ABCD is a rectangle formed by the points A(– 1, – 1), B(– 1, 4), C(5, 4) and D(5, – 1). If P, Q, R and S be the midpoints of AB, BC, CD, and DA respectively, show that PQRS is a rhombus.
The figure is shown below:
P(x,y) = (– 1 – 1)/2 , (4 – 1)/2
= (– 1,3/2)
Q(x,y) = (5 – 1)/2 , (4 + 4)/2
= (2,4)
R(x,y) = (5 + 5)/2 , (– 1 + 4)/2
= (5,3/2)
S(x,y) = (5 – 1)/2 , (– 1 – 1)/2
= (2, – 1)
Coordinates of mid – point of PR = Coordinates of mid – point of QS
Coordinates of mid – point of PR = {(5 – 1)/2 ,(3/2 + 3/2)/2} = (2,3/2)
Coordinates of mid – point of QS = {(2 + 2)/2 , (– 1 + 4)/2 = (2,3/2)
Hence PQRS is a Rhombus.
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