State True or False:
The center of a square is at the point $z_{0} = 1 + i$ and one of its vertices is at the point $z_{1} = 1 - i .$ The other vertices are $3 + i, 1 - 3 i, - 1 + i$ .

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Solution

If $D$ be $(x, y)$ then $O (1, 1)$ is mid-point of $B (1, - 1)$ and $D$
$1 = \frac{x + 1}{2}, 1 = \frac{y - 1}{2}$
$∴ D (x, y) = 1, 3 = 1 + 3 i$
Again rotation in anti-clockwise direction about O gives $O C = O B e^{i π / 2} = - 2 i (i) = 2$
$∴ (x - 1) + i (y - 1) = 2 + 0 i$
$∴ x = 3, y = 1$ or $C = 3 + i$ .
Hence by mid-point A is $- 1 + i$ .

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