Prove that the lines joining the middle points of opposite sides of a quadrilateral and the join of the middle points of its diagonals meet in a point and bisect one another.
Answers
Answer:
Step-by-step explanation:
Let OBCD be the quadrilateral P,Q,R, S be the midpoint off OB,CD,OD and BC.
Let the coordinates of O,B,C,D are (0,0), (x,0), (x,y), (0,y)
Coordinates of midpoint of PQ are
Coordinates of midpoint of RS are
Since, the coordinates of the mid-point of PQ = coordinates of mid-point of RS
Therefore, PQ and RS bisect each other
Answer:
Step-by-step explanation:
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 2 2 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 + y 3 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 2 0+02 = x 1 2 0
Co ordinates of H = x 2 + x 3 2 y 2 + y 3 2
Coordinate of mid point of GH
= x 1 2 + x 2 + x 3 2 2 y 2 + y 3 2 2 = x 1 + x 2 + x 3 4 y 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.
