a rigid motion
Imagine sliding, turning, or flipping a paper triangle on a tabletop. You can move it anywhere and spin it any way, but you never stretch, shrink, bend, or tear it — the shape arrives unchanged in size and form, only in a new spot or orientation. A rigid motion is exactly this idea made precise: a way of moving every point of the plane so that the figure carried along is identical to the original, just relocated.
Precisely, a rigid motion is a transformation of the plane that preserves distance: if it sends point A to A' and point B to B', then |A'B'| = |AB| for every pair of points. Because distances are kept, so is everything built from distances — angles, areas, the lengths of all sides, the whole shape. A handy mental test: a rigid motion is one you could physically act out with a rigid cardboard cut-out, sliding and turning it (a translation or rotation) or, if you allow flipping it over, a reflection. The four basic plane rigid motions are translation, rotation, reflection, and glide reflection.
Why care? Rigid motions are the engine behind the very meaning of 'congruent'. Two figures are congruent precisely when some rigid motion carries one exactly onto the other — that is the modern, motion-based way to say 'same size and shape'. One caution worth stating: a rigid motion need not have any fixed point (a translation moves everything), and it can change orientation (a reflection turns a left hand into a right hand), so 'rigid' means 'distance-preserving', not 'leaves something pinned in place'.
Take triangle ABC with A(0, 0), B(4, 0), C(0, 3). Slide it 5 units right and 2 up: A goes to (5, 2), B to (9, 2), C to (5, 5). Check one side: |AB| = 4 before, |A'B'| = 4 after. Every length matches, so the image is congruent to the original — a rigid motion has occurred.
Same shape, new place: distances survive, so the figure is congruent to where it started.
'Rigid' refers to preserving distance, not to keeping anything still. A translation is a rigid motion yet has no fixed point at all, and reflections flip orientation — both are perfectly rigid.