Question

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

• A. $2MN$
• B. $2NM$
• C. $4MN$
• D. $4NM$

Answer 1

Answer: (C)

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, $AB+AD+CB+CD$
$=(b-a)+(d-a)+(b-c)+(d-c)$
$=2(b+d)-2(a+c)$
$=2\times 2n-2\times 2m$
$=4(n-m)=4MN$