Reflection of Points across the Line y 3

Understanding Reflections through Line y 3

Line y 3 is a horizontal line parallel to the x-axis, located 3 units away from the origin in the y-direction. When reflecting a point across this line, the x-coordinate of the reflected point remains unchanged, while the y-coordinate changes based on the distance of the original point from the line y 3.

Step-by-Step Method to Find the Reflection of a Point

Identify the original point (x, y).

Calculate the distance from the original point to the line y 3. This distance is given by |y - 3|.

To find the y-coordinate of the reflected point, subtract this distance from 6. Hence, the y-coordinate of the reflected point will be 6 - y.

E.g., if the original point is (1, 2), the distance to y 3 is 1. Therefore, the y-coordinate of the reflected point is (6 - 2 4).

The x-coordinate remains the same.

For the point (1, 2), the reflected point will be (1, 4).

Method 1: Using Perpendicular Distance

Recognize that the line y 3 is parallel to the x-axis.

Let the required reflection point be P'.

By common sense, the distance between the line y 3 and the point P is the same as the distance between the line y 3 and the point P'. Since the line joining PP' is perpendicular to y 3, the x-coordinate of P' remains the same as P.

Given point P as (1, 2), the distance to the line y 3 is 1. Thus, the y-coordinate of the reflection point P' is 3 1 4.

Method 2: Using Midpoint Formula

The line y 3 bisects the line segment joining P and P'. Hence, the midpoint of these two points must lie on the line y 3.

If the reflection point P' is (1, b), the midpoint is (1 1/2, b 2/2).

The y-coordinate of the midpoint must equal 3 (since P' lies on y 3). Therefore, (2/2 3), implying (b 2/2 3). Solving for b gives (b 6 - 2 4).

The reflected point P' is (1, 4).

Key Points to Remember

Whenever you reflect a point across a line, the reflected point and the original point will be equidistant from that line. When you reflect a point across a horizontal line, only the y-coordinate will change. When you reflect a point across a vertical line, only the x-coordinate will change.

Conclusion

When reflecting points across the line y 3, the x-coordinate remains unchanged, while the y-coordinate changes based on a simple calculation involving the distance from the line y 3. Understanding these principles will help in accurately reflecting points and solving geometric problems involving reflections.