Review of triangle properties | Special properties and parts of triangles | Geometry | Khan Academy


What I want to do
in this video is review all of the neat
and bizarre things that we’ve learned
about triangles. So first, we learned–
so let me just draw a bunch of triangles
for ourselves. So let’s have a triangle
right over there. The first thing
that we talked about is the perpendicular bisectors
of the sides of the triangles. So if we take, so let’s bisect
this side right over here. And let’s draw a
perpendicular line to it. So this line right over here
would be the perpendicular bisector of this
side right over here. So it’s bisecting and
it is perpendicular. Let’s draw another perpendicular
bisector right over here. So we’re going to– this is
the midpoint of that side. Let’s draw a perpendicular. So this is perpendicular
and this length is equal to this length, and
then let’s do one over here. This looks like the midpoint
of that side right over there. And then we will
draw a perpendicular. It’s perpendicular, and
we know that this length is equal to this
length right over here. And what we learned is where
all of these perpendicular bisectors intersect, and
what’s neat about this, and frankly all the things
that we’re going to talk about in this video, is that they do
intersect in one unique point. That one unique point is
equidistant from the vertices of this triangle. So this distance, is going to
be equal to this distance, which is going to be equal
to that distance. And because it’s
equidistant to the vertices, you could draw a circle
of that radius that goes through the vertices. And that’s why we call
this right over here, that point, that intersection
of the perpendicular bisectors. So let me write
this down, just so that we can keep
track of things. Perpendicular bisectors. We call this point right
over here our circumcenter, because it is the center of
our circumcircle, a circle that can be circumscribed
about this triangle. So this is our circumcircle. And the radius of the
circumcircle, the distance between the circumcenter and the
vertices is the circumradius. So that was the
perpendicular bisectors. Now the next thing we
learned, and the whole point of this video is
just to make sure that we can differentiate
between these things and not get too confused. So let me draw another arbitrary
triangle right over here. The next thing we
thought about is well, what about if we were
to bisect the angles? So we’re not talking
about perpendicular bisecting the
sides, but we’re now talking about bisecting
the angles themselves. So we could bisect this
angle right over here. Let me draw my best
attempt to draw it. And so this angle is going
to be equal to that angle. We could bisect this
angle right over here. We could bisect– I could do
a better version than that. So that looks–
well, one more try. So I could bisect it like that. And them if I’m
bisecting it this angle is going to be
equal to that angle. And then if I
bisect this one, we know that this
angle is going to be equal to that angle over there. And once again, we have
proven to ourselves that they all intersect
in a unique point. And this point, instead of being
equidistant from the vertices, this point is equidistant from
the sides of the triangle. So if you dropped
a perpendicular to each of the sides. So this distance is going to be
equal to that distance, which is going to be equal
to that distance. And because of that,
we can draw a circle that is tangent to the
sides that has this radius. So we could draw a circle
that looks like this. And we call this
circle, because it’s kind of inside the triangle,
we call it an incircle. And this point,
we can call, which is the intersection of
these angle bisectors, we can call this the inradius. Now the other thing we
learned about angle bisectors, and this we just
have to draw one. So let me just draw another
triangle right over here. And let me draw
an angle bisector. So I’m going to
bisect this angle. So this angle is
equal to that angle. And let me label
some points here. So let’s say that this
is– change the colors. Let’s say that is A,
this is B, this is C, and this is D. We learned
that if AC is really the angle bisector of angle BAD, that the
ratio between– that AB over BC is going to be equal to
the ratio of AD to DC. Sometimes this is called
the angle bisector theorem. So that’s neat. So the next thing we
learned is– let’s draw another triangle here. This is just to be an
overview of everything we’ve been covering in
the last few videos. So let me draw
another triangle here. So now instead of drawing the
perpendicular bisector– so let me label everything. This was angle bisectors. And now what I’m going to
think about are the medians. So the perpendicular bisectors
were from the midpoint, were lines that
bisect the sides, and they are perpendicular,
but don’t necessarily go through the vertices. When we talk about
medians, we are talking about points
that bisect the sides, but they go to the
vertices, and they’re not necessarily perpendicular. So let’s draw some medians here. So let’s say this is the
midpoint of that side right over there. So we could draw a
median like that. No, this is going
through the vertices, these did not necessarily
go through the vertices. This right over here is not
necessarily perpendicular. But we do know
that this length is equal to that length
right over there. Let me draw a couple of more
medians right over here. So this, the midpoint looks
like it’s right about here. So this length is
equal to that length, and notice it goes
through the vertex, but it’s not necessarily
perpendicular. And then this one–
see the midpoint looks like it’s
right about there. And once again, all of
these are concurrent. They all intersect at one
point right over here. And so this length
right over here is equal to this
length right over here. There’s a bunch of neat
things about medians. When you draw the three medians
like this, that unique point where they intersect, we
called it the centroid. And as I mentioned, and
you might learn this later on in physics, is if
this was a uniform triangle, if it had a uniform
density, and if you were to throw it or
rotate it in the air, it would rotate around
its centroid, which would essentially be
its center of mass. It would rotate around that as
it’s flying through the air. If it had some type of
rotational, or I guess you could say angular momentum. But the neat thing about this
is it also divides this triangle into six triangles
of equal area. So this triangle has the
same area as that triangle. We proved this in
several videos ago. Each of these six triangles
all have the same area. The other thing that we
learned about medians is that where the centroid
sits on each of the medians is 2/3 along the median. So the ratio of this side, of
this length to this length, is 2 to 1. Or this is 2/3 along
the way of the median. This is 2/3 of the median,
this is 1/3 of the median. So the ratio is 2 to 1. Another related
thing we learned, this wasn’t really
necessarily about medians, but it’s a related concept, was
the idea of a medial triangle. A medial triangle
like this, where you take the midpoint
of each side, and you draw a
triangle that connects the midpoints of each side. We call this triangle
a medial triangle. And we proved to
ourselves that when you draw a medial triangle,
it separates this triangle into four triangles that
not only have equal area, but the four triangles here are
actually congruent triangles. And not only are they
congruent, but we’ve shown that this side is
parallel to this side. That– let me use some
more colors here– this side is parallel. Actually, I shouldn’t
draw two arrows like that. That side is parallel
to that side. This side is parallel
to this side. And then you have this side
is parallel to this side right here. And this length is
1/2 of that length, this length is 1/2
of that length, this length is 1/2
of that length. And it really just
comes out of the fact that these are four
congruent triangles. And then the last thing
that we touched on is drawing altitudes
of a triangle. So there’s medians,
medial triangles, and I’ll draw one last
triangle over here. And here, I’m going to go
from each of the vertex, and I’m not going to go to the
midpoint of the other side. I’m going to drop
a perpendicular to the other side. So here I will drop
a perpendicular, but this isn’t necessarily
bisecting the other side. Once again, going to
drop a perpendicular but not necessarily
bisecting the other side. And then drop a
perpendicular but not necessarily bisecting
the other side. And we’ve also
proven to ourselves– so these are the
altitudes of the triangle. And these also intersect
in a unique point. And I want to be clear,
this unique point does not necessarily have to
be inside of the triangles. And the same thing was true of
the perpendicular bisectors. It actually could be
outside of the triangle. And this unique point
we call an orthocenter. So I’ll leave you there. And hopefully this
was useful, because I know it can get confusing. How’s a median different
than a circumcenter, which is different than an
orthocenter, or an inradius, or any of these type of things? So hopefully this clarified
things a little bit.

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