Definition: A reflection is a transformation that flips a figure over a line, called the line of reflection, so the image is a mirror image of the original.
What is a reflection?
Each point and its image are the same distance from the line of reflection, on opposite sides. The segment joining them is perpendicular to the line.
What are the coordinate rules?
- Over the x-axis: (x, y) becomes (x, -y).
- Over the y-axis: (x, y) becomes (-x, y).
- Over the line y = x: (x, y) becomes (y, x).
What does an example look like?
Reflect P(5, -3) over the y-axis: P’ = (-5, -3). Over the x-axis: (5, 3).
For triangle A(2, 1), B(6, 1), C(2, 4), the reflection over the x-axis gives A'(2, -1), B'(6, -1), C'(2, -4).
What changes and what stays the same?
Lengths and angles are preserved, so the image is congruent to the original. But orientation reverses: clockwise order of vertices becomes counterclockwise.
How do you find the line?
It is the perpendicular bisector of any segment joining a point to its image.
What is a common mistake?
Changing the wrong sign. The x-axis reflection changes only the y-coordinate.