Let's take an example. Say you have a dilation with center (0,0) and scale factor 2. So, k=2. You are dilating a square, whose sides each measure 1 unit.
In a dilation, you multiply each side length of the pre-image by the scale factor (k). The pre-image is the original figure, the square with 1 unit long sides. So each side in the image, the new figure after dilation, will be 2 times as long. 1*2=2, so each side is 2 units long.
Now, you want to return the figure to the original pre-image. The figure we just created has sides of 2 units long each. We want to get it to the original pre-image, which has sides of 1 unit long each. We multiply each side of our original figure (the one with 2 units on each side) by the scale factor to get the dimensions of the new figure (which should be 1 unit long on each side).
Therefore, we can say that 2*x=1. Our original dimension times the scale factor needs to equal the new dimension, which has to be one. Now, just solve for x by dividing by 2 on each side. x=1/2.
What do you notice? The first scale factor (k) is 2, and the scale factor to get it back to the pre-image (x) is 1/2. Notice how k=2, while the denominator (bottom part) of x also equals 2. If you try making k=3, or 4, x will equal 1/3 or 1/4 respectively. There's a pattern here.
We can say that the new scale factor, in relation to k, will be 1/k. That's the dilation you can apply to the image.
Answer: 1/k
