Dilating points on a circular grid

[00:00:00]We're told that four points in their and their dilated images are shown with the center of dilation O. Which point was dilated by a scale factor of 1/3?

[00:00:10]So, pause this video and have a go at that.

[00:00:12]Okay, now let's look at the points. So, the typical way that people do this is whatever the point is, after the dilation, they'll call that same point with the same letter and then put a little prime here. So, for example, N has been dilated as of some in some way to N'.

[00:00:30]Now, we want to identify which one was dilated by a scale factor of 1/3.

[00:00:34]Because the scale factor is less than one, the the image after the dilation, the thing the prime letter that we're going to should be closer to the center of dilation, not further. So, we could rule out N right over here. This actually would be a positive scale factor. It looks like we went from four units away from the origin to six units away from the origin. So, this looks like a scale of 1.5. We went We went one and a half further from the origin. So, we could rule these out. And let's see, this one over here, L looks interesting.

[00:01:10]L is originally 1 2 3 4 5 6 away from the center of dilation. I guess I can still call it center of dilation, not the origin, although it is It does look like an origin in this situation, too.

[00:01:25]And if we were to take that distance six and if we were to multiply it by our scale factor of 1/3, we would get to two. So, it should be in the same direction as L, but 1/3 as far, only two units as far. And that indeed looks like what L' is. So, I like that one right over here. Which point was dilated by a scale factor of 1/3? It is the point L.

[00:01:46]But just for for satisfaction, let's look at these other ones over here. K' or I should say K looks like it has a positive scale or not just positive, it has a scale factor larger than one cuz we're going from two units to two, four, six. So, this right over here is a scale factor.

[00:02:06]So, this is a scale factor of three.

[00:02:12]If they said, "Hey, we're we're we're dilating K prime to K," which wouldn't be the standard way of doing it, then the other way would be a scale factor of 1/3. But, no, if we're talking from K to K prime, this is a scale factor of three.

[00:02:26]And then if we look at M and M prime, M is 1 2 3 4 5 6 away. M prime is two away. So, you might say, "Hey, that feels like a 1/3 scale factor." Except for the fact that they're not in the same direction from the center of dilation. When if we start with M, we would get closer along this same line towards our center of dilation. So, if this was truly a 1/3 scale factor, we would M prime would get right over here.

[00:02:58]It would be along this same direction towards this center. It would not be over there.