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Joriy vaqt:0:00Umumiy vaqti:9:17

When we compare triangle
ABC to triangle XYZ, it's pretty clear that
they aren't congruent, that they have very different
lengths of their sides. But there does seem
to be something interesting about the
relationship between these two triangles. One, all of their corresponding
angles are the same. So the angle right here, angle
BAC, is congruent to angle YXZ. Angle BCA is congruent
to angle YZX, and angle ABC is
congruent to angle XYZ. So all of their corresponding
angles are the same. And we also see that
the sides are just scaled-up versions
of each other. So to go from the length of XZ
to AC, we can multiply by 3. We multiplied by 3 there. To go from the length of XY
to the length of AB, which is the corresponding side,
we are multiplying by 3. We have to multiply by 3. And then to go from the length
of YZ to the length of BC, we also multiplied by 3. So essentially,
triangle ABC is just a scaled-up version
of triangle XYZ. If they were the
same scale, they would be the exact
same triangles. But one is just a bigger,
a blown-up version of the other one. Or this is a miniaturized
version of that one over there. If you just multiply
all the sides by 3, you get to this triangle. And so we can't
call them congruent, but this does seem to be a
bit of a special relationship. So we call this special
relationship similarity. So we can write that triangle
ABC is similar to triangle-- and we want to make sure we
get the corresponding sides right-- ABC is going
to be similar to XYZ. And so, based on
what we just saw, there's actually kind
of three ideas here. And they're all equivalent ways
of thinking about similarity. One way to think
about it is that one is a scaled-up
version of the other. So scaled-up or -down
version of the other. When we talked about
congruency, they had to be exactly the same. You could rotate it, you could
shift it, you could flip it. But when you do all
of those things, they would have to
essentially be identical. With similarity, you can
rotate it, you can shift it, you can flip it. And you can also scale it up
and down in order for something to be similar. So for example, let's
say triangle CDE, if we know that triangle CDE
is congruent to triangle FGH, then we definitely know
that they are similar. They are scaled up
by a factor of 1. Then we know, for a
fact, that CDE is also similar to triangle FGH. But we can't say it
the other way around. If triangle ABC
is similar to XYZ, we can't say that it's
necessarily congruent. And we see, for this
particular example, they definitely
are not congruent. So this is one way to
think about similarity. The other way to
think about similarity is that all of the corresponding
angles will be equal. So if something is similar, then
all of the corresponding angles are going to be congruent. I always have trouble
spelling this. It is 2 Rs, 1 S. Corresponding
angles are congruent. So if we say that triangle ABC
is similar to triangle XYZ, that is equivalent to saying
that angle ABC is congruent-- or we could say that
their measures are equal-- to angle XYZ. That angle BAC is going to
be congruent to angle YXZ. And then finally,
angle ACB is going to be congruent to angle XZY. So if you have two triangles,
all of their angles are the same, then you could
say that they're similar. Or if you find two
triangles and you're told that they are
similar triangles, then you know that all of
their corresponding angles are the same. And the last way
to think about it is that the sides are all
just scaled-up versions of each other. So the sides scaled
by the same factor. In the example we did here,
the scaling factor was 3. It doesn't have to be 3. It just has to be the same
scaling factor for every side. If we scaled this side up by 3
and we only scaled this side up by 2, then we would
not be dealing with a similar triangle. But if we scaled all
of these sides up by 7, then that's still a similar,
as long as you have all of them scaled up or scaled down
by the exact same factor. So one way to think
about it is-- I want to still visualize
those triangles. Let me redraw them right over
here a little bit simpler. Because I'm not talking
in now in general terms, not even for that specific case. So if we say that this is A, B,
and C, and this right over here is X, Y, and Z. I
just redrew them so I can refer them
when we write down here. If we're saying that these
two things right over here are similar, that means
that corresponding sides are scaled-up versions
of each other. So we could say that
the length of AB is equal to some
scaling factor-- and this thing could be
less than 1-- some scaling factor times the length of
XY, the corresponding sides. And I know that AB
corresponds to XY because of the order in which I wrote
this similarity statement. So some scaling factor times XY. We know that the
length of BC needs to be that same scaling
factor times the length of YZ. And then we know
the length of AC is going to be equal to that
same scaling factor times XZ. So that's XZ, and this
could be a scaling factor. So if ABC is larger
than XYZ, then these k's will be larger than 1. If they're the exact same
size, if they're essentially congruent triangles,
then these k's will be 1. And if XYZ is bigger
than ABC, then these [? scaling ?] factors
will be less than 1. But another way to write these
same statements-- notice, all I'm saying is
corresponding sides are scaled-up versions
of each other. This first statement right here,
if you divide both sides by XY, you get AB over XY is equal
to our scaling factor. And then the second
statement right over here, if you divide
both sides by YZ-- let me do it in that
same color-- you get BC divided by YZ is
equal to that scaling factor. And remember, in the
example we just showed, that scaling factor was 3. But now we're saying in the
more general terms, similarity, as long as you have the
same scaling factor. And then finally, if
you divide both sides here by the length between X
and Z, or segment XZ's length, you get AC over XZ is
equal to k, as well. Or another way to think
about it is the ratio between corresponding sides. Notice, this is the
ratio between AB and XY. The ratio between BC and YZ,
the ratio between AC and XZ, that the ratio between
corresponding sides all gives us the same constant. Or you could rewrite
this as AB over XY is equal to BC over YZ is equal
to AC over XZ, which would be equal to some scaling
factor, which is equal to k. So if you have
similar triangles-- let me draw an arrow
right over here. Similar triangles means that
they're scaled-up versions, and you can also flip and
rotate and do all the stuff with congruency. And you can scale
them up or down. Which means all of the
corresponding angles are congruent, which
also means that the ratio between corresponding
sides is going to be the same constant for
all the corresponding sides. Or the ratio between
corresponding sides is constant.