an isometry
/ eye-SOM-uh-tree /
The word isometry comes from Greek roots meaning 'equal measure', and that is exactly its job: a map of the plane (or of space) that keeps all measurements of length intact. If you think of a rigid motion as the physical act of moving a shape without distorting it, an isometry is the same idea stated as a clean mathematical condition on points, with no appeal to picking the cardboard up and putting it down.
Formally, a transformation T of the plane is an isometry if it preserves distance: |T(A) T(B)| = |AB| for all points A and B. From this single requirement a surprising amount follows automatically — an isometry must send straight lines to straight lines, preserve every angle, map circles to circles of the same radius, and carry any figure to a congruent copy. In the plane, the isometries are precisely the four kinds: translations, rotations, reflections, and glide reflections, and any composition of them is again one of these four. They form a group under composition: doing one isometry then another is again an isometry, and every isometry can be undone.
In everyday geometry 'isometry' and 'rigid motion' are used interchangeably, and that is fine. The subtle distinction some books draw is that a rigid motion is an isometry you can achieve by a continuous physical motion within the plane (a slide or a turn), whereas a reflection requires flipping the figure over — lifting it out of the plane. So all rigid motions are isometries; whether a reflection counts as a 'motion' is a matter of definition, not of mathematics.
A reflection across the y-axis sends (x, y) to (-x, y). Check it preserves distance for A(1, 2) and B(4, 6): before, |AB| = sqrt((4-1)^2 + (6-2)^2) = 5. Their images are A'(-1, 2), B'(-4, 6), and |A'B'| = sqrt((-4-(-1))^2 + (6-2)^2) = sqrt(9 + 16) = 5. Equal measure — it is an isometry.
One condition — distance preserved — forces angles, lines, and circles all to be preserved too.
Preserving distance automatically preserves angles and area, but the converse fails: a dilation preserves angles and ratios yet stretches every length, so a dilation is not an isometry.