Geometry (all content)
Sal shows how we can use dilations to map a line into another, parallel, line. Created by Sal Khan.
Want to join the conversation?
- Why is this called a dilation and not a translation? I understand what it means to dilate a figure, to scale it up or down. But dilating a line seems to be simply a matter of changing its location; MOVING it up or down rather than SCALING it up or down.(5 votes)
- Technically, you can manipulate the scale factor and point of dilation to translate a line. This is pretty much all you can do by dilating a line, since a line, being infinite, stays infinite regardless of the scale factor.(6 votes)
- how to you find the scale factor(1 vote)
- The scale factor is not always, given, in fact in most problems you must come up with a scale factor. First, think of two points. One point on the line that you are dilating, another point on the target for that point. Find the scale factor by dividing the distance to the target by the distance to the point on the line that you are dilating. Sorry if this was confusing, I'm not very good at explaining.(11 votes)
- how would we do this with shapes ?(3 votes)
- If you're using polygons with corners, just pick each vertex/corner and find its difference from the point of dilation. Since each edge of a shape is a line segment, just imagine you are dilating all the lines those segments lie on.(3 votes)
- If I am expected to find the scale factor for a dilation without being given numbers and only an image of a square within a square, how would that work?(3 votes)
- I'm really confused with how to do this.(2 votes)
- I watched the video several times and was still confused until speculating that a dilation might be like a shadow (projection). Unfortunately the problems don't have the scratchpad available, but if you could draw lines through corresponding original and image line segment endpoints, the intersection would provide a center of dilation. The scale is the ratio of the distances from the center to the corresponding endpoints.(3 votes)
- Can you use a negative number in your scale factor to make objects smaller? Or do you have to use fractions to make an object smaller?(2 votes)
- You use whole number to make it that much bigger. SF of 3 would make it 3 times bigger. In order to make a shape smaller you use fractions. A SF of 1/3 would make the shape 1/3 of its previous size.(2 votes)
- If you are dilating a line, which factors will be changed?(3 votes)
- Unless your center of dilation does not sit on the line itself, nothing really changes.(1 vote)
- What grade is geometry for? (Just curious, I just moved to US) :)(0 votes)
- Geometry is usually taught in 9th or 10th grade, but in some locations exceptionally gifted students can take it in 8th grade.(4 votes)
- How do you get the coordinates for the center of dilation?(2 votes)
- If you know by how much two coordinates are dilated then draw lines (in your mind) along the trajectory fo those dilating points. Where they meet is the center of dilation!(1 vote)
- What does Dilation mean?(2 votes)
- Dilation is the factor of enlarging or shrinking the distances compared to a specific point in space.(1 vote)
Define a dilation that maps line a onto line b by choosing a center and a scale factor. So let me get my scratch out to think about this a little bit. So let's, just for fun, let's imagine that we pick a point that sits on line a as our center of dilation. And for simplicity, let's pick the origin. So let's imagine that our center of dilation is at 0, 0. And let's just pick an arbitrary scale factor. Let's say our scale factor is 2. And let's think about what happens. This means that each of the points on point a are going to get twice as far from our center of dilation after the dilation than they were before the dilation. Twice as far. So for example, this point right over here, it's 3 away in the x direction before dilation. So it's going to be 6 away in the x direction after dilation. Likewise, it's 3 away in the y direction before dilation. So it's going to be twice as far in the y direction after dilation. So it's going to go from the point (3,3) to the point (6,6). Notice, it got twice as far away from the center of dilation. It essentially got pushed out along the line. Same thing for this point. This point is 3 less in the x direction than our center, and 3 less in the y direction than our center. So after a scaling, after dilation centered at the origin, with a scale factor of 2, it's going to be twice as far away. So it's going to be 6 less in the x direction, and 6 less in the y direction. Once again, just pushed out. So notice when we picked a center of dilation that sits on the line, it really doesn't matter what scale factor we pick, we'll just essentially be stretching and shrinking the points along the same line. And since this line just keeps going on and on and on forever in both directions, it's really just going to map onto itself. If this was a line segment, we would be mapping it to a different line segment that would have the same slope, but its length would change. But a line has an infinite length. So as you stretch and squeeze it, it still has an infinite length, and it's just mapping it onto itself. So this dilation right over here, a, is going to map onto itself. It's just going to map onto a. So that's not going to suit our purposes. But what can suit our purpose is if we pick another point that's not on a, and not on b. So let's say we pick this point right over here. And I like this point, and so let me pick that as my center. I'll tell you why I like it in a second. That is the point 3, 2. The reason why I like that, is we can see that this point on point a right now is 1 away, and if I want to map it to the corresponding point on point b, well that's going to be 3 away from our center of dilation. So I could have a scale factor of 3. If I have a scale factor of 3, it's going to go 3 times as far away. And that'll work for other points as well. For example, this point right over here. It's 1 to the left of our center of dilation. If we have a scale factor of 3, it's going to be 3 to the left. So it's going to map. If we think of it that way, you could think of it as going right over there. So I'm going to go with center at 3, 2, scale factor of 3. And you could have picked many, many other-- in fact an infinite number of other centers, and then you would have had to calculate the appropriate scale factor for each of those. But I like this one because it fit nicely on the grid, and I was able to figure out how much to scale it. So I picked my center, so I want to dilate, I want to pick my center at 3, 2. And I want to scale it by 3. So let's check our answer. We got it right.