Diagram below shows a circle PQRT, centre Oand 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.
(a) the angle x , in terms of p ,
(b) the length , in cm , of the arc ACB ,
(c) the area, in cm2,of the shaded region.
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, Ð QOR= ÐQOP = 60o
Ð POR = 120o
x = 2p/ 3 rad
cos Ð AOQ= OQ / OA
cos 60o = 5 / OA
OA = 10 cm
Length of arc, ACB,
s = r θ
Arc ACB = (10) (2p / 3)
Arc ACB = 20.94 cm
Area of shaded region = 1 2 r 2 ( θ−sinθ ) ( change calculator to Rad mode ) = 1 2 ( 10 ) 2 ( 2π 3 −sin 2π 3 ) =50( 2.094−0.866 ) =61.40 cm 2