Grow or shrink from a point
A dilation makes a figure larger (enlargement) or smaller (reduction) from a fixed center. Every point rides a ray from that center. Distances from the center scale by the same number k, the scale factor.
k is image over pre-image. k > 1 enlarges. 0 < k < 1 reduces. k = 1 leaves size alone.
Angles stay. Orientation stays. Collinearity and betweenness stay: a midpoint is still a midpoint, even though the half-length changed. Side lengths change, so the figures are similar, not congruent.
The origin rule is a function
When the center is (0, 0), every point uses the same machine:
Point B(0, 1) with k = 3 becomes B'(0, 3). Hexagon PATERN with P(−1, 3) and k = 2.5 becomes P'(−2.5, 7.5). Multiply both coordinates. Do not add k. Adding would be a translation.
To recover k from two matching sides, divide: A'B' / AB. Confirm on a second pair so you did not mix vertices.
Find the center
Draw the line through A and A'. Draw the line through B and B'. They meet at the center of dilation. It might be the origin. It might not.
The center is also the point whose distance to each image point is k times its distance to the matching pre-image point, along the same ray.
If a vertex already sits on the center, that vertex does not move, so its name may stay unprimed.
Center not at the origin
Count the vector from the center to a vertex. Multiply that vector by k. Count from the center again.
Center (−4, −4), k = 3, J is 1 right and 3 up from the center. Times 3 is 3 right and 9 up, so J' is (−1, 5).
Check: lines through J and J', K and K', L and L' should still meet at (−4, −4). Each side of the image should be 3 times the matching side.
What happens to segments
A segment that already goes through the center stays on that same line. Its image is a longer or shorter piece of the same line.
A segment that misses the center has an image parallel to it, same slope, length scaled by k.
Those two facts plus "angles match" are why a dilation always produces similar figures.