Question

prove that the lines joining the middle points of the opposite sides of a quadrilateral and the join of the middle points of its diagonals meet in a point and bisect one another

Answer 1

Let ABCD be the quadrilateral such that the diagonal AC is along the  x-axis. Suppose the coordinate of A, B, C and D are  (0,0) , (x2,y2),(x1,0) and (x3,y3) respectively. E and F are the mid-points of its opposite sides AD and BC respectively and G and H are the midpoints of its diagonals AC and BD. Let EF and GH  intersect in I. Co ordinates of E =  0+ x 3 2 0+y 3 2 = x 3 2 y 3 2 Co ordinates of F = x 1 + x 22 0+ y 2 2   = x 1 + x 2 2 y 2 2   Co ordinate of mid point of EF =  x 1 + x 2 2 + x 3 2 2 y 2 2 + y3 2 2 =  x 1 + x 2 + x 3 4 y 2 + y 3 4   ......(1) G and H that is mid point of diagonal AC and BD respectively, then  Co ordinates of G =  0+ x 1 20+02 = x 1 2 0 Co ordinates of H = x 2 + x 3 2 y2 + y 3 2 Coordinate of mid point of GH  =  x 12 + x 2 + x 3 2 2 y 2 + y 3 2 2 = x 1 + x 2 + x 3 4y 2 + y 3 4    .......(2)   From (1) and (2), we observe that the mid points of EF and HG are same.So, EF and HG meets and bisects each other. Therefore, the line joining the middle points of the opposite sides of a quadrilateral and the join of middle points of diagonal meets and bisect each other.



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