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The question in the image involves a geometric problem where line \( XY \) is the perpendicular bisector of \( PQ \). The problem is asking which of the given statements must be true.

### Let's break down the scenario:

1. **Perpendicular Bisector**: If \( XY \) is the perpendicular bisector of \( PQ \), it implies two things:
   - \( XY \) is perpendicular to \( PQ \).
   - \( XY \) intersects \( PQ \) at its midpoint.

### Analysis of the statements:
Assume the points \( P \), \( Q \), \( X \), and \( Y \) are arranged as follows:

- \( XY \) is perpendicular to \( PQ \).
- \( XY \) intersects \( PQ \) at its midpoint \( M \).

Given that \( XY \) is the perpendicular bisector, the following must be true:
- \( \text{Option A: PX = QX} \): This is not necessarily true unless \( X \) is the midpoint of \( PQ \) as well, which is not given.
- \( \text{Option B: PR = QR} \): This must be true because \( XY \) being the perpendicular bisector implies that any point on \( XY \) is equidistant from \( P \) and \( Q \).
- \( \text{Option C: \angle PXR = \angle QXR} \): This must also be true because the perpendicular bisector creates two equal angles.
- \( \text{Option D: \triangle PXR \text{ is congruent to } \triangle QXR} \): This is true by the SAS (Side-Angle-Side) criterion because \( PX = QX \) and \( \angle PXR = \angle QXR \), and the line \( XR \) is common in both triangles.

### Conclusion:
The correct statements that must be true are:
- **B**: \( PR = QR \)
- **C**: \( \angle PXR = \angle QXR \)
- **D**: \( \triangle PXR \text{ is congruent to } \triangle QXR \)

So the correct answers are **B, C, and D**.

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### Follow-up questions:
1. How can the perpendicular bisector theorem be used to solve problems involving circles?
2. What is the significance of congruent triangles in geometry proofs?
3. How does the concept of perpendicular bisectors relate to circumcenters in triangles?
4. Can a perpendicular bisector be used to determine the location of a midpoint in any figure?
5. What other geometric properties are associated with perpendicular bisectors in polygons?

### Tip:
When dealing with perpendicular bisectors, remember that they are not only perpendicular to the segment they bisect, but they also pass through the midpoint of that segment, making them very useful in proving congruence and distance-related properties in geometry.

In the diagram, XY is the perpendicular bisector of PQ. Which of the following statements must be true?

Explore the problem where XY is the perpendicular bisector of PQ. Learn which statements are true based on the Perpendicular Bisector Theorem and the SAS Congruence Criterion. Ideal for students in Grades 8-10.

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