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# Dilating lines

Video transcript

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.