 Hello and welcome to the session. I am Asha and I am going to help you with the following question which says, in a parallelogram A, B, C, D, ELF are the midpoints of site A, B and C, D respectively show that the line segment A, F and EC intersect the diagonal BD. So, let us now start with the solution and we are given a parallelogram A, B, C, D in which our midpoints of sites A, B and C, D and we have to show that the line segments A, F and EC intersect the diagonal BD that is we have to show that D, B is equal to P, Q is equal to Q, B. Now, a coordinate rule is parallel to F, C since we are given that A, B, C, D is a parallelogram and opposite sites of a parallelogram are equal so this implies A, B is parallel to C, D and also E is the midpoint of site A, B so A can be written as half of A, B and since A, B is equal to C, D since opposite sites of a parallelogram are equal we can write half of A, B as half of C, D and half of C, D is C, F since F is midpoint of D, C so this implies A, E is equal to C, F. Now, let this be equation number one and this be equation number two. So, from one to two we can say that A, E, C, F is a parallelogram since in a correlator if a pair of opposite site is equal and parallel also then it is a parallelogram and here in correlator A, E, C, F, A, E is equal to C, F and also A, E is parallel to C, F therefore A, C, F is a parallelogram which further implies that A, F is parallel to E, C since opposite sites of a parallelogram are parallel. Now in triangle E is midpoint of A, B, E, Q is parallel to A, B since we have A, F parallel to E, C in particular this implies that E, Q is parallel to A, B so this implies that B, Q is equal to Q, P and this is the theorem 8.10 of your NCRT textbook of chapter 9 which says that the line drawn through the midpoint of one side of a triangle parallel to another site bisects the third site. Let this be equation number three and now in triangle Q, C, F is midpoint of C, D, E is parallel to Q, C since F, A is parallel to C, E just now we have proved that A, E, C, F is a parallelogram so this implies opposite sites are parallel and this further implies that P is a midpoint of D, Q and as D, P is equal to P, Q and this is equation number four and this is again by theorem 8.10 of your NCRT textbook chapter 9 which says the line segment drawn through the midpoint of one side of a triangle parallel to another site bisects the third site and in triangle D, C, Q, F is the midpoint of site D, C and F, P is parallel to C, Q so this implies F, P bisects the third site that is D, P is equal to P, Q. Now P and four P, Q is equal to P, Q also and P, Q is equal to D, P so this implies D, P is equal to P, Q is equal to P, Q since two lines which are equal to the same given line are equal to each other therefore all the three lines are equal and this implies that the line segment E, C bisects the diagonal BD so this completes the session take care and have a good day.