 Hi and welcome to the session. I am Asha and I am going to help you with the following question which says show that the line segments joining the midpoints of the opposite sides of a quadrilateral bisect each other. So let a, b, c, d be the quadrilateral and p, q, r, s be the midpoints of side a, b, c, d, d respectively and they have to show that p, r and s, q bisect each other. So let's start with the solution and we are given a quadrilateral, let it be a a, b, c, d in which p, q, r and s are midpoints of a, b, b, c, c, d and da and we have to show that the line segment joining the midpoints of opposite sides of a quadrilateral bisect each other that is we have to show that p, o is equal to o, r, s, o is equal to o, q that is lines p, r and s, q bisect each other. So for that we will try to show that p, q, r, s is a parallelogram and if p, q, r, s is a parallelogram then we know that in a parallelogram diagonals bisect each other and p, r and s, q are the diagonals of quadrilateral p, q, r, s so we will try to show that p, q, r, s is a parallelogram. Now in triangle a, b, since we are midpoints of sides a, d and a, b so this implies s, p is parallel to b, d and also s, p is equal to half of p, d and this is the midpoint theorem which is theorem 8.9 of your book which says the line segment joining the midpoints of two sides of a triangle is parallel to the south side. Similarly in triangle d, c, b and q are midpoints of d, c and b, c so this implies that r, q is parallel to b, d and also r, q is equal to half of b, d. Now let this be equation number 1 and this be equation number 2. Now from 1 and 2 we see that b, d line is parallel to s, p also and r, q also and two lines which are parallel to the same line are parallel to each other so this implies s, p is parallel to r, q and also s, p is equal to half of b, d and half of b, d is equal to r, q so this implies s, p is equal to r, q and this is from equation 1 and 2. Now in coordinate rule r, s we have s, p parallel to r, q, s, p is equal to r, q in a coordinate rule so this implies in coordinate rule p, q, r, s where s, p is parallel to r, q and also s, p is equal to r, q so this implies that p, q, r, s is a parallelogram set to each other, parallelogram p, q, r, s the diagonal p, r and s, q will bisect each other, supplies p, r and s, q bisect each other, the line segment joining the midpoints of a coordinate rule each other. This completes the session, take care and have a good day.