What are the coordinates of the reflection of the point (5, -1) in the horizontal line y = 2 in the coordinate plane?

Introduction / Context:
This problem belongs to coordinate geometry and deals with reflection of a point in a horizontal line. Reflections across lines such as y = c or x = c occur frequently in analytical geometry and transformation questions. Understanding how the coordinates change under reflection lets you answer such questions quickly without drawing a detailed graph each time.

Given Data / Assumptions:
- Original point is P(5, -1).
- The line of reflection is the horizontal line y = 2.
- Reflection means that the line y = 2 will act like a mirror.
- We need to find the coordinates of the reflected point P prime.

Concept / Approach:
Reflection of a point across a horizontal line y = c keeps the x coordinate the same, because the point moves straight up or down. The y coordinate changes such that the reflected point is the same perpendicular distance on the other side of the line. If the original y coordinate is y1 and the line is y = c, then the distance from the point to the line is d = c - y1 when the point is below the line. The reflected y coordinate will be c + d. In summary, the y values are symmetric with respect to the line y = c.

Step-by-Step Solution:
Step 1: Identify the x coordinate of the original point P. It is x = 5. Step 2: Identify the y coordinate of P. It is y = -1. Step 3: The line of reflection is y = 2. The vertical distance from P to the line is 2 - (-1) = 3 units. Step 4: The reflected point must lie the same distance above the line, so its y coordinate is 2 + 3 = 5. Step 5: The x coordinate remains unchanged, so the reflected point P prime is (5, 5).

Verification / Alternative check:
Plotting a quick sketch on graph paper or mentally visualising the axes, you see that point (5, -1) lies 3 units below the line y = 2. To reflect it, you move directly upward 3 units above the line, landing at y = 5. The x coordinate does not change, confirming that the reflection is at (5, 5).

Why Other Options Are Wrong:
(5, -5) and (-5, -5) both have y coordinates further below the line y = 2 instead of being above it.
(-5, 5) has the correct y coordinate but the x coordinate has changed sign, which does not happen in reflection across a horizontal line.
(5, 2) lies on the line of reflection itself and would correspond to a point already on the mirror, not the reflection of (5, -1).

Common Pitfalls:
One frequent mistake is to accidentally change both coordinates or only the y coordinate without preserving the distance from the mirror line. Another error is to confuse reflections in horizontal lines with reflections in vertical lines of the form x = c. Always remember that reflections in y = c keep x unchanged and adjust y symmetrically around c.

Final Answer:
The reflection of the point (5, -1) in the line y = 2 is (5, 5).