 Hello and welcome to the session. In this session we discussed the following question which says proves that a parallelogram is formed by the line segments joining the midpoints of the adjacent sides of a quadrilateral. Before we start with its proof, let's recall one result which says that if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. This is the key idea that we use for this question. Now let's see the solution. Consider this quadrilateral p q r s in which p dash q dash r dash s dash are the midpoints of p q q r r s and sp respectively. We need to prove that the line segment joining these midpoints form a parallelogram. That is we have to prove that p dash q dash r dash s dash is a parallelogram. First of all let us join p r. Now let's consider the triangle p q r in this p dash is the midpoint of side p q and q dash is the midpoint of side q r. So this means that the line p dash q dash divides the two sides p q and q r of triangle p q r in the same ratio and we know that if a line divides any two sides of a triangle in the same ratio then the line is parallel to the third side. So as this p dash q dash divides the side p q and q r of triangle p q r in the same ratio. So this means p dash q dash is parallel to p r. Let this be equation one. Now we consider the triangle p s r in this triangle p s r r dash is the midpoint of side s r and s dash is the midpoint of side p s. So this would imply that the line r dash s dash that is the line joining the midpoints of the two sides r s and p s divides the two sides r s and sp of triangle p s r in the same ratio and therefore using the results stated in the key idea we get that r dash s dash is parallel to p r. We take this as equation two. Now as p dash q dash is parallel to p r and r dash s dash is parallel to p r so we have from equations one and two we get that p dash q dash is parallel to r dash s dash. Since we know that the two lines parallel to the same line are parallel to each other and similarly we can show that p dash s dash is parallel to r dash q dash thus p dash q dash is parallel to r dash s dash and p dash s dash is parallel to r dash q dash therefore we say that p dash q dash r dash s dash is a parallelogram since its opposite side is parallel to each other. So we have proved that p dash q dash r dash s dash is a parallelogram hence proved this complete c session would be understood the solution of this question.