Q respectively. It is given that the length of minor arc PQ is 4 cm and
Express in terms of α,
(a) the radius, r, of the circle,
(b) the area, A, of the shaded region.

Solution:
(a)

(b)

Question 6 (3 marks):
Diagram shows two sectors AOD and BOC of two concentric circles with centre O.

Diagram

The angle subtended at the centre O by the major arc AD is 7α radians and the perimeter of the whole diagram is 50 cm.
Given OB = r cm, OA = 2OB and ∠BOC = 2α, express r in terms of α.

Solution:

Long Questions (Question 10)

Question 10 (8 marks):
Diagram shows a circle and a sector of a circle with a common centre O. The radius of the circle is r cm.

It is given that the length of arc PQ and arc RS are 2 cm and 7 cm respectively. QR = 10 cm.
[Use θ = 3.142]
Find
(a) the value of r and of θ,
(b) the area, in cm2, of the shaded region.

Solution
:

(a)

(b)

Long Questions (Question 9)

Question 9 (7 marks):
Mathematics Society of SMK Mulia organized a competition to design a logo for the society.

Diagram shows the circular logo designed by Adrian. The three blue coloured regions are congruent. It is given that the perimeter of the blue coloured region is 20π cm.
[Use π = 3.142]
Find
(a) the radius, in cm, of the logo to the nearest integer,
(b) the area, in cm2, of the yellow coloured region.

Solution:
(a)

(b)

Long Questions (Question 8)

Question 8:

In the diagram above, AXB is an arc of a circle centre O and radius 10 cm with  ∠AOB = 0.82 radian. AYB is an arc of a circle centre P and radius 5 cm with  ∠APB = θ.
Calculate:
(a) the length of the chord AB,
(b) the value of θ in radians,
(c) the difference in length between the arcs AYB and AXB.

Solution:
(a)

(b)

(c)
Using s =
Arcs AXB = 10 × 0.82 = 8.2 cm
Arcs AYB = 5 × 1.848 = 9.24 cm

Difference in length between the arcs AYB and AXB
= 9.24 – 8.2
= 1.04 cm

Long Questions (Question 7)

Question 7:
Diagram below shows a circle PQRT, centre and radius 5 cm. AQB is a tangent to the circle at Q. The straight lines, AO and BO, intersect the circle at P and R respectively.
OPQR is a rhombus. ACB is an arc of a circle at centre O.
Calculate
(a) the angle x , in terms of π ,
(b) the length , in cm , of the arc ACB ,
(c) the area, in cm2,of the shaded region.

Solution:
(a)
Rhombus has 4 equal sides, therefore OP = PQ = QR = OR = 5 cm
OR is radius to the circle, therefore OR = OQ = 5 cm

Triangles OQR and OQP are equilateral triangle,
Therefore,  ∠ QOR=  ∠QOP = 60o
∠ POR = 120o
x = 120o × π/180o

(b)
cos  ∠ AOQ= OQ / OA
cos 60o = 5 / OA
OA = 10 cm

Length of arc, ACB,
s = r θ
Arc ACB = (10) (2π / 3)
Arc ACB = 20.94 cm

(c)

Long Questions (Question 6)

Question 6:
Diagram below shows a circle PQT with centre O and radius 7 cm.

QS
is a tangent to the circle at point Q and QSR is a quadrant of a circle with centre Q. Q is the midpoint of OR and QP is a chord. OQR and SOP are straight lines.
[Use π = 3.142]
Calculate
(b) the perimeter, in cm ,of the shaded region,
(c) the area, in cm2 ,of the shaded region.

Solution:
(a)

(b)

(c)

Long Questions (Question 5)

Question 5:
Diagram below shows a semicircle PTS, centre O and radius 8 cm. PTR is a sector of a circle with centre P and Q is the midpoint of OS.

[Use π = 3.142]
Calculate
(b) the length , in cm , of the arc TR ,
(c) the area, in cm2 ,of the shaded region.

Solution:
(a)

(b)

(c)

Long Questions (Question 4)

Question 4:
Diagram below shows two circles. The larger circle has centre A and radius 20 cm. The smaller circle has centre B and radius 12 cm. The circles touch at point R. The straight line PQ is a common tangent to the circles at point P and point Q.

[Use π = 3.142]
Given that angle PAR = θ radians,
(a) show that θ = 1.32 ( to two decimal places),
(b) calculate the length, in cm, of the minor arc QR,
(c) calculate the area, in cm2, of the shaded region.

Solution:
(a)

(b)
Angle QBR = 3.142 – 1.32 = 1.822 rad
Length of minor arc QR
= 12 × 1.822
= 21.86 cm

(c)
= Area of trapezium PQBA– Area of sector QBR – Area of sector PAR
½ (12 + 20) (30.98) – ½ (12)2 (1.822) – ½ (20)2(1.32)
= 495.68 – 131.18 – 264
= 100.5 cm2

Short Questions (Question 5 & 6)

Question 5:
Diagram below shows a sector QOR of a circle with centre O.

It is given that PS = 8 cm and QP = PO= OS = SR = 5 cm.
Find
(a) the length, in cm, of the arc QR,
(b) the area, in cm2, of the shaded region.

Solution:
(a) Length of arc QR = θ = 10 (1.75) = 17.5 cm

(b)
= Area of sector QOR – Area of triangle POS
½ (10)2 (1.75) – ½ (5) (5) sin 1.75 (change calculator to Rad mode)
= 87.5 – 12.30
= 75.2 cm2

Question 6:
Diagram below shows a circle with centre O and radius 12 cm.

Given that A, B and C are points such that OA = AB and  ∠OAC = 90°, find
(b)  the area, in cm2, of the shaded region.

Solution:
(a) For triangle OAC,
cos  ∠AOC = 6/12