The scale factor of translation is always 1, as translations only shift a figure and do not change its size. Therefore, no scaling occurs. A scale factor greater or less than 1 would represent a dilation, not a translation.
While the term "scale factor" is typically associated with dilations (enlargements or reductions), a translation simply slides a figure without altering its dimensions. In the context of transformations, the scale factor relates to how much the size of a figure changes. Since a translation preserves the size, the scale factor effectively defaults to 1.
Understanding Transformations
To understand this better, let's define some key terms:
- Translation: A transformation that slides a figure without changing its size, shape, or orientation.
- Dilation: A transformation that enlarges or reduces a figure by a scale factor.
- Scale Factor: The ratio of the length of a side in the image (new figure) to the length of the corresponding side in the pre-image (original figure).
Why the Scale Factor of Translation is 1
If you were to measure the length of a side in the original figure (pre-image) and the corresponding side in the translated figure (image), you would find they are identical. The ratio of the new length to the original length would therefore always be 1.
Example:
Imagine a square with side length 5 units. You translate this square 3 units to the right.
- Original side length: 5 units
- Translated side length: 5 units
- Scale factor: 5/5 = 1
Because the dimensions are unchanged, the "scale factor" is always 1 in a translation. The transformation vector specifies how far the object has translated but translation itself involves no scaling.
