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  4. M and N are the midpoints of the diagonals AC and BD respectively of quadrilateral ABCD, then AB +AD+ CB +CD =

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Q. $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$ respectively of quadrilateral $ABCD$, then ${AB} +{AD}+ {CB} +{CD} = $ _______________

MHT CETMHT CET 2016Vector Algebra

Solution:

Let the position vectors of $A, B, C, D, M$ and $N$ are $a , b , c , d , m$ and $n $
Since, $M$ and $N$ are the mid-points of $AC$ and $BD$.
$m =\frac{ a + c }{2}, n =\frac{ b + d }{2}$
Now, $ A B+A D+C B+C D $
$=(b-a)+(d-a)+(b-c)+(d-c) $
$=2(b+d)-2(a+c)$
$=2 \times 2 n-2 \times 2 m$
$=4(n-m)=4 \,MN $