A point P(5, 2) is reflected in the vertical line x = −3 in the coordinate plane. After this reflection, what are the coordinates of the image of point P?

Introduction / Context:
This problem checks understanding of reflections of points in vertical lines in coordinate geometry. Reflecting across a vertical line changes the x coordinate but preserves the y coordinate, and the new x is symmetric with respect to the given line.

Given Data / Assumptions:

• Original point P has coordinates (5, 2)
• Reflection line is x = −3
• We assume the usual Cartesian coordinate system
• We must find coordinates of the reflected point

Concept / Approach:
For reflection across a vertical line x = k, the y coordinate remains unchanged while the x coordinate is mirrored across k. If the original x is x₁, then the distance from x₁ to k equals the distance from the reflected x coordinate to k, but on the opposite side. Algebraically, reflected x is x′ = 2k − x₁.

Step-by-Step Solution:
Line of reflection is x = −3 Original point P has x coordinate 5 and y coordinate 2 Distance from P to the line along x axis is 5 − (−3) = 8 units The reflected point must lie 8 units on the other side of x = −3 So new x coordinate x′ = −3 − 8 = −11 The y coordinate does not change, so y′ = 2 Reflected point is (−11, 2)

Verification / Alternative check:
We can use the formula x′ = 2k − x₁ directly. Here k = −3 and x₁ = 5. So x′ = 2(−3) − 5 = −6 − 5 = −11. The y coordinate remains 2. This matches the coordinate pair (−11, 2) obtained from the distance based reasoning, confirming the result.

Why Other Options Are Wrong:
(−11, −2) incorrectly changes the y coordinate. (11, −2) and (11, 2) lie on the opposite side of the origin and are not symmetric about x = −3 relative to point (5, 2). The option (5, 2) is just the original point and represents no reflection at all.

Common Pitfalls:
Students sometimes reflect across the y axis x = 0 instead of the given line x = −3. Others mistakenly negate both coordinates or only change the sign of x without considering the exact distance from the line of reflection. Remember that reflection preserves perpendicular distance to the mirror line.

Final Answer:
The image of P after reflection in x = −3 is (−11, 2).