Question
Mon July 09, 2012 By: Nevin
 

sir/madam using vectors prove that the line segment joining the midpoint of non-parallel sides of a trapezium is parallel to the base and is equal to half the sum of the parallel sides

Expert Reply
Wed July 11, 2012
Answer: Given : a line segment joining the midpoint of non-parallel sides of a trapezium
to prove :   the line segment joining the midpoint of non-parallel sides of a trapezium is parallel to the base and is equal to half the sum of the parallel sides
 
Let us represent the vertices A by vector a, B by vector b , C by vector c, D by vector d.
 let EF be the line segment joining the midpoint of two non parallel  sides of the given trapezium.
hence the side AB can be represented by vector a- vector b and DC by vector d - vector c
since the two sides AB and CD are parallel, therefore
=>vector a- vector b = k(vector d - vector c) , k is a constant. ............................(1)
 
 
E is the mid point of AD , therefore can be represented as 
=> vector e= (vector a + vector d) /2 .....................(2)
 
 
F is the mid point of BC , therefore can be represented as 
=> vector f= (vector b + vector c) /2..........................(3)
 
 
then EF can be represented as 
= vector e  -  vector f
=  ((vector a + vector d) /2 ) - ( (vector b + vector c) /2 )    {from eq (2) and eq(3) }
= ( (vector a - vector b ) + ( vector d - vector c) ) /2 .........(4)
= k (vector a - vector b ) /2               { using eq 1) 
 
 
therefore EF is parallel to side AB , and hence to side DC 
also from eq 4 ,EF is equal to half the sum of the parallel sides
 
Hence proved 
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