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.

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