Final answer:
In geometry, rotations, translations, and reflections maintain the congruence of figures. Transformations that include dilations (with a scale factor other than 1) or stretches do not produce congruent figures because they alter the sizes of the figures. Therefore, only a combination of rotations, translations, and reflections will always yield congruent figures.
Step-by-step explanation:
In the realm of geometry, certain transformations can be applied to figures without altering their size and shape, thus preserving congruence. When seeking combinations of transformations that will always yield congruent figures, we must consider only those that maintain the original dimensions and orientation of the figure.
- A rotation followed by a translation preserves the shape and size because neither of these transformations alters the figure's dimensions.
- Similarly, a translation followed by a reflection, or another reflection, will maintain congruency. Each reflection is like a flip over a line, which does not change the size or shape.
However, any transformation that includes a dilation (except when the scale factor is 1) or a stretch will not produce a congruent figure, as these transformations change the figure's size. Therefore, a dilation followed by a rotation, a reflection followed by a dilation, and a translation followed by a rotation and then a stretch would not produce congruent figures.