In coordinate geometry, find the reflection of the point (5, −2) across the horizontal line y = −1, and give the coordinates of the reflected point.

Introduction / Context:
This problem tests understanding of reflections in coordinate geometry, specifically reflection across a horizontal line. When a point is reflected across a horizontal line y = k, its x coordinate stays the same, while its y coordinate is mirrored with respect to k. Being able to compute such reflections quickly is important in geometry, transformations, and graphical reasoning questions.

Given Data / Assumptions:

• The original point is P(5, −2).
• The line of reflection is the horizontal line y = −1.
• We need to find the coordinates of the reflected point P' after reflection in this line.
• The coordinate system is the standard Cartesian plane.

Concept / Approach:
When reflecting a point across a horizontal line y = k, the x coordinate does not change because the reflection is vertical. The y coordinate transforms according to a simple formula: if the original y coordinate is y, then the reflected y coordinate y' is given by y' = 2k − y. This formula captures the idea that the line y = k is exactly halfway between the point and its reflection. Applying this rule with k = −1 and y = −2 will give the new y coordinate directly.

Step-by-Step Solution:
1) Identify k from the line y = k. Here k = −1. 2) For the point P(5, −2), the x coordinate is x = 5 and the y coordinate is y = −2. 3) Under reflection across y = −1, the x coordinate remains unchanged, so x' = 5. 4) Compute the reflected y coordinate using y' = 2k − y. 5) Substitute k = −1 and y = −2: y' = 2(−1) − (−2) = −2 + 2 = 0. 6) Therefore the reflected point is P'(5, 0).

Verification / Alternative check:
You can check the result geometrically by measuring vertical distances. The line y = −1 lies between y = −2 and y = 0. The distance from y = −2 up to y = −1 is 1 unit. The distance from y = −1 up to y = 0 is also 1 unit. Since the line of reflection should be the perpendicular bisector of the segment joining a point and its reflection, having equal distances on both sides confirms that (5, 0) is the correct reflected point of (5, −2) across y = −1.

Why Other Options Are Wrong:
Option b (−5, 0) changes the x coordinate, which should stay the same in reflection across a horizontal line. Options c (−7, −2) and d (−7, 2) modify both coordinates and do not preserve vertical alignment with the original point. Option e (5, −4) flips the point in the wrong direction, moving it further down away from y = −1 instead of above it. Only option a, (5, 0), maintains the correct x coordinate and symmetry about the line y = −1.

Common Pitfalls:
A frequent mistake is to confuse reflection across a horizontal line with reflection across a vertical line, leading to changes in x instead of y. Some learners also mis calculate the distance when the line has a negative y value, forgetting that the formula y' = 2k − y works uniformly for all real k. Drawing a quick sketch of the point and the line or imagining the vertical distances on the y axis helps avoid these errors.

Final Answer:
The reflection of the point (5, −2) in the line y = −1 is the point (5, 0).