If ABCD is a square with A(2,3), B(1,2), C(–1,–2), then what are the coordinates of D?

The correct answer is (0, –1).

• In a square, the diagonals bisect each other. This means the midpoint of diagonal AC must be the same as the midpoint of diagonal BD.
• First, find the midpoint of AC: M = ((2 + (-1))/2, (3 + (-2))/2) = (1/2, 1/2).
• Let the coordinates of D be (x, y). The midpoint of BD is ((1 + x)/2, (2 + y)/2).
• Equating the x-coordinates: (1 + x)/2 = 1/2 => 1 + x = 1 => x = 0.
• Equating the y-coordinates: (2 + y)/2 = 1/2 => 2 + y = 1 => y = -1.
• Thus, the coordinates of D are (0, -1).

This method ensures that the geometric properties of a square are maintained.