1.5.2 Circular Measure Long Questions (Question 3 & 4 )

Question 3:

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)

$\begin{array}{l} In △ B S A \\ \cos θ = \frac{8}{32} = \frac{1}{4} \\ θ = 1.32 rad (2 d .p .) \end{array}$

(b)

Angle QBR = 3.142 – 1.32 = 1.822 rad

Length of minor arc QR

= 12 × 1.822

= 21.86 cm

(c)

$P Q = \sqrt{32^{2} - 8^{2}} = 30.98 cm$

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

Question 4:
Diagram below shows a sector QPR with centre P and sector POQ, with centre O.

It is given that OP = 17 cm and PQ = 8.8 cm.
[Use π = 3.142]
Calculate
(a) angle OPQ, in radians,
(b) the perimeter, in cm, of sector QPR,
(c) the area, in cm², of the shaded region.

Solution:

$\begin{array}{l} (a) \\ ∠ O P Q = ∠ O Q P \\ x + x + 30 = 180 \\ 2 x = 150 \\ x = 75 \\ ∠ O P Q = \frac{75 \times 3.142}{180} \\ = 1.3092 radians \end{array}$

$\begin{array}{l} (b) \\ Length of arc Q R = r θ \\ = 8.8 \times 1.3092 \\ = 11.52 cm \\ Perimeter of sector Q P R \\ = 11.52 + 8.8 + 8.8 \\ = 29.12 cm \end{array}$

$\begin{array}{l} (c) \\ 30^{o} = \frac{30 \times 3.142}{180} = 0.5237 rad \\ Area of segment P Q \\ = \frac{1}{2} r^{2} (θ - \sin θ) \\ = \frac{1}{2} \times 17^{2} \times (0.5237 - \sin 30) \\ = \frac{1}{2} \times 289 \times (0.5237 - 0.5) \\ = 3.4247 {cm}^{2} \\ Area of sector Q P R \\ = \frac{1}{2} r^{2} θ \\ = \frac{1}{2} \times {8.8}^{2} \times 1.3092 \\ = 50.692 {cm}^{2} \\ Area of shaded region \\ = 3.4247 + 50.692 \\ = 54.1167 {cm}^{2} \end{array}$

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