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Question
If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] and \[\vec{d}\] are the position vectors of points A, B, C, D such that no three of them are collinear and \[\vec{a} + \vec{c} = \vec{b} + \vec{d} ,\] then ABCD is a
Options
rhombus
rectangle
square
parallelogram
Solution
Given:
\[\vec{a} + \vec{c} = \vec{b} + \vec{d} \]
\[ \Rightarrow \vec{c} - \vec{d} = \vec{b} - \vec{a} \]
\[ \Rightarrow \overrightarrow{AB} = \overrightarrow{DC} \]
\[\text{ And }\vec{a} + \vec{c} = \vec{b} + \vec{d} \]
\[ \Rightarrow \vec{c} - \vec{b} = \vec{d} - \vec{a} \]
\[ \Rightarrow \overrightarrow{AD} = \overrightarrow{BC} \]
\[\text{ Also, since }\vec{a} + \vec{c} = \vec{b} + \vec{d} \]
\[ \Rightarrow \frac{1}{2}( \vec{a} + \vec{c)} = \frac{1}{2}( \vec{b} + \vec{d} )\]
so, position vector of mid point of BD = position vector of mid point of AC .]
hence diagonals bisect each other .
the given ABCD is a paralle log ram .
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