In geometry, what is the rule for dilation?

In geometry, a transformation is an operation that moves, flips, or changes a shape to create a new shape. A dilation is a type of transformation that enlarges or reduces a figure (called the preimage) to create a new figure (called the image).

For a dilation to maintain its proportionality of sides, the two variables must be multiplied by a constant value, k, known as the scale factor.

A notation rule has the following form Dk(x,y) = (kx,ky) and tells you that the preimage has undergone a dilation about the origin by scale factor k.

The following are preserved between the pre-image and its image when dilating:

Angle measure (angles stay the same)

Parallelism (things that were parallel are still parallel)

Collinearity (points on a line, remain on the line)

Distance is not preserved

When the dilation is an enlargement, the scale factor is greater than 1 (k > 1). What this means is that if you have a scale factor of 3, or sometimes written as 1:3, the image is three times bigger proportionally to the pre-image.

When the dilation is a reduction, the scale factor is between 0 and 1 (k > 1). What this means is that if you have a scale factor of ½, or sometimes written as 2:1, that the image is half the size proportionally to the pre-image.

So what happens if the scale factor is 1 (k = 1)? Nothing. To multiply anything by 1 maintains the pre-image. It is like rotating a shape 360° - the shape is not altered in any way – it is an identity transformation.

After a dilation, the pre-image and image have the same shape but not the same size.

To learn more about dilation, visit Geometrycommoncore.com and Ck12.org.