Circular Measure Long Questions (Question 2)

Question 2:

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 PQis a common tangent to the circles at point P and point Q.
[Use p = 3.142]
Given that Ð PAR = q radians,
(a) show that q = 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)

In △BSA cosθ= 8 32 = 1 4       θ=1.32 rad (2 d.p.)

(b)

Ð QBR = 3.142 – 1.32 = 1.822 rad
Length of minor arc QR
= 12 × 1.822

= 21.86 cm

(c)
PQ= 32 2 − 8 2 =30.98 cm

Area of the shaded region
= 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

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