Q. 31

A quadrilateral has the vertices at the point (-4,2), (2,6), (8,5) and (9,-7). Show that the mid-point of the sides of this quadrilateral are the vertices of a parallelogram.

Answer :


Let the vertices of quadrilateral be P(-4,2), Q(2,6), R(8,5) and S(9,-7)


Let A, B, C and D are the midpoints of PQ, QR, RS and SP respectively.


Now, since A is the midpoint of P(-4, 2) and Q(2, 6)


Coordinates of A are



Coordinates of B are



Coordinates of C are



and


Coordinates of D are



Now,


we find the distance between A and B










Now, since length of opposite sides of the quadrilateral formed by the midpoints of the given quadrilateral are equal .i.e.


AB = CD and AD = BC


it is a parallelogram


Hence Proved


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