Rigid — Definition, Formula & Examples
Rigid describes a transformation that moves a figure without changing its size or shape. Translations, reflections, and rotations are all rigid transformations.
A rigid transformation (also called a rigid motion or isometry) is a mapping of points in the plane that preserves the distance between every pair of points, ensuring the pre-image and image are congruent.
How It Works
To determine whether a transformation is rigid, check if every length and angle in the original figure is unchanged in the image. If a triangle has sides of 3 cm, 4 cm, and 5 cm before the transformation, those same measurements must appear after it. Translations slide figures, reflections flip them, and rotations turn them — all without stretching or shrinking. Because rigid transformations preserve distances, the original figure and its image are always congruent.
Worked Example
Problem: Triangle ABC has vertices A(1, 2), B(4, 2), and B(4, 6). It is translated 5 units to the right. Is this a rigid transformation?
Step 1: Find the side lengths of the original triangle.
Step 2: Apply the translation by adding 5 to each x-coordinate: A'(6, 2), B'(9, 2), C'(9, 6).
Step 3: Find the side lengths of the translated triangle.
Answer: All side lengths are preserved, so the translation is a rigid transformation and triangle A'B'C' is congruent to triangle ABC.
Why It Matters
Rigid transformations are the foundation of congruence in geometry courses. Two figures are congruent precisely when one can be mapped onto the other by a sequence of rigid motions. Understanding this idea is essential for writing geometric proofs and reasoning about symmetry.
Common Mistakes
Mistake: Thinking a dilation is rigid because the shape looks the same.
Correction: A dilation changes the size of a figure, so distances between points are not preserved. Only translations, reflections, and rotations are rigid.