In the diagram, PQRS is a circle with centre O and radius 7cm. SQ and PR intersect at K and < SKR = 90^\circ. If the length of the arc SR is four times that of arc PQ, find the length of the arc SR. [Take \(\pi = \frac{22}{7}\)].
Verified Correct Answer: .
Step-by-Step Mathematical Solution
Let < POQ = \(\theta\)
\(\implies\) SR = \(4\theta\).
\(< PSQ = \frac{\theta}{2} ; < SPR = 2\theta\)
In \(\Delta PSK\),
\(2\theta + \frac{\theta}{2} = 90^\circ\)
\(\frac{5\theta}{2} = 90^\circ \implies 5\theta = 180^\circ\)
\(\theta = \frac{180^\circ}{5} = 36^\circ\)
\(\therefore < SOR = 4\theta = 4 \times 36^\circ = 144^\circ\)
Length of arc = \(\frac{\theta}{360^\circ} \times 2\pi r\)
\(\frac{144}{360} \times 2 \times \frac{22}{7} \times 7 \)
= \(17.6 cm\).
Source: Myschool Past Questions Archive (Record #53267)