Definition: A rigid motion is a transformation that preserves distance and angle measure, so the image is congruent to the original; reflections, rotations, and translations are rigid motions.
What stays the same?
Lengths, angle measures, and the shape and size of the figure. What may change is its position or orientation. A rigid motion is sometimes called an isometry.
Which transformations qualify?
- Translation: slides every point the same distance in the same direction.
- Reflection: flips the figure across a line.
- Rotation: turns the figure about a point by a given angle.
Dilations are not rigid unless the scale factor is 1, because they change size.
Example. Triangle ABC has A(1, 1), B(4, 1), C(1, 5). Translate by (x + 6, y – 2): A'(7, -1), B'(10, -1), C'(7, 3). The side lengths remain 3, 4, and 5, so the image is congruent. Two figures are congruent exactly when a sequence of rigid motions maps one onto the other.
What error is typical?
Calling a stretch or squeeze a rigid motion. Also, orientation may reverse in a reflection but the shape is unchanged.