What is the reflection of the point (1, 2) in the horizontal line y = 3 in the Cartesian plane?

Introduction / Context:
This coordinate geometry question involves reflecting a point in a horizontal line. Reflection problems help strengthen understanding of symmetry in the Cartesian plane and of how x and y coordinates change under such transformations.

Given Data / Assumptions:

• Original point is A(1, 2).
• Line of reflection is the horizontal line y = 3.
• We want the coordinates of the reflected point across this line.

Concept / Approach:
Reflection in a horizontal line y = k keeps the x coordinate the same and adjusts the y coordinate so that the line is the perpendicular bisector between the point and its image. Algebraically, if a point (x, y) is reflected across y = k, the image is (x, 2k - y). This comes from the fact that the vertical distances of the point and its image from the line must be equal and on opposite sides.

Step-by-Step Solution:
Step 1: Identify k from the line y = 3, so k = 3.Step 2: The original point has coordinates (x, y) = (1, 2).Step 3: For reflection in y = k, use the formula image (x', y') = (x, 2k - y).Step 4: Substitute values: x' = 1 and y' = 2 * 3 - 2 = 6 - 2 = 4.Step 5: Thus, the reflected point is (1, 4).

Verification / Alternative check:
You can visually verify: the original point (1, 2) is 1 unit below the line y = 3. Its reflection should therefore be 1 unit above that line, which is at y = 4, and the x coordinate stays at 1. This simple geometric reasoning confirms the algebraic result.

Why Other Options Are Wrong:
Points like (1, -4), (-1, -4), and (-1, 4) either change the x coordinate or place the point at the wrong vertical distance relative to the line y = 3. For a reflection in a horizontal line, the x coordinate must remain unchanged, ruling out all options except (1, 4). The point (3, 1) moves both coordinates and is not a reflection across y = 3.

Common Pitfalls:
Students sometimes mistakenly reflect across the x axis instead of the given horizontal line, or they subtract the y value from k instead of computing 2k - y. Remember that the line of reflection is the perpendicular bisector, so the vertical distances must be equal on both sides of the line.

Final Answer:
The reflection of point (1, 2) in the line y = 3 is (1, 4).