For a regular polygon let r and R be radii of the inscribed and the circumscribed circles. Which one of the following is not true?

Options

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

Answer

To solve the problem regarding the relationship between the radii of the inscribed and circumscribed circles of a regular polygon, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find which statement regarding the ratio \( \frac{r}{R} \) (where \( r \) is the radius of the inscribed circle and \( R \) is the radius of the circumscribed circle) for a regular polygon is not true.

2. **Formulas for \( r \) and \( R \)**:
- For a regular polygon with \( n \) sides, the radius of the inscribed circle \( r \) is given by:
\[ r = \frac{a}{2} \cot\left(\frac{\pi}{n}\right) \]
- The radius of the circumscribed circle \( R \) is given by:
\[ R = \frac{a}{2} \frac{1}{\cos\left(\frac{\pi}{n}\right)} \]

3. **Finding the Ratio \( \frac{r}{R} \)**:
- We can find the ratio \( \frac{r}{R} \) by substituting the formulas:
\[ \frac{r}{R} = \frac{\frac{a}{2} \cot\left(\frac{\pi}{n}\right)}{\frac{a}{2} \frac{1}{\cos\left(\frac{\pi}{n}\right)}} \]
- The \( \frac{a}{2} \) cancels out:
\[ \frac{r}{R} = \cot\left(\frac{\pi}{n}\right) \cdot \cos\left(\frac{\pi}{n}\right) \]
- Using the identity \( \cot(x) = \frac{\cos(x)}{\sin(x)} \):
\[ \frac{r}{R} = \frac{\cos\left(\frac{\pi}{n}\right)}{\sin\left(\frac{\pi}{n}\right)} \cdot \cos\left(\frac{\pi}{n}\right) = \frac{\cos^2\left(\frac{\pi}{n}\right)}{\sin\left(\frac{\pi}{n}\right)} \]

4. **Simplifying Further**:
- We can express this as:
\[ \frac{r}{R} = \cos\left(\frac{\pi}{n}\right) \cdot \cot\left(\frac{\pi}{n}\right) \]
- This gives us:
\[ \frac{r}{R} = \cos\left(\frac{\pi}{n}\right) \]

5. **Evaluating the Options**:
- We need to evaluate the given options for \( \frac{r}{R} \) for different values of \( n \) (starting from \( n = 3 \) since \( n = 2 \) does not form a polygon).
- For \( n = 3 \): \( \frac{r}{R} = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \)
- For \( n = 4 \): \( \frac{r}{R} = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \)
- For \( n = 5 \): \( \frac{r}{R} = \cos\left(\frac{\pi}{5}\right) \) (not a standard value)
- For \( n = 6 \): \( \frac{r}{R} = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \)

6. **Identifying the False Statement**:
- Among the calculated values, we check which of the given options does not match the calculated ratios.
- The value \( \frac{2}{3} \) is not equal to any of the calculated values for \( n = 3, 4, 5, 6 \).

### Conclusion:
The statement that \( \frac{r}{R} = \frac{2}{3} \) is not true for any regular polygon.