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
waec 1991mathematics

The correct answer is C. 1:9

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

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