Circular Measure Long Questions (Question 1)


Question 1:
Diagram shows a circle, centre O and radius 8 cm inscribed in a sector SPT of a circle at centre P.  The straight lines, SP and TP, are tangents to the circle at point Q and point R, respectively.

[Use p = 3.142]
Calculate
(a) the length, in cm, of the arc ST,
(b) the area, in cm2, of the shaded region.            

Solution:
(a)
For triangle OPQ sin 30 = 8 OP OP= 8 sin 30 =16 cm Radius of sector SPT=16+8=24 cm SPT=60× 3.142 180 =1.047 radian Length of arc ST=24×1.047=25.14 cm
(b)
For triangle OPQ: tan 30 = 8 QP PQ= 8 tan 30 =13.86 cm QOR=2( 60 )= 120 Reflex angle QOR= 360 120 = 240 240 = 3.142 180 × 240 =4.189 radian Area of shaded region =(   Area of  sector SPT )( Area of major    sector OQR )( Area of triangle  OPQ and OPR ) = 1 2 ( 24 ) 2 ( 1.047 ) 1 2 ( 8 ) 2 ( 4.189 )2( 1 2 ×8×13.86 ) =301.54134.05110.88 =56.61  cm 2