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# Dilating lines

## Video transcript

define a dilation that maps line a on to line B by choosing a center and a scale factor so let me get my scratch pad out to think about this a little bit so let's just for just for fun let's imagine that we pick a point 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 comma 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 then they were before the dilation twice as far so for example point 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 so twice as far away from the center of dilation twice as far away from the central dilation essentially got pushed out along the line same thing for this point this point is 3 less and the wider three less in the x-direction than our Center and 3 less in the y-direction than our Center so after a scaling after a dilation centered at the origin with a scale factor of 2 is 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 on to 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 let's say we pick at this point right over here and I like 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 comma 2 3 comma 2 the reason why I like that is we can see that this point on point a right now is one away and I want to if I want to map it to the corresponding point on point B well that's going to be that's going to be three away from our center of dilation so I can have a scale factor of 3 if I have a scale factor of 3 is going to go three times as far away scale factor of three and that will work for other points as well for example this point right over here it's one to the left of our center of dilation if we have a scale factor of three 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 as going right over there so I'm going to go with Center 3 to scale factor of 3 and you could have picked many many other in fact an infinite number of other centers 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 let me so I picked my Center so I want to dilate I want to pick my Center at 3 comma 2 and I want to scale it by 3 so let's check our answer got it right