Statement -1 : if ` 1-i,1+i, z_(1) and z_(2)` are the vertices of a square taken in order in the anti-clockwise sense then ` z_(1) " is " i-1` and Statement -2 : If the vertices are `z_(1),z_(2),z_(3),z_(4)` taken in order in the anti-clockwise sense,then ` z_(3) =iz_(1) + (1+i)z_(2)`

Options

A. Statement -1 is True, Statement -2 is True, Statement -2 is a correct explanation for statement -7
B. Statement -1 is True, Statement -2 is True , Statement -2 is NOT a correct explanation for Statement -7
C. Statement -1 is True, Statement -2 is False
D. Statement -1 is Flase, Statement -2 is True

Answer

To solve the problem, we need to analyze both statements regarding the complex numbers representing the vertices of a square.

### Step-by-Step Solution:

**Step 1: Identify the vertices of the square.**
- The given vertices are \( z_1 = 1 - i \), \( z_2 = 1 + i \), and we need to find \( z_1 \) and \( z_2 \) in the context of the square's properties.

**Step 2: Determine the coordinates of the vertices.**
- The points \( 1 - i \) and \( 1 + i \) correspond to the coordinates \( (1, -1) \) and \( (1, 1) \) respectively on the Argand plane.

**Step 3: Find the coordinates of the other two vertices of the square.**
- Since the square is oriented in an anti-clockwise direction, the other two vertices \( z_1 \) and \( z_2 \) can be determined by rotating the points \( z_1 \) and \( z_2 \) by 90 degrees.
- The vertex opposite to \( z_1 \) (which is \( 1 - i \)) can be found by rotating it 90 degrees counter-clockwise. The point \( z_1 \) can be expressed as \( z_1 = -1 + i \) or \( z_1 = i - 1 \).

**Step 4: Verify Statement 1.**
- Statement 1 claims that \( z_1 \) is \( i - 1 \). Since we have determined that \( z_1 = i - 1 \), Statement 1 is **true**.

**Step 5: Analyze Statement 2.**
- Statement 2 states that \( z_3 = i z_1 + (1 + i) z_2 \).
- We need to substitute the values of \( z_1 \) and \( z_2 \) into this equation.

**Step 6: Substitute the values into Statement 2.**
- We know \( z_1 = i - 1 \) and \( z_2 = 1 + i \).
- Substitute these into the equation:
\[ z_3 = i(i - 1) + (1 + i)(1 + i) \]
- Calculate \( i(i - 1) = i^2 - i = -1 - i \).
- Calculate \( (1 + i)(1 + i) = 1 + 2i + i^2 = 1 + 2i - 1 = 2i \).
- Therefore, \( z_3 = (-1 - i) + 2i = -1 + i \).

**Step 7: Verify Statement 2.**
- We need to check if \( z_3 = -1 + i \) is consistent with the properties of the square.
- Since the vertices are taken in anti-clockwise order, \( z_3 \) should indeed be one of the vertices of the square. The calculated \( z_3 \) is valid.

### Conclusion:
- Both Statement 1 and Statement 2 are **true**.