Distinguish rigid and non-rigid transformations

A rigid transformation of the plane—such as a translation, rotation, reflection, or composition of these—preserves distances, angle measures, and overall shape, so the image is congruent to the original even when its position or orientation changes. A non-rigid transformation changes at least some distances, as in a dilation, stretch, or shear; a dilation preserves angle measures and shape while changing size, whereas other non-rigid transformations may change shape. The scope is plane geometry and coordinate representations, not abstract or higher-dimensional transformations.

Detailed Explanation: Distinguish rigid and non-rigid transformations

A rigid transformation preserves distances and angle measures. The figure may move or turn, but it stays the same size and shape. Translations, rotations, reflections, and combinations of these are rigid.

A non-rigid transformation changes at least one distance. For example, a dilation, stretch, or shear can change the size or shape.

Worked example

Triangle $ABC$ has vertices

A(0,0),\qquad B(3,0),\qquad C(0,4).

The transformation is

(x,y)\rightarrow(2x+1,\;2y-3).

Is the transformation rigid or non-rigid?

Step 1: Find the image points.

Apply the rule to each vertex:

\begin{aligned} A(0,0)&\rightarrow A'(1,-3),\\ B(3,0)&\rightarrow B'(7,-3),\\ C(0,4)&\rightarrow C'(1,5). \end{aligned}

Step 2: Compare corresponding side lengths.

In the original triangle:

$AB=3$
$AC=4$

In the image:

$A'B'=6$
$A'C'=8$

The corresponding side lengths have doubled:

3\rightarrow 6,\qquad 4\rightarrow 8.

Step 3: Classify the transformation.

Because the distances changed, the transformation is non-rigid. The rule doubles the size of the triangle and then moves it, so the image is not congruent to the original.

To distinguish transformations, compare corresponding distances: if all distances stay the same, the transformation is rigid; if any distance changes, it is non-rigid.

Learn by doing: Distinguish rigid and non-rigid transformations

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Multiple Transformations - Grid Image & Transformations to Similar or Congruent

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