 Hello students, let's work out the following problem. It says using vectors prove that the line segment joining the midpoints of the non-parallel sides of a trapezium is parallel to the base and is equal to half the sum of the parallel sides So let's now move on to the solution and let a, b, c, d be the trapezium in which a, b is parallel to dc. So this is the trapezium in which a, b is parallel to dc. Now let a be the origin, e and f be the midpoints of a, d. Now let the position vector of n, d, v vector b and vector d So a is the origin and position vector of b is b and position vector of d is vector d therefore a, b is equal to vector d, a, d is equal to vector d. Now we know that a, b is parallel to dc therefore is equal to m times vector a, b where m is a scalar So we have vector dc is equal to m times vector a, b that is m into vector b. Now is the resultant vector a, d and dc So vector a, c is equal to a, d plus dc Now a, d is vector d and dc is m times vector b right, so therefore position vector of c is d plus m times vector b. Now as b is the midpoint of a, d is the midpoint of b, c we have position vector of half of a, d that is half of vector a, d that is half of vector d and the position vector of f is half of b, c. Now position vector of b is b and position vector of c is vector d plus m times vector b and since f is the midpoint of b, c so this would be half of vector b plus the position vector of c that is vector d plus m times vector b. So this is equal to half of 1 plus m times vector b plus vector d therefore vector ef is given by position vector of e minus position vector of e and here we have position vector of f. Now since we have to prove that the line joining the midpoints of the non-parallel sides is parallel to the base we need to prove that position vector that is vector ef is equal to some scalar into vector b, right? Now position vector of f is 1 by 2 into vector b plus vector b plus m times vector b minus position vector of e which is 1 by 2 to vector d. So this is equal to 1 by 2 vector b plus 1 by 2 vector d plus 1 by 2 into m times vector b minus 1 by 2 vector d. Now this cancels and we have 1 by 2 into 1 plus m times vector b. So this implies ef that is the line joining the midpoints of the non-parallel sides of the trapezium is parallel to the base and base is ab that is ef is parallel to ab. Now we also have to prove that the line joining the midpoints of the non-parallel sides is equal to half the sum of the parallel sides. That is we have to prove that ef is equal to 1 by ab plus dc. So now mod of mod of 1 by 2 into 1 plus m times vector b. Now this is 1 by 2 into 1 plus m into ab since position vector of v is this b. So now this is equal to 1 by 2 into 1 plus m ab. This is equal to 1 by 2 into ab plus 1 by 2 into m into ab. Now m times ab is dc. Let's name this as 1. So here we have 1 by 2 ab plus 1 by 2 m into ab is dc. This is using 1. So this is equal to half of ab plus dc. Hence prove. So this completes the session and the question. Bye for now. Take care. Have a good day.