If \(\overline{\mathbf{P O}}+\overline{\mathbf{O Q}}=\overline{\mathbf{Q O}}+\overline{\mathbf{O R}}\), then

• Correct answer
  A

  \(Q\) is the mid-point of \(\overline{P R}\)
• B

  \(Q\) divides \(\overline{\mathrm{PR}}\) in \(2: 1\)
• C

  \(Q\) divides \(\overline{\mathrm{PR}}\) in \(1: 2\)
• D

  \(Q\) divides \(\overline{\mathrm{PR}}\) in \(-1: 2\)

Solution

From diagram below,
Using triangle law, \(\begin{aligned} & \quad \mathbf{P O}+\mathbf{O Q}=\mathbf{Q P} \text { and } \mathbf{Q O}+\mathbf{O R}=\mathbf{Q R} \\ & \text {Given; } \mathbf{P O}+\mathbf{O Q}=\mathbf{Q O}+\mathbf{O R} \Rightarrow \mathbf{Q P}=\mathbf{Q R} \\ & \Rightarrow \quad|\mathbf{Q P}|=|\mathbf{Q R}| \Rightarrow Q P \text { length }=Q R \text { length } \\ & \therefore Q \text { is mid point of } \mathbf{P R} \end{aligned}\)

Hence, the correct answer is option: 'A'.