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# Performing sequences of transformations

Sal reflects and then translates a given pentagon, and determines whether the resulting figure is congruent to the source figure. Created by Sal Khan.

Video transcript

Starting with the figure
shown, perform the following transformations--
reflection over the line y equals x minus 2, translation
by 3 in the x direction. So we're going to
shift to the right by 3 and then shift down by 9. Is a transformed
figure congruent to the original figure? And actually, we could
answer this question before we even do
the transformation, because this transformation
only involves reflections and translations. As long as we're only
reflecting, translating, and rotating, we're going to
get to a congruent figure. The thing that will make it
not necessarily congruent is when we dilate it. So dilating it is going to
actually scale it up or down. The figure will
still be similar, but it won't necessarily
be congruent anymore. So I could already say yes,
the figures are congruent. But let's actually
perform the transformation that they want us to. So a reflection over the
line y equals x minus 2. And so let's click on translate,
and this isn't popping up-- this isn't allowing us
just to do it by hand, or do it with our mouse. They want us to think about how
much-- actually we don't want to translate first, we
want to reflect first. So once again, the reflection
line tool doesn't show up. Instead, for this
exercise, they want us to specify two
points on the line that we want to reflect around--
because two points define a line. So what are two points that are
on the line y equals x minus 2? Well when x is equal
to 0, y is negative 2. And then another
point on it that might jump out at you--
when x is 2, y is 0. So we just need any two
different points on that line. And then notice,
y equals x minus 2 is a line that looks like this. I'm just going to
kind of trace it. I don't have my drawing
tool out right now, so it's a line that looks
something like that. Actually, let me do
it, just because I think it'll be
interesting to look at. Let me copy and paste
this right over here. And then let me put it
onto my actual scratch pad. So that's from a
previous problem. So let me clear all
of this business out. So let me paste it
right over here. And notice, we just
reflected around the line y equals x minus 2.
y equals x minus 2. Let's see, it has its
y-intercept right over there, it has a slope of 1. So this is the line that
we just reflected around. And it looks pretty clear
that that is what happened. So just like that, I know
my line isn't as straight as it could be, but you
get the general idea. That is the line y is
equal to x minus 2. Notice each of these points--
you drop a perpendicular from that point,
whatever distance that is, go the same
distance on the other side, and we have reflected over. Drop a perpendicular, same
distance on the other side. And so that is our
reflection about the line y equals x minus 2. But we are not done yet. We now have to perform the
translation by 3, negative 9. So we're going to translate. So when we do it
by 3, it's going to shift this to the right by 3. And then negative 9, which is
going to shift it down by 9. And we see that it did. And we already said, these
are definitely congruent. We haven't scaled
this up or down, we've just translated it,
rotated it-- actually, we didn't even rotate it. We've just translated it, and
reflected-- or reflected it, and translated it. But let's check our answer, just
to feel good about ourselves. And we got it right.