The big idea A dilation keeps shape and changes size by multiplying every distance from one fixed center by the same scale factor.
Grow or shrink from a point
A dilation makes a figure larger or smaller from a fixed center. Every point rides a ray from that center. Its distance from the center is multiplied by the scale factor k.
k is image divided by pre-image. k > 1 enlarges. 0 < k < 1 reduces. k = 1 leaves size alone.
Angles, orientation, collinearity, and betweenness stay. Side lengths change, so the image is similar to the pre-image and is congruent only when the size stays the same.
Rigid versus dilation: translations, reflections, and rotations keep length. A dilation usually does not.
The origin rule is a function
When the center is (0, 0), every point uses the same rule:
(x, y) → (kx, ky)
B(0, 1) with k = 3 becomes B'(0, 3). P(−1, 3) with k = 2.5 becomes P'(−2.5, 7.5). Multiply both coordinates. Adding k would be a translation.
To recover k from matching sides, divide image length by pre-image length. Confirm the result on a second pair.
Find the center
Draw a line through A and A'. Draw another through B and B'. Those lines meet at the center of dilation. The center may be the origin, but it does not have to be.
The image point lies on the same ray from the center as its pre-image point, at k times the original distance. A vertex already at the center does not move.
Center not at the origin
Find the vector from the center to a vertex. Multiply that vector by k, then count the new vector from the center.
Center (−4, −4), k = 3. J is 1 right and 3 up from the center, so the scaled vector is 3 right and 9 up. J' is (−1, 5).
As a check, lines through each pair of corresponding points should still meet at the center, and every image side should be three times its matching side.
What happens to segments
A segment passing through the center stays on that same line. Its image is a longer or shorter segment on the line.
A segment missing the center has an image parallel to it, with the same slope and a length multiplied by k.
Those facts, along with preserved angles, are why a dilation always produces similar figures.
k versus 1/k: read which figure is the image. A dilation from a large figure to a half-size image uses k = 1/2, not 2.
Your turn
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