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Rigid Transformation

A rigid transformation is a transformation that moves a figure without changing its size or shape. The three types are translations (slides), rotations (turns), and reflections (flips).

A rigid transformation (also called an isometry) is a geometric transformation that preserves the distance between every pair of points in a figure. Because distances are preserved, angles and side lengths remain unchanged, meaning the pre-image and image are always congruent. Rigid transformations include translations, rotations, and reflections, as well as any combination of these.

Worked Example

Problem: Triangle ABC has vertices A(1, 2), B(4, 2), and B(4, 6). Apply a reflection over the y-axis to find the image, then verify that the transformation is rigid.
Step 1: Apply the reflection rule. Reflecting over the y-axis changes each point (x,y)(x, y) to (−x,y)(-x, y).
A(1,2)→A′(−1,2),B(4,2)→B′(−4,2),C(4,6)→C′(−4,6)A(1,2) \to A'(-1,2), \quad B(4,2) \to B'(-4,2), \quad C(4,6) \to C'(-4,6)
Step 2: Find the side lengths of the original triangle ABC.
AB=4−1=3,BC=6−2=4,AC=(4−1)2+(6−2)2=9+16=5AB = 4 - 1 = 3, \quad BC = 6 - 2 = 4, \quad AC = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9+16} = 5
Step 3: Find the side lengths of the image triangle A'B'C'.
A′B′=∣−4−(−1)∣=3,B′C′=∣6−2∣=4,A′C′=(−4+1)2+(6−2)2=5A'B' = |-4 - (-1)| = 3, \quad B'C' = |6-2| = 4, \quad A'C' = \sqrt{(-4+1)^2+(6-2)^2} = 5
Step 4: Compare the two triangles. All corresponding side lengths are equal, so the triangles are congruent and the transformation is rigid.
AB=A′B′=3,BC=B′C′=4,AC=A′C′=5AB = A'B' = 3, \quad BC = B'C' = 4, \quad AC = A'C' = 5
Answer: The image triangle has vertices A′(−1,2)A'(-1,2), B′(−4,2)B'(-4,2), C′(−4,6)C'(-4,6). Since all side lengths are preserved, the reflection is a rigid transformation.

Why It Matters

Rigid transformations are foundational to how congruence is defined in modern geometry: two figures are congruent if and only if one can be mapped onto the other by a sequence of rigid transformations. This idea appears throughout geometry proofs, engineering design, and computer graphics whenever objects need to be moved or mirrored without distortion.

Common Mistakes

Mistake: Including dilations as rigid transformations.
Correction: A dilation changes the size of a figure, so it does not preserve distances. Dilations are transformations, but they are not rigid. Only translations, rotations, and reflections qualify.
Mistake: Thinking that a combination of rigid transformations is no longer rigid.
Correction: Any sequence of rigid transformations (for example, a reflection followed by a translation) is itself a rigid transformation. Distances stay the same no matter how many rigid steps you chain together.

Related Terms

  • Isometry — Another name for a rigid transformation
  • Reflection — A type of rigid transformation (flip)
  • Rotation — A type of rigid transformation (turn)
  • Transformations — The broader category that includes rigid and non-rigid types