In coordinate geometry, what are the coordinates of the reflection of the point (5, −2) in the vertical line x = −1?

Introduction / Context: This question tests reflection of a point in a vertical line in the coordinate plane. Understanding how reflections affect x and y coordinates is fundamental in coordinate geometry and helps in reasoning about symmetry and transformations.

Given Data / Assumptions:

• Original point P has coordinates (5, −2).
• The mirror line is the vertical line x = −1.
• We seek the reflected point P prime across this line.
• The coordinate system is the standard Cartesian plane.

Concept / Approach: Reflection in a vertical line x = k preserves the y coordinate and changes the x coordinate in such a way that the point and its reflection are equally distant from the line on opposite sides. If the original x coordinate is x1, then the reflected x coordinate x2 satisfies x2 − k = −(x1 − k), which simplifies to x2 = 2k − x1. The y coordinate remains unchanged because the reflection occurs horizontally.

Step-by-Step Solution: The mirror line is x = −1, so k = −1. The original point is (5, −2), so x1 = 5 and y1 = −2. Under reflection in a vertical line, the y coordinate remains the same: y2 = −2. Compute the horizontal distance from the point to the line: 5 − (−1) = 6 units. The reflected point lies 6 units on the other side of the line, so its x coordinate is −1 − 6 = −7. Alternatively, use the formula x2 = 2k − x1 = 2 * (−1) − 5 = −2 − 5 = −7. Thus the reflected point has coordinates (−7, −2).

Verification / Alternative check: Check that the line x = −1 is the perpendicular bisector of the segment joining (5, −2) and (−7, −2). The midpoint of these points is ((5 + (−7))/2, (−2 + (−2))/2) = (−2/2, −4/2) = (−1, −2). The x coordinate of the midpoint equals −1, which lies on the mirror line x = −1, and the line segment from the original point to the reflection is horizontal, confirming a correct reflection in a vertical line.

Why Other Options Are Wrong: The point (7, −2) lies on the opposite side of the origin and is farther from x = −1 than the original point. The points (5, 0) and (5, 2) change the y coordinate instead of the x coordinate, which would correspond to reflection in a horizontal line, not a vertical one. The point (−3, −2) is only 2 units away from x = −1, not 6 units, so it does not maintain equal distance on both sides of the line.

Common Pitfalls: A frequent mistake is to move the point in the vertical direction when reflecting in a vertical line, or to subtract the distance only once rather than mirroring it across the line. Using the formula x2 = 2k − x1 and keeping y unchanged provides a quick and reliable way to avoid such errors.

Final Answer: The reflection of (5, −2) in the line x = −1 is (−7, −2).