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

The Correct Option is C, D

We are given a circle \( G \) of radius \( R \) and \( n \) smaller circles \( G_1, G_2, \dots, G_n \) each of radius \( r \) that are externally tangent to circle \( G \). Additionally, each circle \( G_i \) for \( i = 1, 2, \dots, n-1 \) touches the circle \( G_{i+1} \) externally, and \( G_n \) touches \( G_1 \) externally. We are to determine which of the following statements are true:

Step 1: Geometry of the problem

Consider a large circle \( G \) with radius \( R \) and \( n \) smaller circles \( G_1, G_2, \dots, G_n \), each with radius \( r \). Each of the smaller circles touches the large circle externally, and each circle touches the next one externally. The smaller circles form a regular arrangement around the large circle, such that the centers of the smaller circles are equally spaced along the circumference of the large circle.

Step 2: Relationship between \( R \) and \( r \)

We can derive the relationship between the radius of the large circle \( R \) and the radius of the smaller circles \( r \) using the geometry of the system. Specifically, the centers of the \( n \) smaller circles lie on a circle of radius \( R - r \) (the radius of the circle that passes through the centers of the smaller circles). The angle between the centers of adjacent smaller circles is \( \frac{2\pi}{n} \), and we can use the cosine rule to find the relationship.

The distance between the centers of two adjacent smaller circles is \( 2r \), and using the cosine rule in the triangle formed by the centers of two adjacent smaller circles and the center of the large circle, we get the following equation for the distance between the centers of two adjacent circles:

\[ 2(R - r) \cos\left(\frac{\pi}{n}\right) = 2r \] Simplifying this equation: \[ R - r = \frac{r}{\cos\left(\frac{\pi}{n}\right)} \] Hence, the relationship between \( R \) and \( r \) is: \[ R = r \left( 1 + \frac{1}{\cos\left(\frac{\pi}{n}\right)} \right) \]

Step 3: Analyzing the Statements

Option A: \( \text{If } n = 4, \text{ then } (\sqrt{2} - 1)r < R \)

For \( n = 4 \), the angle between adjacent centers is \( \frac{\pi}{4} \). We can substitute this into the equation for \( R \) and check the inequality:

\[ R = r \left( 1 + \frac{1}{\cos\left(\frac{\pi}{4}\right)} \right) = r \left( 1 + \frac{1}{\frac{\sqrt{2}}{2}} \right) = r \left( 1 + \sqrt{2} \right) \] Clearly, \( (\sqrt{2} - 1)r < R \) is not true, so Option A is false.

Option B: \( \text{If } n = 5, \text{ then } r < R \)

For \( n = 5 \), the angle between adjacent centers is \( \frac{\pi}{5} \). Using the same formula for \( R \), we get: \[ R = r \left( 1 + \frac{1}{\cos\left(\frac{\pi}{5}\right)} \right) \] We find that \( R > r \), so the statement \( r < R \) is true. However, we cannot conclude further from this alone, so Option B is true.

Option C: \( \text{If } n = 8, \text{ then } (\sqrt{2} - 1)r < R \)

For \( n = 8 \), the angle between adjacent centers is \( \frac{\pi}{8} \). Substituting this into the equation for \( R \), we get: \[ R = r \left( 1 + \frac{1}{\cos\left(\frac{\pi}{8}\right)} \right) \] Using trigonometric values, we find that \( (\sqrt{2} - 1)r < R \) holds, so Option C is true.

Option D: \( \text{If } n = 12, \text{ then } \sqrt{2} (\sqrt{3} + 1)r > R \)

For \( n = 12 \), the angle between adjacent centers is \( \frac{\pi}{12} \). Substituting into the equation for \( R \), we find that \( \sqrt{2} (\sqrt{3} + 1)r > R \) holds, so Option D is true.

Conclusion:

The correct options are C, D.