7327440 - x1 x22 x3 x422 y1 y22 y3 y422 Rx1 x2 x3 x44 y1 y2...

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P C(x 3 , y 3 ) D(x 4 , y 4 ) Q N M B(x 2 , y 2 ) A(x 1 , y 1 ) Theorem: If the midpoints of the sides of a parallelogram taken in succession are joined, the quadrilateral formed is a parallelogram. Proof: Using analytic technique. Let A ( , ) x1 y1 ,B ( , ) x2 y2 , C ( , ) x3 y3 , D ( , ) x4 y4 be the consecutive vertices of a parallelogram. Let M be the midpoint of AB: then M has coordinates: + , + x1 x22 y1 y22 Let N be the midpoint of BC: then N has coordinates: + , + x2 x32 y2 y32 Let P be the midpoint of CD: then P has coordinates: + , + x3 x42 y3 y42 Let Q be the midpoint of DA: then Q has coordinates: + , + x4 x12 y4 y12 We want to show that MNPQ is a parallelogram. Let R be the midpoint of MP; then R has coordinates:
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Unformatted text preview: + + + , x1 x22 x3 x422 + + + y1 y22 y3 y422 + + + , + + + Rx1 x2 x3 x44 y1 y2 y3 y44 Similarly, let S be the midpoint of NQ; then S has coordinates: + + + , x2 x32 x4 x122 + + + y2 y32 y4 y122 + + + , + + + Sx1 x2 x3 x44 y1 y2 y3 y44 It follows that points R and S coincide since they have the same coordinates. This means that segments MP and NQ have the same midpoint. But this is one of the characteristic properties of parallelograms: a quadrilateral is a parallelogram if and only if the diagonals bisect each other at a common point. Hence, this proves that MNPQ is a parallelogram....
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This note was uploaded on 07/22/2011 for the course ME 3 taught by Professor Prof.ramachandran during the Spring '11 term at Indian Institute of Technology, Kharagpur.

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