 Hello and welcome to the session. Let us solve the following question which says, A B C D is a trapezius in which A B is parallel to D C, B D is a diagonal as E is the midpoint of A D. A line is drawn through E parallel to A B intersecting B C and F show that F is the midpoint of B C. Now before we show that F is the midpoint of B C, let us first learn theorem 8.10 of chapter 9 of your book which says, the line drawn to the midpoint, one side of a triangle and to another side, bifrates the third side. So we will use this theorem to show that F is the midpoint of B C. So this theorem is our key idea. Let us now start with the solution and we are given a trapezium A B C D in which A B is parallel to C D. Also we are given that E is midpoint of A D and a line through E is drawn such that it is parallel to A B. So we have E is parallel to A B and since A B is parallel to C D, so this implies that E F is parallel to C D also. And let this be equation number one, since two lines parallel to the same given line are parallel to each other, here the lines A B is parallel to C D also and E F also therefore C D and E F are parallel to each other. And we have to show that F is a midpoint of, now let the point of intersection of line E F and D B B denoted by G. So in triangle A D B, E is a midpoint of A D and since E F is parallel to A B this implies E G is parallel to A B. And by a key idea we know that if a line is drawn from the midpoint of one side of a triangle parallel to the other side it bisects the third side. So this implies D G is equal to G B since E is a midpoint of D A in triangle D A B and E G is parallel to A B so the remaining side is D B. So this line which is parallel to the second side will bisect the third side. So we have D G is equal to G B and let this be equation number two. Now in triangle D B C G is midpoint of D B in one we have shown that E F is parallel to C D so this implies G A F is also parallel to C D and this implies that C F is equal to F B which is by theorem 8.10 which says in a triangle if a line is drawn from the midpoint of one side of a triangle parallel to the other side it will bisect the third side. So here G is the midpoint of D B and the line is drawn through this point which is parallel to the other side so it will bisect the third side. So we have C F is equal to F B which further implies that F is midpoint of B C this is what we have to show so this completes the session take care and have a good day.