Reflection symmetry of points on a line.

Problem Description:

The image shows a geometric configuration with points \( X \), \( Y \), and \( Z \) on a horizontal line. Point \( X \) is labeled as \( X(Z') \), indicating that it is the reflection of point \( Z \) across the vertical dashed line passing through point \( Y \). The task is to solve the problem and explain the solution.

Solution:

#### Step 1: Understand the Reflection
- Reflection: When a point is reflected across a line, the line acts as the perpendicular bisector of the segment joining the original point and its reflected image.
- In this case, the vertical dashed line passing through point \( Y \) is the line of reflection. This means that:
- Point \( X \) is the reflection of point \( Z \) across this vertical line.
- The vertical line through \( Y \) is equidistant from both \( X \) and \( Z \).

#### Step 2: Analyze the Geometry
- Let the coordinates of the points be as follows:
- \( Y \) is at the origin for simplicity: \( Y = (0, 0) \).
- \( Z \) is to the right of \( Y \) on the horizontal line, so its coordinates are \( Z = (d, 0) \), where \( d > 0 \).
- Since \( X \) is the reflection of \( Z \) across the vertical line through \( Y \), \( X \) must be the same distance from \( Y \) as \( Z \) but on the opposite side. Therefore, the coordinates of \( X \) are \( X = (-d, 0) \).

#### Step 3: Verify the Reflection Property
- The distance from \( Y \) to \( Z \) is \( d \).
- The distance from \( Y \) to \( X \) is also \( d \).
- The vertical line through \( Y \) is the perpendicular bisector of the segment \( XZ \). This confirms that \( X \) is indeed the reflection of \( Z \) across this line.

#### Step 4: Conclusion
- The problem involves understanding the concept of reflection and applying it to determine the position of the reflected point.
- From the analysis, we see that if \( Z \) is at \( (d, 0) \), then \( X \) (the reflection of \( Z \)) is at \( (-d, 0) \).

Final Answer:

\[
\boxed{X \text{ is the reflection of } Z \text{ across the vertical line through } Y.}
\]

This solution explains the geometric relationship and confirms the correctness of the reflection property.