To determine the scale factor of the dilation in the case where the image of a triangle is congruent to the pre-image, we should first understand what it means for two triangles to be congruent.
Congruent triangles are triangles that are identical in shape and size. This means that not only are their corresponding angles equal, but also their corresponding side lengths are equal.
When dealing with dilation, the scale factor is the ratio by which the image is enlarged or reduced relative to the pre-image. Specifically, the scale factor is defined as the ratio of any side length of the image to the corresponding side length of the pre-image.
For two triangles to be congruent, every corresponding side length between the pre-image and image must be exactly the same. Therefore, the ratio of any side length of the image to the corresponding side length of the pre-image must be 1.
Thus, if the image of the triangle is congruent to the pre-image, the scale factor of the dilation is:
[tex]$\boxed{1}$[/tex]
