Representing dilations algebraically, k greater than 1

0:00- [Instructor] We're told triangle ABC was dilated with the origin as the center of dilation to create triangle A prime, B prime, C prime. And we could see that ABC around its origin as the center of dilation, it was now dilated and essentially scaled to triangle A prime, B prime, C prime.

0:21Write a rule to represent this dilation. So if you are so inspired, pause the video to figure that out yourself before we do this together. Now, a rule to represent the dilation will look something like this. For any x, y coordinate on the original shape, we are now going to dilate, or you could even think about scaling it.

0:43So X is now going to be KX and Y is going to be KY. We're assuming that we're scaling in both dimensions the same. That's what we're talking about when we're talking about a dilation, especially a dilation around the center here. Now, there's a couple ways you could approach it.

0:59You could compare the lengths of corresponding sides and think about how much they've been scaled up, or you could actually look at the coordinates themselves. So let's do the first technique. Let's say we wanna compare, say this side right over here. So the length of AB, AB is equal to, we can count the units, 1, 2, 3, 4, 5, 6, 7 units long.

1:22And now let's look at the corresponding side on A prime, B prime, C prime. So A prime, B prime is equal to, what's that length going to be? Well, that looks like it is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and it looks like 10-1/2 if I were to eyeball it. So let's just go with that, 10.5.

1:55So what are we scaling this by? To go from seven to 10.5, what are we scaling that by? Well, it looks like we're scaling seven, if we take half of seven and add it to that, you get to 10.5. So it looks like we're multiplying seven by three-halves. So let's see, is three-halves going to be our scaling factor?

2:15Is that what our K is going to be? Well, we could try that with a few other sides just to make sure. So let's look at AC. AC is equal to 1, 2, 3, 4, 5, 6 units long. And then A prime, B prime, if it's this scaling factor, three-halves times six, we should see A prime, or not A prime, B prime.

2:39A prime, C prime should be three-halves times this, which would be nine, so let's see if that's true. 1, 2, 3, 4, 5, 6, 7, 8, 9. So yes, our scaling factor is looking good. So it looks like we're multiplying by three-halves. As I mentioned, you could also look at points.

3:02For example, the point A right over here that has the coordinates, A has the coordinates it looks like negative four, four, negative four comma four, and A prime over here, A prime, has the coordinates negative six, six, negative six, six. So once again, we are multiplying each of these times three-halves, times three-halves.

3:30So I feel pretty comfortable that that K there is three-halves. And so the rule right over here that represents this dilation is you give me any x, y and that's going to get mapped to K times X and K, I shouldn't write K, I already knew it was K, three-halves, three-halves x, three halves y, and we are done.