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.
Let us consider a Cartesian plane having a parallelogram OABC in which O is the origin.
We have to prove that middle point of the opposite sides of a quadrilateral and the join of the mid-points of its diagonals meet in a point and bisect each other.
Let coordinates be A(0, 0).
So other coordinates will be B(x1 + x2, y1), C(x2, 0) ... refer figure.
Let P, Q, R and S be the mid-points of the sides AB, BC, CD, DA respectively.
By midpoint formula,
x = 
, y = 
For midpoint P on AB,
x =
, y = 
∴ x = 
, y = 
∴ Coordinate of P is ( 
, 
)
For midpoint Q on BC,
x =
, y = 
∴ x = 
, y = 
∴ Coordinate of Q is ( 
, 
)
For R, we can observe that, R lies on x axis.
∴ Coordinate of R is ( 
, 
)
For midpoint S on OA,
x =
, y = 
∴ x = 
, y = 
∴ Coordinate of S is ( 
, 
)
For midpoint of PR,
x = 
, y = 
∴ x = 
, y = 
∴ Midpoint of PR is ( 
, 
)
Similarly midpoint of QS is ( 
, 
)
Also, similarly midpoint of AC and OA is ( 
, 
)
Hence, midpoints of PR, QS, AC and OA coincide
∴We say that middle point of the opposite sides of a quadrilateral and the join of the mid-points of its diagonals meet in a point and bisect each other.