Reflections

When a point is reflected in a line, the line is the perpendicular bisector of the segment joining the point and its image.  We will only consider coordinate rules for reflections in horizontal and vertical lines, and in the lines y = x and y = –x since the rules for lines in general involve messy details beyond the scope of this course.

Example 3:  Give a coordinate rule for reflecting in the line vertical line x = 3.

Solution:  Consider a point (x, y) and its image (p, q):

 

The y-coordinate of the image is the same as the y-coordinate of the preimage, so q = y.  Since the line x = 3 bisects the segment from the point to its image, the horizontal distances from the point to the line and its image to the line are equal, so

                         3 – x = p – 3

 

Adding 3 to both sides tells us that p = 6 – x.  Therefore the coordinate rule is:

(x, y) --> (6 – x, y)