The scale factor of dilation is a number that describes how much a figure is enlarged or reduced during a dilation transformation. It is a ratio that compares the size of the image (the new figure) to the size of the original figure.
- If the scale factor is greater than 1, the figure is enlarged.
- If the scale factor is between 0 and 1, the figure is reduced.
- If the scale factor is exactly 1, the figure remains the same size.
The scale factor can be represented mathematically as:
\[ \text{Scale Factor} = \frac{\text{Size of the Image}}{\text{Size of the Original Figure}} \]
For example, if you have a triangle with side lengths of 3, 4, and 5, and it undergoes a dilation with a scale factor of 2, the resulting triangle will have side lengths of 6, 8, and 10. Conversely, if the scale factor is 0.5, the side lengths would become 1.5, 2, and 2.5.
In coordinate geometry, if a point \( (x, y) \) is dilated from the origin with a scale factor \( k \), the new point \( (x', y') \) is given by:
\[ (x', y') = (kx, ky) \]