ACT® · Math: Geometry — Transformations, Proof, and Constructions
Reflections Across a Vertical or Horizontal Line
Reflecting a point across a vertical or horizontal line keeps one coordinate fixed and flips the other to the same distance on the opposite side of the mirror line.
A reflection is a flip over a line, called the line of reflection or mirror line. Across a vertical line, left-right positions swap but up-down positions never change, because the mirror only cares about horizontal distance. Across a horizontal line, it is the opposite: up-down flips, left-right stays put. The mirror line always sits exactly halfway between a point and its image, which is the whole mechanism behind the formula.
For a vertical line x = c, the y-coordinate of a point never moves, and the new x-coordinate is 2c minus the original x. That formula comes straight from the midpoint idea: if the mirror is halfway between x and its image x', then c = (x + x')/2, so x' = 2c - x. The same logic flipped sideways handles horizontal lines: for y = c, the x-coordinate stays fixed and the new y-coordinate is 2c minus the original y.
This sits right next to reflections across the x-axis and y-axis, which are just the special cases where c = 0 — that is why negating a coordinate only works when the mirror is one of the axes. It also connects to translations and rotations: chain a reflection across the x-axis with one across the y-axis and you get the same result as a 180-degree rotation about the origin, since both coordinates end up negated. Seeing reflections as a special case of a general rule, rather than a trick with negative signs, makes the rest of the transformations unit click faster.
Key terms
- Line of reflection
- The mirror line that a figure flips across, with every point and its image equally distant from it.
- Image
- The new point or figure produced after a transformation is applied to the original.
- Vertical line x = c
- A line made of every point with x-coordinate equal to c, running straight up and down.
- Horizontal line y = c
- A line made of every point with y-coordinate equal to c, running straight left and right.
Watch it explained
- How it worksHow does reflecting across the vertical line x = 3 send a point to the same distance from that line on the opposite side?
- Worked exampleFind the image of (7, -2) after a reflection across the vertical line x = 4.
- Worked exampleFind where (-5, 6) lands after a reflection across the horizontal line y = 1.
- Common mistakeWhy isn't reflecting across the line x = 3 the same thing as making the x-coordinate negative?
- Going furtherWhat happens to a point when you reflect it across the x-axis and then reflect that result across the y-axis?
Check yourself
End of topic.