To solve the problem, we need to find the ratio of the areas of two polygons defined by their vertices. Let's break it down step by step.
### Step 1: Define the vertices of the polygons
For \( n = 4 \):
- The vertices of polygon \( A \) are given by:
\[
A_0 = 1, \quad A_1 = \omega, \quad A_2 = \omega^2, \quad A_3 = \omega^3
\]
where \( \omega = \cos\left(\frac{2\pi}{4}\right) + i \sin\left(\frac{2\pi}{4}\right) = 0 + i(1) = i \).
Thus, the coordinates of the vertices are:
- \( A_0 = (1, 0) \)
- \( A_1 = (0, 1) \)
- \( A_2 = (-1, 0) \)
- \( A_3 = (0, -1) \)
For polygon \( B \):
- The vertices are:
\[
B_0 = 1, \quad B_1 = 1 + \omega, \quad B_2 = 1 + \omega^2, \quad B_3 = 1 + \omega^3
\]
Thus, substituting \( \omega \):
- \( B_0 = (1, 0) \)
- \( B_1 = (1, 1) \)
- \( B_2 = (0, 0) \)
- \( B_3 = (1, -1) \)
### Step 2: Calculate the area of polygon \( A \)
The vertices of polygon \( A \) form a square with vertices at \( (1, 0) \), \( (0, 1) \), \( (-1, 0) \), and \( (0, -1) \). The length of the side of the square can be calculated as:
\[
\text{Length} = \sqrt{(1 - 0)^2 + (0 - 1)^2} = \sqrt{1 + 1} = \sqrt{2}
\]
The area of polygon \( A \) is:
\[
\text{Area}(A) = \text{Length}^2 = (\sqrt{2})^2 = 2 \text{ square units}
\]
### Step 3: Calculate the area of polygon \( B \)
The vertices of polygon \( B \) form a triangle with vertices at \( (1, 0) \), \( (1, 1) \), and \( (0, 0) \). The base of the triangle is the distance between \( (1, 0) \) and \( (1, 1) \) which is \( 1 \) unit, and the height is the distance from \( (0, 0) \) to the line \( x = 1 \), which is also \( 1 \) unit.
The area of polygon \( B \) is given by:
\[
\text{Area}(B) = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2} \text{ square units}
\]
### Step 4: Calculate the ratio of the areas
Now we can find the ratio of the areas of polygon \( A \) to polygon \( B \):
\[
\lambda = \frac{\text{Area}(A)}{\text{Area}(B)} = \frac{2}{\frac{1}{2}} = 2 \text{ (unitless)}
\]
### Conclusion
The value of \( \lambda \) is \( 2 \).
