Question

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 and the mid points of AB, BC CD and DA respectively. Is the quadrilateral PQRS square, a rectangle or a rhombus. Justify your answer. 

Answer 1

According to question the given point are A(-1, -1), B(-1, 4), C(5, 4) and D(5, -1).

Co-ordinate of mid point P of side AB

Co-ordinate of mid point Q of side BC

= ((5 - 1)/2, (4 + 4)/2)

= (4/2, 8/2)

= (2, 2)

Co-ordinate of mid point R of side CD

= ((5 + 5)/2, (4 + 4)/2)

= (10/2, 3/2)

= (5, 3/2)

Co-ordinate of mid point S of side DA

= ((5 - 1)/2, (-1 -1)/2)

= (4/2, -2/2)

= (2, -1)

Now P (-1, 3/2), Q(2, 4), R (5, 3/2) and S(2, -1)

In quadrilateral PQRS, 

PQ = QR = RS = SP and diagonal PR ≠ diagonal QS

Hence, quadrilateral PQRS is a rhombus.