Question

ABCDE is a pentagon, prove that
\[\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CD} + \overrightarrow{DE} + \overrightarrow{EA} = \overrightarrow{0}\]

Answer 1

Given: ABCDE is a pentagon.
 To Prove: 
\[\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CD} + \overrightarrow{DE} + \overrightarrow{EA} = \overrightarrow{0} .\]
Proof: We have,
\[LHS = \overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CD} + \overrightarrow{DE} + \overrightarrow{EA} \]
\[= \overrightarrow{AC} + \overrightarrow{CD} + \overrightarrow{DA}\]                             [ ∵ \[\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}\]  and  \[\overrightarrow{DE} + \overrightarrow{EA} = \overrightarrow{AD}\]]
\[= \overrightarrow{AD} + \overrightarrow{DA}\]
∵[\[\overrightarrow{AC} + \overrightarrow{CD} = \overrightarrow{AD}]\]
\[= \overrightarrow{0}\] = RHS 
Hence proved.