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Geometry Difficulty 3.5 AMC 10/12 Find the answer South Africa

The circle centre OO has radius 44 cm. If the area of sector OPQOPQ is 12π\frac{1}{2}\pi, and if the probability that a point chosen randomly in the circle lies in the shaded sector is 1n\frac{1}{n}, then the value of nn is

Figure 1

A number or a short expression. Spacing and $ signs are ignored.

Solution

The area of the whole circle is π⋅42=16π\pi \cdot 4^2 = 16\pi, but the area of the sector is 12π\frac{1}{2}\pi, so the probability is 12π16π=132\frac{\frac{1}{2}\pi}{16\pi} = \frac{1}{32}

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