Question

Let $G$ be a circle of radius $R>0$. Let $G _1, G _2, \ldots, G _{ n }$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G _1, G _2, \ldots, G _{ n }$ touches the circle $G$ externally. Also, for $i =1,2, \ldots, n -1$, the circle $G _{ i }$ touches $G _{ i +1}$ externally, and $G _{ n }$ touches $G _1$ externally. Then, which of the following statements is/are TRUE ?

• A. If $n =4$, then $(\sqrt{2}-1) r < R$
• B. If $n =5$, then $r < R$
• C. If $n =8$, then $(\sqrt{2}-1) r < R$
• D. If $n =12$, then $\sqrt{2}(\sqrt{3}+1) r > R$

Answer 1

Answer: (C), (D)

$ 2( R + r ) \sin \frac{\pi}{ n }=2 r $
$ \frac{ R + r }{ r }=\operatorname{cosec} \frac{\pi}{ n }$

(A) $n =4, R + r =\sqrt{2} r$
(B) $n =5, \frac{ R + r }{ r }=\operatorname{cosec} \frac{\pi}{5}<\operatorname{cosec} \frac{\pi}{6}$
$R + r <2 r \Rightarrow r > R$
(C) $n =8, \frac{ R + r }{ r }=\operatorname{cosec} \frac{\pi}{8}>\operatorname{cosec} \frac{\pi}{4}$
$R + r >\sqrt{2} r$
(D) $n =12, \frac{ R + r }{ r }=\operatorname{cosec} \frac{\pi}{12}=\sqrt{2}(\sqrt{3}+1)$
$R + r =\sqrt{2}(\sqrt{3}+1) r$
$ \sqrt{2}(\sqrt{3}+1) r > R$