For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is There is a regular polygon with `r/R=1/(sqrt(2))` (17) There is a regular polygon with `r/R=2/3` (30) There is a regular polygon with `r/R=(sqrt(3))/2` (47) There is a regular polygon with `r/R=1/2` (60)

Options

A. There is a regular polygon with `(r)/(R) = (sqrt3)/(2)`
B. There is a regular polygon with `(r)/(R) = (1)/(2)`
C. There is a regular polygon with `(r)/(R) = (1)/(sqrt2)`
D. There is a regular polygon with `(r)/(R) = (2)/(3)`

Explanation

To solve the problem, we need to analyze the relationship between the radii of the inscribed circle (r) and the circumscribed circle (R) for a regular polygon. The ratio \( \frac{r}{R} \) can be expressed as \( \cos\left(\frac{\pi}{n}\right) \), where \( n \) is the number of sides of the polygon.

### Step-by-Step Solution:

1. **Understanding the Ratios**:
For a regular polygon, the radius of the inscribed circle \( r \) and the radius of the circumscribed circle \( R \) are related by the formula:
\[ \frac{r}{R} = \cos\left(\frac{\pi}{n}\right) \]
where \( n \) is the number of sides of the polygon.

2. **Evaluating Each Option**:
We need to check if each given ratio can be represented as \( \cos\left(\frac{\pi}{n}\right) \) for some integer \( n \).

- **Option 1**: \( \frac{r}{R} = \frac{\sqrt{3}}{2} \)
- This corresponds to \( \cos\left(\frac{\pi}{6}\right) \), which is valid for \( n = 12 \) (a regular dodecagon).

- **Option 2**: \( \frac{r}{R} = \frac{1}{2} \)
- This corresponds to \( \cos\left(\frac{\pi}{3}\right) \), which is valid for \( n = 6 \) (a regular hexagon).

- **Option 3**: \( \frac{r}{R} = \frac{\sqrt{2}}{2} \)
- This corresponds to \( \cos\left(\frac{\pi}{4}\right) \), which is valid for \( n = 8 \) (a regular octagon).

- **Option 4**: \( \frac{r}{R} = \frac{2}{3} \)
- We need to check if \( \cos\left(\frac{\pi}{n}\right) = \frac{2}{3} \) is possible for some integer \( n \). The cosine function ranges from -1 to 1, and we need to find \( n \) such that \( \cos\left(\frac{\pi}{n}\right) = \frac{2}{3} \).

3. **Conclusion**:
The value \( \frac{2}{3} \) does not correspond to any angle \( \frac{\pi}{n} \) for integer \( n \) because the cosine function does not take this value for angles in the range \( [0, \pi] \). Thus, option 4 is the false statement.

### Final Answer:
The false statement among the options is:
**There is a regular polygon with \( \frac{r}{R} = \frac{2}{3} \)**.