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)

If $n =8$, then $(\sqrt{2}-1) r < R$

Answer 2

So, the correct option is
(C) If $n =8$, then $(\sqrt{2}-1) r < R$
(D) If $n =12$, then $\sqrt{2}(\sqrt{3}+1) r > R$