Quadrilateral form a Parallelogram
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 What is Coordinate Geometry? Rectangular Cartesian Coordinates Polar Coordinates Relation between Cartesian and Polar CoOrdinates Distance between Two given Points Distance between Two Points in Polar Coordinates Division of Line Segment: Internal & External Area of the Triangle Formed by Three coordinate Points Condition of Collinearity of Three Points Medians of a Triangle are Concurrent Apollonius' Theorem Quadrilateral form a Parallelogram Problems on Distance Between Two Points Area of a Triangle Given 3 Points Worksheet on Quadrants Worksheet on Rectangular – Polar Conversion Worksheet on LineSegment Joining the Points Worksheet on Distance Between Two Points Worksheet on Distance Between the Polar Coordinates Worksheet on Finding MidPoint Worksheet on Division of LineSegment Worksheet on Centroid of a Triangle Worksheet on Area of Coordinate Triangle Worksheet on Collinear Triangle Worksheet on Area of Polygon Worksheet on Cartesian Triangle
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