Representing dilations algebraically, k less than 1 | Grade 8 (TX) | Khan Academy

We are told quadrilateral WXYZ was dilated with the origin as the center of dilation to create quadrilateral W' X' Y' Z'. So, we started off with this black quadrilateral, and then it looks like it was dilated down.

One way to think about it, centered at the origin, it was scaled down. Write a rule to represent this dilation. So, like always, pause the video, have a go at it on your own before we do this together.

All right, so let's just remind ourselves what a rule that represents a dilation even looks like. A rule would look something like this: You take any (x, y) coordinate on the original shape, and it's going to get mapped to another (x, y) coordinate, which will now be on the new shape, on the shape in green.

Actually, why don't I write that in green just to make it clear what's going on? So, it's going to be scaled in the x-direction by K and scaled in the y-direction by K. And so the key is we have to figure out what this scaling factor actually is.

Now, there's a couple of ways you could do it. You could look at a corresponding side, especially one that runs horizontal or vertical, so that you can actually just count how long it is, so you can know its dimensions or its length. For example, we could look at that length right over there.

So, we know that WZ is equal to— it looks like 1, 2, 3, 4, 5, 6, 7, 8 units. And now, let's see what W' Z' is. So, W' Z' looks like it is 1, 2, 3, 4, 5, 6 units. So, it looks like when we went from WZ to W' Z', it looks like we scaled; we multiplied by 2 over 3.

So, that gives us a pretty good clue that the scaling factor is 2/3. And that makes sense. If the scaling factor is less than one, the shape that we are mapping to after the dilation is going to be smaller. If the scaling factor is greater than one, then we're going to enlarge it.

But let's see if we can find other confirmation of that. Well, there aren't any other sides that are horizontal or vertical, but we could actually also confirm that by looking at a point where we can clearly get the coordinates of that point.

So, for example, we see that point Z right over here. It has the coordinates—this looks like (-9, -3). Now, Z'—if we believe this scaling factor; if we believe that it is 2/3 for both the x and the y, then if we multiply this by 2/3, it should be -6. And if we multiply this by 2/3, it should be -2.

Let's see—Z' is indeed the coordinates (-6, -2). So, once again we have multiplied by 2/3 in either of these situations. Thus, we feel very comfortable that the rule to represent this dilation is for any (x, y) on the original shape.

It is going to get mapped to—instead of a K, we now know that K is 2/3 of the original x and 2/3 for the new y-coordinate of the original y. And we are done.