a reflection
A reflection is a flip across a line, exactly like the image you see in a flat mirror. The line plays the role of the mirror; every point on one side is copied to a matching point the same distance away on the other side, straight across the line. Your reflection in a mirror raises its left hand when you raise your right — that swap of left and right is the unmistakable signature of a reflection.
Precisely, a reflection across a line m (the axis or mirror line) sends each point P to the point P' such that m is the perpendicular bisector of the segment PP' — that is, PP' crosses m at a right angle and is cut by m into two equal halves. Points lying on m do not move at all; they are all fixed, so a reflection's set of fixed points is the entire mirror line, not just one point. Two clean coordinate cases: reflecting across the x-axis sends (x, y) to (x, -y); reflecting across the line y = x sends (x, y) to (y, x). A reflection preserves distance (it is an isometry) but reverses orientation: a counterclockwise-labelled triangle becomes clockwise. Doing the same reflection twice returns every point to where it began.
Reflections are the true building blocks of plane symmetry: the three-reflections theorem says every plane isometry is a product of at most three reflections, so reflections generate all the rigid motions. A figure has line symmetry (also called reflective or mirror symmetry) when some reflection maps it onto itself — the human face, a butterfly, and a regular polygon all carry such mirror lines. The honest caveat: because reflections flip orientation, a shape and its mirror image are congruent yet may not be slidable onto each other within the plane (think of a left and a right shoe).
Reflect triangle A(1, 1), B(4, 1), C(4, 3) across the x-axis. Each (x, y) becomes (x, -y): A' = (1, -1), B' = (4, -1), C' = (4, -3). The image sits below the axis as a mirror twin; its lengths match the original, but if ABC was labelled counterclockwise, A'B'C' now reads clockwise — orientation has flipped.
The mirror line is the perpendicular bisector of every point-to-image segment; orientation reverses.
Reflection reverses orientation, so it is an opposite isometry — a single flip can never be reproduced by any slide-and-turn within the plane. That is why a left glove and a right glove are mirror images but not slide-congruent.