Statement of the Theorem: Prove that the lines joining the middle points of the adjacent sides of a quadrilateral form a parallelogram.
Proof: Let ABCD be a quadrilateral and length of its side AB is 2a.
Let us choose origin of rectangular cartesian coordinates at the vertex A and xaxis along the side AB and AY as the yaxis. Then, the coordinates of A and B are (0, 0) and (2a, 0) respectively. Referred to the chosen axes, let (2b, 2c) and (2d, 2e) be the coordinates of the vertices C and D respectively. If J, K, L, M be the midpoints of the sides AB, BC, CD, and, DA, respectively, then the coordinates of J, K, L and M are (a, 0 ), (a + b, c), (b + d, c + e) and (d, e) respectively.
Now, the coordinates of the midpoint of the diagonal JL of the quadrilateral JKLM are {(a + b + d)/2, (c + e)/2}
Again, the coordinates of the midpoint of the diagonal MK of the same quadrilateral are {(a + b + d)/2, (c + e)/2}.
Clearly, the diagonals JL and MK of the quadrilateral JKLM bisect each other at ((a + b + d)/2, (c + e)/2). Hence, the quadrilateral JKLM is a parallelogram. Proved.
● Coordinate Geometry
11 and 12 Grade Math
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