Answer:
Refer to the attached image.
Given: A quadrilateral ABCD, where P, Q, R and S are the mid points of the sides AD, AB, BC, CD respectively.
To prove: The segments joining the mid points of the opposite sides of a quadrilateral bisect each other that is OP=OR and OQ=OS.
Proof: Join the segment BD.
Consider the triangle ABD,
Since P is the mid point of AD and Q is the mid point of AB.
Therefore, .... (Equation 1)
[Line segments joining the mid points of two sides of a triangle is parallel to the third side and is also half of it]
Consider the triangle BCD,
Since S is the mid point of CD and R is the mid point of BC.
Therefore, .... (Equation 2)
[Line segments joining the mid points of two sides of a triangle is parallel to the third side and is also half of it]
By equations 1 and 2, we get
Therefore, PQRS is a parallelogram as opposite sides are equal and parallel.
In a parallelogram, diagonals bisect each other.
Since, PR and QS are the diagonals of the parallelogram PQRS.
Therefore, OP=OR and OQ=OS.
Hence, proved.