What are isometries/congruence transformations?

Isometries, also known as congruence transformations, are transformations that preserve the shape and size of a figure. This means that when a figure undergoes an isometry, its dimensions and the distances between points remain unchanged, leading to figures that are congruent to each other.

For instance, when a shape is translated, rotated, or reflected, each point of the shape moves in such a way that the original dimensions and angles are maintained. The resulting figure is identical in terms of size and shape to the original figure, although its position may change.

In contrast, transformations that increase the size of a figure would change the dimensions and violate the concept of isometry. Similarly, transformations that change the shape or create perspective distortion do not correspond to congruence transformations, as these would alter either the proportions or the visual characteristics of the figure in a way that does not preserve congruence. Thus, the focus on maintaining the same measures is what categorizes these transformations as isometries or congruence transformations.