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If \(\overline{\mathbf{P O}}+\overline{\mathbf{O Q}}=\overline{\mathbf{Q O}}+\overline{\mathbf{O R}}\), then
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1947 Upvotes
Verified Answer
The correct answer is:
\(Q\) is the mid-point of \(\overline{P R}\)
From diagram below,
Using triangle law,
\(\begin{aligned}
& \quad \mathbf{P O}+\mathbf{O Q}=\mathbf{Q P} \text { and } \mathbf{Q O}+\mathbf{O R}=\mathbf{Q R} \\
& \text {Given; } \mathbf{P O}+\mathbf{O Q}=\mathbf{Q O}+\mathbf{O R} \Rightarrow \mathbf{Q P}=\mathbf{Q R} \\
& \Rightarrow \quad|\mathbf{Q P}|=|\mathbf{Q R}| \Rightarrow Q P \text { length }=Q R \text { length } \\
& \therefore Q \text { is mid point of } \mathbf{P R}
\end{aligned}\)
Using triangle law,
\(\begin{aligned}
& \quad \mathbf{P O}+\mathbf{O Q}=\mathbf{Q P} \text { and } \mathbf{Q O}+\mathbf{O R}=\mathbf{Q R} \\
& \text {Given; } \mathbf{P O}+\mathbf{O Q}=\mathbf{Q O}+\mathbf{O R} \Rightarrow \mathbf{Q P}=\mathbf{Q R} \\
& \Rightarrow \quad|\mathbf{Q P}|=|\mathbf{Q R}| \Rightarrow Q P \text { length }=Q R \text { length } \\
& \therefore Q \text { is mid point of } \mathbf{P R}
\end{aligned}\)
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