 Hi and welcome to the session. I am Asha and I am going to help you with the following question which says, A, B, C, D is a quadrilateral in which P, Q, R, S are midpoints of the sites A, B, C, C, D and D, A and A, C is a diagonal. Show that first S, R is parallel to A, C and S, R is equal to half of A, C. Second is P, Q is equal to S, R and third is P, Q, R, S is a parallelogram. Let us now begin with the solution and here we are given a quadrilateral A, B, C, D in which P is a midpoint of A, B, so this implies A, P is equal to P, P, Q is midpoint of B, C, so this implies B, Q is equal to Q, C. Also we are given that R is midpoint of D, C, so this implies D, R is equal to R, C and S is midpoint of D, so this implies D, S is equal to S, A and we have to show that first S, R is parallel to A, C and S, R is equal to half of A, C. Now in triangle A, D, C midpoints of sites A, D and D, C respectively, so this implies that S, R is parallel to A, C and S, R is equal to half of A, C and this is the midpoint theorem which says the line segment joining the midpoints of two sites of a triangle is parallel to the third site and there is half of it. So here the midpoints are S and R and the line segment joining these two points will be parallel to the third site which is site A, C and there will be half of it. So we have S, R is parallel to A, C and S, R is equal to half of A, C. So this shows the first part and next we have to show that Q is equal to S, R. Now from part first we have S, R is equal to half of A, C and let this be equation number one. Now in triangle A, B, C points P and Q are midpoints of A, B and B, C respectively. So this implies that PQ is parallel to A, C and also PQ is half of A, C and this is by midpoint theorem which says that the line segment joining the two sites of a triangle is parallel to the third site and there is half of it. So let this be equation number two. Now from equation number one and two we find that S, R is equal to half of A, C and half of A, C is also equal to PQ. So this in turn implies that S, R is equal to PQ. So this shows the second part and in the third part we have to show that PQ, R, S is a parallelogram. Now this construct diagonal DB are the points which are midpoints of sites D, C and V, C. This implies by midpoint theorem that RQ is equal to half of DB also RQ is parallel to DB. Similarly in triangle DAB are midpoints of again that midpoint theorem we have SP is equal to half of DB parallel to DB. Now DB is parallel to RQ also and DB is parallel to SP also. So this implies that RQ is parallel to SP since half of DB is equal to RQ and half of DB is equal to SP. So this implies that RQ is equal to SP and similarly from part one and two we have is parallel to PQ and SR is equal to PQ. So let this be equation number three and this be equation number four. Now since in correlator PQ, RS the opposite sites are parallel also and equal also. So this implies correlator PQ, RS is a parallelogram. Hence we have shown that first SR is parallel to AC, SR is equal to half of AC, second PQ is equal to SR and third PQ RS is a parallelogram. This completes the session. Take care and have a good day.