Progression (Sequence and Series)HardBloom L3
Question
A convex $n$-sided polygon has a circumcircle and an inscribed circle, its area is $B$ and the area of its circumcircle and inscribed circle are A and C respectively, then
Options
A.$B = \frac{A + C}{2}$
B.$B > \frac{A + C}{2}$
C.$B < \frac{A + C}{2}$
D.Nothing can be said
Solution
Sol. [Key Concept: Convex polygon with both circum- and in-circle: relate its area to the two circle areas.]
Let $r \rightarrow$ inradius, $R \rightarrow$ circumradius, $P \rightarrow$ perimeter of polygon
$$\begin{matrix} P < 2\pi R \\ \text{ }B = \frac{1}{2}rP < \frac{1}{2}r(2\pi R) = \pi rR \\ \Rightarrow \ \text{ }B < \pi rR < \pi\left( \frac{r^{2} + R^{2}}{2} \right) = \frac{A + C}{2} \end{matrix}$$
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