**In ΔPQR, PQ = QR and O is an interior point of ΔPQR such that ∠OPR = ∠ORP. Consider the following statements:**

(i) ΔPOR is an isosceles triangle.

(ii) O is the centroid of ΔPQR.

(iii) ΔPQQ is congruent to ΔRQO.

**Which of the above statements are correct?**

Question

**In ΔPQR, PQ = QR and O is an interior point of ΔPQR such that ∠OPR = ∠ORP. Consider the following statements:**

(i) ΔPOR is an isosceles triangle.

(ii) O is the centroid of ΔPQR.

(iii) ΔPQQ is congruent to ΔRQO.

**Which of the above statements are correct?**

Accepted Answer

**Solution:**

1. Given ΔPQR with PQ = QR, so ΔPQR is isosceles with PQ = QR.

2. Point O is inside ΔPQR such that ∠OPR = ∠ORP.

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**Step 1: Evaluate statement (i)**

- In ΔPOR, ∠OPR = ∠ORP.
- By the converse of the Isosceles Triangle Theorem, if two angles are equal, the sides opposite those angles are equal.
- Therefore, OP = OR.
- Hence, ΔPOR is isosceles.

**Statement (i) is correct.**

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**Step 2: Evaluate statement (ii)**

- The centroid is the intersection of medians.
- No information is given about medians or O being the intersection of medians.
- Hence, we cannot conclude O is the centroid.

**Statement (ii) is incorrect.**

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**Step 3: Evaluate statement (iii)**

- Consider ΔPQQ and ΔRQO (assuming a typo and it means ΔPQO and ΔRQO).
- Given PQ = QR.
- O is common in both triangles.
- Since OP = OR (from step 1), and PQ = QR, by Side-Angle-Side (SAS) criterion, ΔPQO ≅ ΔRQO.

**Statement (iii) is correct.**

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**Answer:** Only statements (i) and (iii) are correct.

**Correct option:** 2. Only (i) and (iii)

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