Suppose PQ=QR. Your friend says that Q is always the midpoint of segment PR. Is he correct? Explain.

No, your friend is not correct. The statement "PQ = QR" does not necessarily imply that Q is the midpoint of segment PR.

### Explanation:
- **PQ = QR** means that the distances between \( P \) and \( Q \), and \( Q \) and \( R \), are equal. This suggests that \( Q \) divides the segment \( PR \) into two equal parts, but this is only true if \( P \), \( Q \), and \( R \) are collinear (i.e., they lie on the same straight line).

- If \( P \), \( Q \), and \( R \) are **not collinear**, then \( Q \) may not lie on the line segment \( PR \), meaning \( Q \) would not be a midpoint. In such a case, even though the distances \( PQ \) and \( QR \) are equal, \( Q \) would not divide the segment \( PR \) in half.

Thus, for \( Q \) to be the midpoint of segment \( PR \), the points must be **collinear** in addition to having equal distances. Therefore, \( PQ = QR \) alone is not enough to conclude that \( Q \) is always the midpoint of \( PR \).

### Do you have any questions, or would you like more details?

Here are 5 related questions for further exploration:
1. What is the definition of a midpoint in geometry?
2. How can we prove that a point is the midpoint of a line segment?
3. What other conditions must hold for \( PQ = QR \) to imply that Q is the midpoint?
4. What happens if \( PQ = QR \) but the points are not collinear?
5. How does the concept of vectors help to prove midpoint relationships?

**Tip:** When solving geometric problems, always check for additional conditions like collinearity or specific orientations of the points.

Suppose PQ = QR. Your friend says that Q is always the midpoint of segment PR. Is he correct? Explain.

Explore the geometric question of whether Q is always the midpoint of segment PR when PQ = QR. Learn about key concepts such as midpoints, collinearity, and when equal distances do not imply a midpoint. Suitable for students in Grades 6-8.

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