In the diagram above, PQ and XY are two concentric arc; center O, the ratio of the length of the two arc is 1:3, find the ratio of the areas of the two sectors OPQ and OXY
A. 1:3 B. 1:6 C. 1:9 D. 2:3 E. 4:9
Correct Answer: C
Explanation
Let the radius of the arc PQ = r and the radius of the arc XY = R. Length of arc PQ = \(\frac{\theta}{360} \times 2\pi r = 1\) Length of arc XY = \(\frac{\theta}{360} \times 2\pi R = 3\) Ratio of the arc = \(\frac{r}{R} = \frac{360 \times 2\pi \theta}{2\pi \theta \times 360 \times 3}\) = \(\frac{1}{3}\) Ratio of their area = \((\frac{1}{3})^2 = \frac{1}{9}\) = 1 : 9
