Dilating points on a circular grid
About this video
Sal works out which of four points on a circular grid was dilated by a scale factor of 1/3 from center O. He compares each point's distance from the center before and after, checks that the image lies in the same direction, and finds the answer is L.
You'll be able to
- find a scale factor by comparing distances from the center of dilation before and after.
- explain that a scale factor less than 1 moves points closer to the center.
- explain why a dilated image must lie on the same ray from the center as the original point.
Key words
- dilation
- A transformation that moves every point toward or away from a center by the same scale factor.
- center of dilation
- The fixed point that distances are measured from in a dilation.
- scale factor
- The number you multiply each distance from the center by.
- image
- Where a point ends up after the transformation, written with a prime, like L prime.
Transcript
- [Instructor] We're told that four points and their dilated images are shown with the center of dilation, O. Which point was dilated by a scale factor of 1/3? So, pause this video. Have a go at that. 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 in some way to N prime. Now, we want to identify which one was dilated by a scale factor of 1/3. Because the scale factor is less than 1, 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 and write 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 1 1/2 further from the origin, so we could rule these out.
And let's see, this one over here, L looks interesting. L is originally one, two, three, four, five, six away from the center of dilation. I guess I could still call it center of dilation, not the origin, although it is. It does look like an origin in this situation too.
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 2. So, it should be in the same direction as L, but 1/3 is far, only 2 units is far. And that indeed looks like what L prime 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. But just for satisfaction, let's look at these other ones over here. K prime, or I should say K, looks like it has a positive scale, or not just positive, it has a scale factor larger than one because we're going from two units to two, four, six.
So, this right over here is a scale factor. So, this is a scale factor of 3. If they said, "Hey, 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 3.
And then if we look at M and M prime, M is one, two, three, four, five, six 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. It would be along this same direction towards this center. It would not be over there.
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