 Hi and welcome to our session. Let us discuss the following question. The question says, three vertices of a parallelogram A, B, C, D are A having coordinates 3 minus 1, 2, B having coordinates 1, 2, minus 4, and C having coordinates minus 1, 1, 2. Find the coordinates of the fourth vertex. Before solving this question we should first with the first with section formula which says that if having coordinates x1, y1, z1 and q having coordinates x2, y2, z2 are two points then the coordinates of the point let's say are which divides the line segment pq in the ratio m is to n are given by mx2 plus nx1 upon m plus n, my2 plus ny1 upon m1 plus m2 mz2 plus nz1 upon m1 plus m2. The knowledge of this formula is a key idea in this question given with the solution. This is a parallelogram A, B, C, D. We are given coordinates of three vertices of this parallelogram we have to find the coordinates of the fourth vertex that means we have to find the coordinates of D. Let the coordinates of fourth vertex B, B, x, y, z, the diagonals of a parallelogram bisect each other parallelogram A, B, C, D also bisect each other. This implies A, C, B, D will have the same middle point divides the line segment in the ratio 1 is to 1. So bisection formula coordinates of middle point in segment joining the points having coordinates x1, y1, z1, q having coordinates x2, y2, z2, x1 plus x2 by 2, y1 plus y2 by 2, z1 plus z2 by 2. Now using this we will find coordinates of middle point of diagonal A, C. Now coordinates of A are 3 minus 1, 2 and coordinates of C are minus 1, 1, 2. So coordinates of middle point of diagonal A, C are 3 minus 1 by 2, minus 1 plus 1 by 2, 2 plus 2 by 2. This implies coordinates of middle point of diagonal A, C are 1, 0 and 2. So coordinates of middle point of diagonal A, C are 1, 0, 2. Now we will find coordinates middle point of B, D. So V are 1, 2, minus 4 and coordinates of D are x, y, z. So coordinates of middle point of diagonal V, D are 1 plus x by 2, 2 plus y by 2, minus 4 plus z by 2. The middle point of the line segments is C and V, D of C plus x by 2 is equal to 1, 2 plus y by 2 is equal to 0 and minus 4 plus z by 2 is equal to 2. Now this implies x is equal to 1, y is equal to minus 2 and z is equal to, so required coordinates of vertex D are 1 minus 2, 8. This is our required answer. So this completes the session. I take care.