Skill: 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.