If `z and bar z` represent adjacent vertices of a regular polygon of `n` sides where centre is origin and if `(Im(z))/(Re(z)) = sqrt(2) - 1`, then `n` is equal to:

Options

A. (A) `8`
B. (B) `16`
C. (C) `24`
D. (D) `32`

Answer

To solve the problem, we will follow a systematic approach step by step.

### Step 1: Understanding the given information
We are given that \( z \) and \( \bar{z} \) are adjacent vertices of a regular polygon with \( n \) sides, and the center of the polygon is at the origin. We also know that:

\[ \frac{\text{Im}(z)}{\text{Re}(z)} = \sqrt{2} - 1 \]

### Step 2: Expressing \( z \) in terms of its components
Let \( z = x + iy \), where \( x = \text{Re}(z) \) and \( y = \text{Im}(z) \). The given ratio can then be rewritten as:

\[ \frac{y}{x} = \sqrt{2} - 1 \]

### Step 3: Relating the angle to the tangent
From the ratio of the imaginary part to the real part, we can express the angle \( \theta \) that \( z \) makes with the positive real axis:

\[ \tan(\theta) = \sqrt{2} - 1 \]

### Step 4: Finding the angle \( \theta \)
To find \( \theta \), we take the arctangent:

\[ \theta = \tan^{-1}(\sqrt{2} - 1) \]

### Step 5: Calculating \( \theta \)
Using the known value, we find:

\[ \theta = \frac{\pi}{8} \]

### Step 6: Finding the angle subtended by one side of the polygon
Since \( z \) and \( \bar{z} \) are adjacent vertices, the angle subtended by one side of the polygon at the center is:

\[ 2\theta = 2 \times \frac{\pi}{8} = \frac{\pi}{4} \]

### Step 7: Relating the angle to the number of sides \( n \)
The total angle around the center of the polygon is \( 2\pi \). If one side subtends an angle of \( \frac{\pi}{4} \), then the number of sides \( n \) can be calculated as:

\[ n \cdot \frac{\pi}{4} = 2\pi \]

### Step 8: Solving for \( n \)
To find \( n \), we rearrange the equation:

\[ n = \frac{2\pi}{\frac{\pi}{4}} = 2\pi \cdot \frac{4}{\pi} = 8 \]

### Conclusion
Thus, the number of sides \( n \) of the polygon is:

\[ \boxed{8} \]