Dilations and Similarity

A dilation is a transformation that produces an image that is the same shape as the original, but a different size. Think of it like zooming in or out on a photo. The 'center' of this zoom in our lessons is usually the origin $(0, 0)$. To change the size, we use a scale factor, represented by the letter $k$. If $k > 1$, the shape gets larger (an enlargement). If $0 < k < 1$, the shape gets smaller (a reduction). The most important rule to remember is that every coordinate $(x, y)$ of the original shape is multiplied by $k$ to find the new coordinates: $(kx, ky)$.

When we dilate a figure, the result is a similar figure. In geometry, 'similar' has a very specific meaning. Two figures are similar if: 1. Their corresponding angles are congruent (exactly the same). 2. Their corresponding side lengths are proportional. This means if you divide the length of a side on the new image by the length of the matching side on the original, you will always get the scale factor $k$. For example, if the original side is $s$ and the new side is $s'$, then $\frac{s'}{s} = k$.

Sometimes you aren't given the side lengths directly and must calculate them using the coordinates. To verify similarity, you can use the distance formula to find the lengths of corresponding sides. If the ratio of every pair of corresponding sides equals the same scale factor $k$, the figures are similar. Remember: Dilation changes the perimeter by a factor of $k$, but it changes the area by a factor of $k^2$!