In coordinate geometry, what is the reflection of the point (5, 3) across the horizontal line y = -2? Give the coordinates of the reflected point.

Introduction / Context:
This transformation problem involves reflecting a point across a horizontal line in the coordinate plane. Reflection is a basic geometric transformation where each point is mapped to a mirror image across a given line. Here, the mirror line is y = -2, which is parallel to the x-axis.

Given Data / Assumptions:

• Original point P has coordinates (5, 3).
• The line of reflection is y = -2.
• The line y = -2 is horizontal, located 2 units below the x-axis.
• We must find coordinates of the reflected point P'.

Concept / Approach:
For reflection across a horizontal line y = k: x-coordinate stays the same. y-coordinate transforms using y' = 2k − y. This formula ensures that the line y = k is exactly halfway between the point and its reflection, so the vertical distances to the line are equal and opposite.

Step-by-Step Solution:
Given P(5, 3) and reflection line y = k = -2. Keep x the same: x' = 5. Compute new y-coordinate using y' = 2k − y. So y' = 2 * (-2) − 3 = -4 − 3 = -7. Thus, the reflected point P' is (5, -7).

Verification / Alternative check:
Check the vertical distances. Original y = 3, reflection line y = -2. The distance from P to the line is 3 − (-2) = 5 units. For P' at y = -7, the distance to the line is -2 − (-7) = 5 units. Since these distances are equal and on opposite sides of the line, the reflection is correct.

Why Other Options Are Wrong:
(-9, 3) and (-9, -3) change the x-coordinate, which should remain 5 under reflection across a horizontal line. (-5, -7) changes both x and y, corresponding to some other transformation, not reflection across y = -2.

Common Pitfalls:
Learners often confuse reflections across horizontal and vertical lines. For a horizontal line y = k, only the y-coordinate changes; for a vertical line x = k, only the x-coordinate changes. Another mistake is to subtract k from y instead of using the correct 2k − y formula, which ensures the line is the perpendicular bisector of the segment joining point and its image.

Final Answer:
The reflection of (5, 3) in the line y = -2 is (5, -7).