https://file.aiquickdraw.com/mathbot/file/1727974601018vypbvwf5.jpg

The problem in the image involves a triangle \( \triangle MON \) with lines \( s \) and \( t \) acting as the perpendicular bisectors of sides \( NO \) and \( MO \), respectively. This geometric setup leads us to a few key observations about the relationships between the points and angles.

### Given Information:
- Lines \( s \) and \( t \) are perpendicular bisectors.
- \( s \) bisects side \( NO \), meaning that it cuts \( NO \) into two equal parts at point \( P \).
- \( t \) bisects side \( MO \), cutting it into two equal parts at point \( R \).
- Point \( Q \) is the intersection of \( s \) and \( t \), and is the circumcenter of \( \triangle MON \), meaning that it is equidistant from the vertices \( M \), \( O \), and \( N \).

### Statement Analysis:

1. **\( QR = RN \)**: Since \( Q \) is the circumcenter, it is equidistant from all the vertices of the triangle, including \( N \). Therefore, \( QR \) cannot equal \( RN \), as \( Q \) is equidistant from \( R \) and \( N \), but these distances differ. **(False)**

2. **\( MP = PO \)**: Since \( s \) is the perpendicular bisector of side \( NO \), point \( P \) is the midpoint of \( NO \), making \( MP \neq PO \) in this context. **(False)**

3. **\( \angle QNR \cong \angle QNM \)**: This statement suggests that the two angles formed by point \( Q \) are congruent. Since point \( Q \) is the circumcenter, the angles between these points are equal. **(True)**

4. **\( \angle QOR \cong \angle QOP \)**: Since \( s \) and \( t \) are perpendicular bisectors, and the angles formed around the circumcenter are symmetrical, this is a plausible statement. **(True)**

### Conclusion:
The correct answer, based on the information, is **\( \angle QNR \cong \angle QNM \)**, as this geometric relationship must always hold true when \( Q \) is the circumcenter of the triangle.

Would you like further clarification or details on how the other answers were determined?

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Here are five follow-up questions to expand on this topic:
1. What properties do circumcenters have in different types of triangles (e.g., acute, obtuse)?
2. How are the perpendicular bisectors of a triangle related to its circumcenter?
3. Can a circumcenter lie outside a triangle? If so, under what conditions?
4. What is the relationship between the circumcenter and the orthocenter of a triangle?
5. How do the perpendicular bisectors help in constructing the circumcircle?

**Tip**: Remember, the circumcenter is always equidistant from all the vertices of the triangle, making it a key point in circumscribing circles.

Lines s and t are perpendicular bisectors of sides NO and MO respectively. Which of the following is always true?

Solve a geometry problem involving perpendicular bisectors in a triangle and find out which statement is always true. This problem covers the properties of circumcenters and the relationship between the sides of a triangle. Suitable for Grades 8-10.

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