Reflections

Unit 9: Transformations · ~15–30 min

What a reflection does

A reflection flips a figure across a line, called the line of reflection, the way a mirror would. Every point of the image is the same distance from the line as the corresponding point of the preimage, but on the opposite side — the line of reflection is the perpendicular bisector of every segment connecting a point to its image. Unlike a translation, a reflection reverses orientation: a figure and its mirror image are congruent, but you can't slide one onto the other without flipping it.

Coordinate rules for common lines

Three lines of reflection come up constantly enough to memorize as rules:

• Reflect over the x-axis: (x, y) → (x, −y). The x-coordinate stays, the y-coordinate flips sign.
• Reflect over the y-axis: (x, y) → (−x, y). The y-coordinate stays, the x-coordinate flips sign.
• Reflect over the line y = x: (x, y) → (y, x). The coordinates swap places.

For a line of reflection that isn't one of these three, use the perpendicular-bisector property directly: the line must be perpendicular to segment PP′ and pass through its midpoint.

Guided practice

• 1.Reflect point B(−3, 8) over the x-axis.
  Show answer

  (x, y) → (x, −y) gives B′(−3, −8).
• 2.Reflect point C(5, −2) over the y-axis.
  Show answer

  (x, y) → (−x, y) gives C′(−5, −2).
• 3.Reflect point D(7, 1) over the line y = x.
  Show answer

  (x, y) → (y, x) gives D′(1, 7).
• 4.Quadrilateral has vertex P(3, 4). After a reflection, P′(3, −4). Which line was the figure reflected over?
  Show answer

  The x-coordinate stayed the same and the y-coordinate flipped sign, so the reflection was over the x-axis.
• 5.Does a reflection preserve orientation? Explain briefly.
  Show answer

  No. A reflection reverses orientation — the image is a mirror flip of the preimage, so you can't slide or turn one onto the other without lifting it off the plane, even though the two are congruent.