What is the reflection of the point (6, −3) in the horizontal line y = 2 in the Cartesian plane?

Introduction / Context:
This coordinate geometry question asks for the reflection of a point across a horizontal line. Reflecting a point across a line involves finding a point on the opposite side of the line at the same perpendicular distance. Here, we reflect the point (6, −3) across the line y = 2.

Given Data / Assumptions:
Original point P has coordinates (6, −3).Line of reflection is the horizontal line y = 2.We seek the coordinates of the reflected point P'.

Concept / Approach:
For reflection across a horizontal line y = k, the x coordinate remains unchanged and only the y coordinate is adjusted. The distance from the original point to the line is measured vertically, and the reflected point lies the same distance on the other side of the line. If the original y coordinate is y₁, the reflected y coordinate y₂ satisfies y₂ − k = −(y₁ − k).

Step-by-Step Solution:
Step 1: Identify the distance from point P to the line y = 2.Step 2: The original y coordinate is −3. The line y = 2 has y = 2.Step 3: Compute the vertical distance: 2 − (−3) = 2 + 3 = 5.Step 4: The reflected point must lie 5 units above the line if the original point is 5 units below it.Step 5: Therefore, the new y coordinate is 2 + 5 = 7.Step 6: Reflection across a horizontal line does not change the x coordinate, so x remains 6.Step 7: The reflected point has coordinates (6, 7).

Verification / Alternative check:
We can also use the formula for reflection across y = k: if the original y coordinate is y₁, then the reflected y coordinate is y₂ = 2k − y₁. Here k = 2 and y₁ = −3, so y₂ = 2 × 2 − (−3) = 4 + 3 = 7. This matches the coordinate found earlier. The x coordinate remains 6, so the reflected point is (6, 7).

Why Other Options Are Wrong:
Options with x = −2 or x = −6 incorrectly change the x coordinate, which should remain constant when reflecting across a horizontal line. The point (−2, 3) is not the correct symmetric point about y = 2 and also changes x. Only (6, 7) maintains the correct horizontal position and symmetric vertical distance.

Common Pitfalls:
A common mistake is to reflect about the x axis instead of y = 2, or to change both coordinates instead of only the y coordinate. Another error is to subtract the distance twice, leading to an incorrect y value. Always calculate the vertical distance from the point to the line and then move the same distance on the opposite side.

Final Answer:
The reflection of (6, −3) in the line y = 2 is the point (6, 7).