When determining the scale factor of a dilation which results in an image that is congruent to the pre-image, you need to consider the properties of congruence.
1. Definition of Congruence: Two figures or shapes are congruent if and only if one can be transformed into the other via rigid motions such as translations (slides), rotations (turns), reflections (flips), and congruence-preserving dilations. Rigid motions do not change the size or shape of the figure.
2. Dilation: Dilation is a transformation that produces an image that is the same shape as the original, but is a different size. The dilation has a center and a scale factor. A scale factor greater than 1 enlarges the figure, whereas a scale factor between 0 and 1 reduces the size of the figure.
3. Congruent Figures through Dilation: For an image produced by dilation to be congruent to the pre-image, it implies that there is no change in the size of the figure. Thus, neither resizing nor distortion occurs. Therefore, the scale factor must maintain the original dimensions.
4. Scale Factor of 1: The scale factor that maintains the original size of the figure through dilation (thus preserving congruence) is 1. This means each point of the image is exactly the same distance from the center of dilation as the corresponding point of the pre-image.
Thus, the scale factor of the dilation that results in a congruent image of a triangle is:
[tex]\[ \boxed{1} \][/tex]
