If one vertex and centre of a square are z and origin then which of the following cannot be the vertex of the square ?

Options

A. iz
B. `-z`
C. `-iz`
D. 2z

Answer

To solve the problem of identifying which of the given options cannot be a vertex of the square when one vertex is \( z \) and the center is the origin, we will follow these steps:

### Step 1: Understand the Geometry of the Square
Since the origin is the center of the square, the vertices of the square can be determined based on the given vertex \( z \). The vertex \( z \) can be expressed in the form \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 2: Identify the Other Vertices
The other vertices of the square can be derived from the properties of symmetry. If \( z \) is one vertex, the opposite vertex (which is symmetric with respect to the center) will be \( -z \). The other two vertices can be found by rotating \( z \) by \( 90^\circ \) and \( 270^\circ \) around the origin.

- The vertex obtained by rotating \( z \) by \( 90^\circ \) is \( iz \).
- The vertex obtained by rotating \( z \) by \( 270^\circ \) is \( -iz \).

Thus, the four vertices of the square are:
1. \( z = x + iy \)
2. \( -z = -x - iy \)
3. \( iz = i(x + iy) = -y + ix \)
4. \( -iz = -i(x + iy) = y - ix \)

### Step 3: Analyze the Options
Now we need to check the given options to see which one does not fit the form of the vertices derived above. The options will typically be in the form of complex numbers, and we need to see if they can be expressed as one of the vertices listed above.

### Step 4: Check Each Option
Let’s denote the options as follows:
- Option A: \( iz \)
- Option B: \( -z \)
- Option C: \( -iz \)
- Option D: \( 2z \)

1. **Option A: \( iz \)** - This is one of the vertices.
2. **Option B: \( -z \)** - This is also one of the vertices.
3. **Option C: \( -iz \)** - This is again one of the vertices.
4. **Option D: \( 2z \)** - This is not a vertex of the square. The distance from the center to any vertex is equal to half the diagonal of the square, and \( 2z \) represents a point that is twice as far from the center as any vertex could be.

### Conclusion
The vertex that cannot be part of the square is \( 2z \).

### Final Answer
Thus, the answer is \( 2z \).

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