A chord of a circle makes an angle of \(90^\circ\) at the centre of the circle. If the area of the major segment is \(k\) times the area of the minor segment, then what is the value of \(k\)? (Take \(\pi = \frac{22}{7}\))
\(10\)
Let radius \(= r\). Minor segment area \(=\) sector area \(-\) triangle area \(= \frac{90}{360}\pi r^2 - \frac{1}{2}r^2\sin 90^\circ = \frac{\pi r^2}{4} - \frac{r^2}{2} = \frac{r^2(\pi - 2)}{4}\). Major segment area \(= \pi r^2 - \frac{r^2(\pi-2)}{4} = \frac{r^2(3\pi+2)}{4}\). So \(k = \frac{3\pi+2}{\pi-2}\). With \(\pi = \frac{22}{7}\): \(3\pi+2 = \frac{80}{7}\) and \(\pi-2 = \frac{8}{7}\), so \(k = \frac{80/7}{8/7} = 10\).
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