Reflections
Overview
A reflection creates a mirror image of a figure across a line, informally called a flip. This lesson gives the coordinate rules for the reflections you will use most — across the x-axis, the y-axis, horizontal and vertical lines, and the lines y = x and y = −x — and shows how to reflect any point across a line by using the fact that the line of reflection is the perpendicular bisector of the segment joining a point to its image.
Vocabulary
- Reflection
- A transformation that creates a mirror image of a figure about a specified line. The preimage and the image are equidistant from the line of reflection.
- Line of Reflection
- The line across which a figure is reflected. It is the perpendicular bisector of the segment joining any point of the preimage to its image.
- Rigid Motion
- Any transformation that maintains congruence. A reflection is a rigid motion, but it reverses the orientation of the figure.
- Composition
- Two or more transformations combined and applied in a specific order, such as a translation followed by a reflection.
Main Ideas
- Reflection in the x-axis: (x, y) → (x, −y). Reflection in the y-axis: (x, y) → (−x, y). Only the coordinate perpendicular to the line changes sign.
- Reflection in the horizontal line y = c: (x, y) → (x, 2c − y). Reflection in the vertical line x = c: (x, y) → (2c − x, y).
- Reflection in the line y = x swaps the coordinates: (x, y) → (y, x). Reflection in the line y = −x: (x, y) → (−y, −x).
- The line of reflection is the perpendicular bisector of the segment from a point to its image, so the image is the same distance from the line as the original point, on the opposite side.
- To reflect across any line: draw the perpendicular from the point to the line, find the intersection point (the midpoint), and extend the same distance beyond it.
Lesson Notes
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Lesson Video
Watch the lesson video.
Interactive Activity
Reflect figures over axes, lines, and any line, and compose reflections with translations.
Practice Problems
Want More Practice?
Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.