To determine the scale factor of the dilation for an image of a triangle that is congruent to its pre-image, let's understand what "congruent" means in the context of geometry:
1. Definition of Congruence:
- Two shapes are congruent if they are exactly identical in shape and size. This means every corresponding angle and side length in the two shapes are equal.
2. Effect of Dilation on Congruence:
- A dilation transformation changes the size of a shape but preserves its overall form. The side lengths of the shape are multiplied by a scale factor, denoted as [tex]\( k \)[/tex].
- If the scale factor [tex]\( k \)[/tex] is not equal to 1, the size of the shape changes, and it is no longer congruent to its pre-image.
3. Congruent Triangles and Dilation:
- For the image of the triangle to remain congruent to the original triangle (pre-image), the side lengths must remain unchanged.
- This means that the scale factor must be such that it does not alter the side lengths, which indicates that [tex]\( k = 1 \)[/tex].
Therefore, the scale factor of the dilation that results in the image of a triangle being congruent to its pre-image is:
[tex]\[ \boxed{1} \][/tex]
