To determine the scale factor of the dilation for a triangle whose image is congruent to its pre-image, we need to understand a few key concepts:
1. Congruence: Two triangles are congruent if they have the same shape and size, i.e., corresponding sides are equal in length and corresponding angles are equal in measure.
2. Dilation: A dilation transformation scales an object by a certain factor. If the object remains congruent to its original form after the transformation, the scaling does not change the size of the object.
Given that the image of the triangle is congruent to the pre-image, the scale factor must be such that the sizes of the triangles remain unchanged. Mathematically, this means:
- If the pre-image and image are congruent, all corresponding side lengths are equal.
- Therefore, the scale factor must be 1, because scaling by 1 simply means the figure retains its original size.
So, the correct scale factor of the dilation, which ensures that the image is congruent to the pre-image, is:
[tex]\[ \boxed{1} \][/tex]
