Circular Measure Long Questions (Question 5)


Question 5:
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
x = 2π/ 3 rad

(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)
Area of shaded region = 1 2 r 2 ( θsinθ ) ( change calculator to Rad mode ) = 1 2 ( 10 ) 2 ( 2π 3 sin 2π 3 ) =50( 2.0940.866 ) =61.40  cm 2

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