What are the coordinates of the reflection of the point (6, −1) in the line y = 2? This is a reflection across a horizontal line, so only the y-coordinate changes while x remains the same.

Introduction / Context:
This coordinate-geometry problem tests reflection of a point across a line, specifically a horizontal line of the form y = k. Reflecting across a horizontal line is simpler than reflecting across a slanted line because the x-coordinate does not change. Only the y-coordinate shifts so that the reflected point is the same perpendicular distance from the line y = k but on the opposite side. A reliable method is to use the idea that the line y = k is the midpoint (average) of the original y and the reflected y. That means (y + y_reflected)/2 = k, so y_reflected = 2k − y. Students commonly make mistakes by changing the x-coordinate (which should remain unchanged) or by adding k instead of doubling it in the reflection formula.

Given Data / Assumptions:

• Original point P = (6, −1) • Mirror line: y = 2 (a horizontal line) • Required: reflected point P' across y = 2

Concept / Approach:
For reflection across y = k: x_reflected = x (unchanged) y_reflected = 2k − y This works because the line y = k is the midpoint between y and y_reflected. Here k = 2 and y = −1, so y_reflected = 2*2 − (−1) = 5.

Step-by-Step Solution:
1) Identify k from the line y = 2: k = 2 2) Keep the x-coordinate the same (horizontal reflection): x' = 6 3) Compute the reflected y-coordinate: y' = 2k − y = 2*2 − (−1) = 4 + 1 = 5 4) Therefore the reflected point is: (6, 5)

Verification / Alternative check:
Distance check: The original y-coordinate is −1. The line y = 2 is 3 units above −1 (because 2 − (−1) = 3). The reflected point should be 3 units above y = 2, which gives y = 5. This matches the computed result and confirms the reflection is correct. Also, x remains 6 in both points, consistent with reflection across a horizontal line.

Why Other Options Are Wrong:
• (6, 1): would reflect across y = 0, not y = 2. • (−6, 5): incorrectly changes x, which should not happen for reflection across y = 2. • (−2, 1) and (−2, −1): change x with no basis and do not preserve perpendicular symmetry to y = 2.

Common Pitfalls:
• Changing x when reflecting across a horizontal line. • Using y' = k − y instead of y' = 2k − y. • Forgetting that the line of reflection must be the midpoint between the original and reflected y-values.

Final Answer:
(6, 5)