In coordinate geometry, what are the coordinates of the reflection of the point (2, 3) in the horizontal line y = 4?

Introduction / Context:
This problem checks understanding of reflections of points in a coordinate plane, specifically reflection in a horizontal line. Such questions are very common in school level coordinate geometry and aptitude tests, because they quickly reveal whether a learner understands how vertical and horizontal distances behave under reflection symmetry.

Given Data / Assumptions:

• Original point P has coordinates (2, 3).
• The mirror line is the horizontal line y = 4.
• We need the coordinates of the reflection P prime after reflecting P across the line y = 4.
• The coordinate system is the usual Cartesian plane.

Concept / Approach:
A horizontal line y = k reflects points by preserving the x coordinate and changing the y coordinate so that the reflected point is at the same perpendicular distance from the line on the opposite side. If a point has y coordinate y1, its reflection across y = k will have y coordinate y2 such that the vertical distance from y1 to k equals the distance from k to y2. Algebraically this means y2 = 2k − y1, while x remains unchanged because the reflection is vertical.

Step-by-Step Solution:
The mirror line is y = 4, so k = 4. The original point has coordinates (2, 3), so x1 = 2 and y1 = 3. Under reflection in a horizontal line, the x coordinate remains the same, so x2 = 2. Compute the vertical distance from the point to the line: 4 − 3 = 1 unit. The reflected point lies 1 unit on the other side of the line, so its y coordinate is 4 + 1 = 5. Hence the reflected point P prime has coordinates (2, 5).

Verification / Alternative check:
We can check by using the formula y2 = 2k − y1. With k = 4 and y1 = 3, we get y2 = 2 * 4 − 3 = 8 − 3 = 5. The x coordinate stays 2, so again the image point is (2, 5). Additionally, the midpoint between (2, 3) and (2, 5) is (2, 4), which lies exactly on the line y = 4, confirming that the line is the perpendicular bisector of the segment joining the point and its reflection.

Why Other Options Are Wrong:
(2, −5) reflects in the x axis rather than in y = 4. The points (−2, −5) and (−2, 5) change the x coordinate improperly because reflection in a horizontal line does not alter the x coordinate. The point (2, 1) is one unit below y = 2 rather than y = 4 and is not symmetric about the given line. None of these satisfy the requirement that y = 4 be the perpendicular bisector of the segment.

Common Pitfalls:
Learners sometimes confuse reflection in a horizontal line with reflection in a vertical line, inadvertently changing the wrong coordinate. Another common mistake is to subtract the distance only once instead of adding it on the other side, leading to y2 = k − d instead of y2 = k + d. Remember that reflection keeps the perpendicular distance equal but reverses direction relative to the mirror line.

Final Answer:
The reflection of the point (2, 3) in the line y = 4 is (2, 5).