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Let \mathrm{ABCD} be a quadrilateral. If \mathrm{E} and \mathrm{F}are the mid points of the diagonals \mathrm{AC} and \mathrm{BD} respectively and(\overrightarrow{\mathrm{AB}}-\overrightarrow{\mathrm{BC}})+(\overrightarrow{\mathrm{AD}}-\overrightarrow{\mathrm{DC}})=k \overrightarrow{\mathrm{FE}}  then k is equal to

Option: 1

4


Option: 2

2


Option: 3

-2


Option: 4

-4


Answers (1)

best_answer

\begin{aligned} & \overrightarrow{\mathrm{AB}}-\overrightarrow{\mathrm{BC}}+\overrightarrow{\mathrm{AB}}-\overrightarrow{\mathrm{DC}}=\mathrm{k} \overrightarrow{\mathrm{FE}} \\ & (\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}})-(\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{b}})+(\overrightarrow{\mathrm{d}}-\overrightarrow{\mathrm{a}})-(\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{d}})=\mathrm{k} \overrightarrow{\mathrm{FE}} \\ & 2(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{d}})-2(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{c}})=\mathrm{k} \overrightarrow{\mathrm{FE}} \\ & 2(2 \overrightarrow{\mathrm{f}})-2(2 \overrightarrow{\mathrm{e}})=\mathrm{kFE} \\ & 4(\overrightarrow{\mathrm{f}}-\overrightarrow{\mathrm{e}})=\mathrm{k} \overrightarrow{\mathrm{FE}} \\ & -4 \overrightarrow{\mathrm{FE}}=\mathrm{k} \overrightarrow{\mathrm{FE}} \\ & \mathrm{k}=-4 \end{aligned}

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