Dilation is not considered a rigid transformation because it does not preserve length. When we think of a rigid transformation, such as translation, rotation, or reflection, we imagine that the image is exactly the same size as the pre-image. However, with dilation, the image can be either larger or smaller than the pre-image.
To understand why dilation is not rigid, let’s consider a real-life example. Imagine you have a rubber band and you want to stretch it. As you pull the rubber band, it becomes longer and thinner. The overall shape of the rubber band changes, but it still retains its proportions. This is similar to what happens with dilation.
In mathematics, dilation refers to the transformation of an object by scaling it up or down, while maintaining its shape. The scale factor determines how much the object is stretched or compressed. If the scale factor is greater than 1, the object is enlarged, and if it is between 0 and 1, the object is reduced in size.
When we apply a dilation to a figure, every point in the figure is stretched or compressed according to the scale factor. The distance between any two points on the figure changes proportionally, but not necessarily by the same amount. Some points may move farther apart, while others may move closer together.
For example, let’s consider a rectangle and apply a dilation with a scale factor of 2. The length of the sides of the rectangle will be doubled, but the angles between the sides will remain the same. However, the length of the diagonal will also be doubled, resulting in a larger overall shape.
Similarly, if we apply a dilation with a scale factor of 0.5 to the same rectangle, the length of the sides will be halved, but the angles between the sides will remain the same. The length of the diagonal will also be halved, resulting in a smaller overall shape.
In both cases, the shape of the rectangle is distorted, as the sides and angles are not preserved. This is why dilation is not considered a rigid transformation.
To summarize, dilation involves stretching or compressing an object while maintaining its shape. The scale factor determines the amount of stretching or compression. Unlike rigid transformations, dilation does not preserve length, as the image can be either larger or smaller than the pre-image.