ABCD is a rectangle formed by the points A(–1, –1), B(– 1, 4), C(5, 4) and D(5, – 1). P, Q, R and S are the mid-points of AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.

The mid-point of AB is P. Therefore,
The coordinates of P = ((-1, 1)/2, (-1+4)/2) = (-1, 3/2)
Similarly, the coordinates of points Q, R and S are (2, 4). (5, 3/2) and (2, -1) respectively.
Length of PQ = √((-1-2)²+(3/2 -4)²) = √(9+25/4) = √(61/4)
Length of QR = √((2-5)²+(4-3/2)²) = √(9+25/4) = √(61/4)
Length of PS = √((5-2)²+(3/2+1)²) = √(9+25/4) = √(61/4)
Length of SP = √((2+1)²+(-1-3/2)²) = √(9+25/4) = √(61/4)
Length of PR = √((-1-5)²+(3/2-3/2)²) = √(36+0) = 6
Length of SQ = √((2-2)²+(4+1)²) = √(0+25) = 5
All the sides of quadrilateral PQRS are equal (PQ = QR = RS = ST) but diagonals are not equal (PR ≠ SQ). So, PQRS is a rhombus.

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