Question : In $\triangle PQR, PQ=QR$ and $O$ is an interior point of $\triangle PQR$ such that $\angle OPR=\angle ORP$. Consider the following statements: (i) $\triangle POR$ is an isosceles triangle. (ii) $O$ is the centroid of $\triangle PQR$. (iii) $\triangle PQO$ is congruent to $\triangle RQO$. Which of the above statements is/are correct?
Option 1: Only (i) and (ii)
Option 2: Only (i) and (iii)
Option 3: Only (ii) and (iii)
Option 4: Only (ii)
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Correct Answer: Only (i) and (iii)
Solution : Given: In $\triangle PQR, PQ=QR$ and $O$ is an interior point of $\triangle PQR$ such that $\angle OPR=\angle ORP$. Since $PQ = QR$, so $\triangle PQR$ is an isosceles triangle. From the given data we cannot say that $O$ is the centroid of the $\triangle PQR$. Also, $PQ=QR, OP=OR$ and $OQ=OQ$ So, $\triangle PQO$ and $\triangle RQO$ are congruent. Hence, the correct answer is Only (i) and (iii).
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