The points 1 + i, 1 - i, - 1 + i and - 1 - i are

Options

A. collinear
B. concyclic
C. four vertices of a regular polygon
D. lie within a circle of radius 1

Answer

To determine the nature of the points \(1 + i\), \(1 - i\), \(-1 + i\), and \(-1 - i\), we will analyze their positions in the complex plane and see if they form a specific geometric shape.

### Step-by-Step Solution:

1. **Identify the Points**:
The given points are:
- \(A = 1 + i\)
- \(B = 1 - i\)
- \(C = -1 + i\)
- \(D = -1 - i\)

2. **Plot the Points on the Argand Plane**:
- The real part of each point corresponds to the x-coordinate, and the imaginary part corresponds to the y-coordinate.
- Plotting these points:
- \(A(1, 1)\)
- \(B(1, -1)\)
- \(C(-1, 1)\)
- \(D(-1, -1)\)

3. **Visualize the Points**:
- On the Argand plane, the points will be plotted as follows:
- \(A\) is in the first quadrant.
- \(B\) is in the fourth quadrant.
- \(C\) is in the second quadrant.
- \(D\) is in the third quadrant.

4. **Connect the Points**:
- If we connect these points in order, we will see that they form a square:
- \(A\) to \(B\) is a vertical line.
- \(B\) to \(D\) is a horizontal line.
- \(D\) to \(C\) is a vertical line.
- \(C\) to \(A\) is a horizontal line.

5. **Determine the Shape**:
- Since all sides are equal and all angles are right angles, the figure formed by connecting these points is a square.

6. **Conclusion**:
- Therefore, the points \(1 + i\), \(1 - i\), \(-1 + i\), and \(-1 - i\) are the vertices of a square.

### Final Answer:
The points \(1 + i\), \(1 - i\), \(-1 + i\), and \(-1 - i\) are the vertices of a square.

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