Adilation with center (0,0) and scale factor k is applied to a polygon. what dilation can you apply to the image to return it to the original preimage?

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.

1/k