In coordinate geometry, find the reflection of the point (4, 7) in the horizontal line y = -1 by using the concept of equal perpendicular distances from the line, and choose the correct reflected coordinates.

Introduction / Context:
Reflection of points in lines is a fundamental concept in coordinate geometry and is often tested in aptitude questions. Here, you are asked to reflect a point in a horizontal line, which is easier than a general line because only the y coordinate changes. Understanding how distances behave under reflection is the key to solving such problems quickly and accurately.

Given Data / Assumptions:

• Original point P has coordinates (4, 7).
• The mirror line is the horizontal line y = -1.
• We want the coordinates of the reflected point P prime.
• The reflection is across a horizontal line, so the x coordinate remains unchanged.
• The vertical distance from the point to the line is equal on both sides after reflection.

Concept / Approach:
For reflection across a horizontal line y = k, the x coordinate of a point does not change, while its y coordinate moves so that the line lies exactly midway between the original point and its image. In other words, the line y = k is the perpendicular bisector of the segment joining the original point and its reflection. Therefore, the y coordinates of the original and the reflected point must be symmetric with respect to k, meaning their average is k.

Step-by-Step Solution:
The original point is P(4, 7) and the mirror line is y = -1. For a reflection in a horizontal line, the x coordinate stays the same, so x prime = 4. Let the reflected point be P prime with coordinates (4, y prime). The line y = -1 is the midpoint of the vertical segment between 7 and y prime. Therefore (7 + y prime)/2 = -1. Solve for y prime: 7 + y prime = -2, so y prime = -9. Thus the reflected point is (4, -9).
Verification / Alternative check:
You can check the vertical distances from both the original point and the reflected point to the line y = -1. From (4, 7) to y = -1, the distance is 7 - (-1) = 8 units upward. From (4, -9) to y = -1, the distance is -1 - (-9) = 8 units upward if you consider absolute distance. The line y = -1 lies exactly halfway between y = 7 and y = -9, confirming that the reflection is correct.

Why Other Options Are Wrong:
Option b ( -4, -9 ) changes both x and y coordinates, which is not correct for a reflection in a horizontal line. Options c and d change the x coordinate but do not maintain symmetry about y = -1. Option e (4, 1) maintains the x coordinate but places the point only 2 units above y = -1, whereas the original is 8 units above, so the distances are not equal.

Common Pitfalls:
A frequent error is to reflect across the x axis (y = 0) instead of the given line y = -1, which leads to incorrect y coordinates such as -7. Another pitfall is changing both coordinates when reflecting across a horizontal or vertical line, even though only the coordinate perpendicular to the line should change. Always identify whether the line is horizontal or vertical and adjust only the relevant coordinate accordingly.

Final Answer:
The reflection of the point (4, 7) in the line y = -1 is (4, -9).