GeoProof — Reflection across a Line explorer

the mirror line is the perpendicular bisector of each point–image segment

Drag the purple handles to tilt the mirror, or drag a vertex. The image is congruent, with orientation reversed.

Reflection across a Line

A reflection flips a figure over a mirror line. Each point P maps to the point P′ on the opposite side of the line, the same distance away:

the mirror is the perpendicular bisector of every segment PP′

So to reflect a point, drop a perpendicular to the mirror and continue the same distance past it.

What stays the same, what flips

Reflection is a rigid motion (an isometry): it preserves all lengths and angles, so the image is congruent to the original. But it reverses orientation — if ABC runs clockwise, A′B′C′ runs counter-clockwise. That's why your reflection in a mirror has left and right swapped.

Points on the mirror

Any point lying on the mirror line maps to itself — it is "fixed." Slide a vertex onto the line and watch its image meet it.

Building block of symmetry

A shape has line symmetry when reflecting it across some line gives back the same shape. Two reflections in a row equal a rotation (about the lines' intersection) or a translation (if the mirrors are parallel).