Statement-1: If ABCD is a quadrilateral and E and F are the mid-points of AC and BD respectively, then $\vec{AB}+ \vec{AD}+\vec{CB}+\vec{CD} = 4\vec{EF}$
Statement-2: If O, A, B are three points and C is the mid-point of AB, then $\vec{OA}+\vec{OB}=2\vec{OC}$
Answer & explanation
Correct answer: option 1
Using triangle law of addition in triangles OAC and OBC, we have
$\vec{OA}+ \vec{AC}=\vec{OC}$ and $\vec{OB} + \vec{BC}=\vec{OC}$
$⇒\vec{OA}+ \vec{AC} +\vec{OB}+ \vec{BC}=\vec{OC} + \vec{OC}$
$⇒\vec{OA}+\vec{OB}(\vec{AC} +\vec{BC})=2\vec{OC}$
$⇒\vec{OA}+\vec{OB}=2\vec{OC}$ [∵ C is the mid-point of AB $∴ \vec{AC} +\vec{BC} = 0$]
So, statement-2 is true.
Using statement-2, we have
$\vec{AB}+\vec{AD}=2\vec{AF}$ and $\vec{CB}+\vec{CD}=2\vec{CF}$
$⇒\vec{AB}+\vec{AD} +\vec{CB}+\vec{CD}=2(\vec{AF}+\vec{CF})$
$⇒\vec{AB}+\vec{AD} +\vec{CB}+\vec{CD}=-2 (\vec{FA}+\vec{FC})$
$⇒\vec{AB}+\vec{AD} +\vec{CB}+\vec{CD}=-2(2\vec{FC})$ [Using statement-2]
$⇒\vec{AB}+\vec{AD} +\vec{CB}+\vec{CD}=4\vec{EF}$
So, statement-1 is true and statement-2 is a correct explanation for statement-1.