Euler line

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:03.830
So I've drawn an arbitrary
triangle over here

00:00:03.830 --> 00:00:07.430
that we will assume is not
an equilateral triangle.

00:00:07.430 --> 00:00:10.050
And let's draw some of
the interesting properties

00:00:10.050 --> 00:00:11.230
of this triangle.

00:00:11.230 --> 00:00:14.770
So, let's first start with
the perpendicular bisectors.

00:00:14.770 --> 00:00:17.200
So let's say this is the
midpoint of this side

00:00:17.200 --> 00:00:18.040
right over here.

00:00:18.040 --> 00:00:20.920
So-- draw a little bit closer
to the midpoint-- looks

00:00:20.920 --> 00:00:22.550
like right about there.

00:00:22.550 --> 00:00:26.610
So this length is equal to
this length right over here.

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And so let me draw a
perpendicular bisector.

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So perpendicular bisector
would look something

00:00:32.180 --> 00:00:34.060
like that for that side.

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Now let me do it on this side.

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So just trying to
eyeball it, this

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looks roughly like the midpoint.

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So this length is going to be
equal to this length, and then

00:00:42.990 --> 00:00:46.020
once again, let me draw
a perpendicular bisector,

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so it's going to go
at a 90 degree angle.

00:00:48.630 --> 00:00:51.090
I can draw a straighter
line than that.

00:00:51.090 --> 00:00:53.610
So that looks pretty good.

00:00:53.610 --> 00:00:54.790
And it is perpendicular.

00:00:54.790 --> 00:00:57.150
And then you can imagine I
will do it with this side,

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with side AB.

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That looks like the midpoint.

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This length is going to
be equal to that length

00:01:03.360 --> 00:01:04.560
right over there.

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And then if we draw a
perpendicular-- my best attempt

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to draw this-- it looks
something like that.

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And we've already learned
that these three lines will

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definitely intersect in a unique
point right over there, which

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we call the circumcenter.

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So let me write it.

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So this right over there
is our circumcenter.

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This is nothing new.

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Now let's draw the medians.

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So, the medians go from
each of the vertices

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to the midpoint of
the opposite side.

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So let me draw.

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So this is a median, this is
a median, this is a median,

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and then this is-- and all
the errors in my drawing

00:01:56.200 --> 00:01:58.060
will start to become
apparent, now that I'm

00:01:58.060 --> 00:01:59.910
trying to do multiple
things with it.

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But we know all of the medians
intersect at the centroid.

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So this right over
here is the centroid.

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Now let's draw the
altitudes of this triangle.

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So let's draw the altitude.

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So I'm going to drop a
perpendicular right over here.

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This is one altitude.

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I will draw another altitude.

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So remember, it's going
to go down like this.

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Draw another altitude that
looks something like that.

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And then I can draw another
altitude from this side,

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and we know it's going to
intersect at this unique point

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right over here.

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So I'll try to draw
it as neatly as I can.

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And this is going
to be perpendicular,

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although the way I've drawn
it, it doesn't necessarily

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look that way.

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And we know that this
thing right over here

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is called an orthocenter.

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Now the whole reason
why I'm doing this,

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and this is frankly kind
of a mind-boggling idea.

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It's neat enough that each
of these points even exists,

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that three altitudes will
intersect in one point,

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that the three medians
intersect in one point,

00:03:01.660 --> 00:03:03.326
and they have all
these neat properties,

00:03:03.326 --> 00:03:05.202
and that the three
perpendicular bisectors

00:03:05.202 --> 00:03:06.160
intersect in one point.

00:03:06.160 --> 00:03:08.319
That by itself is neat enough.

00:03:08.319 --> 00:03:10.110
But what's really neat--
and it's maybe not

00:03:10.110 --> 00:03:12.443
completely apparent by the
way I drew it, because I just

00:03:12.443 --> 00:03:16.020
drew it free hand-- is that
these three points will

00:03:16.020 --> 00:03:18.530
definitely be on the same line.

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If this was an
equilateral triangle,

00:03:20.030 --> 00:03:21.860
they would actually
be the same point.

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But for any other triangle
there'll be different points,

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and they will be
on the same line.

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Which is just kind of crazy,
because obviously two points

00:03:29.370 --> 00:03:32.310
define a line, but three
points being on the same line

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is kind of a-- it seems
like a very unlikely thing,

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if you were to just guess it.

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And I'm going to try to
draw it as best as I can.

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If you were to draw this
with a straight edge ruler,

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it would come out a lot cleaner.

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But all three points
sit on this unique line

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that, it seems something
special or magical about it.

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And because it's something
special or magical,

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there's one famous
mathematician who

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tends to get all of the most
special and mathematical things

00:03:56.870 --> 00:03:58.750
named after him, because
he's the guy who's

00:03:58.750 --> 00:04:00.270
really explored these things.

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And we call this the Euler line.

00:04:05.620 --> 00:04:09.510
And I say that Leonhard
Euler-- that looks like ENTER.

00:04:15.980 --> 00:04:18.750
I say that he gets all
of the cool and magical

00:04:18.750 --> 00:04:21.470
mystical things named after
him, because he's also

00:04:21.470 --> 00:04:25.340
responsible for
Euler's identity, which

00:04:25.340 --> 00:04:29.750
is e to the i pi is
equal to negative 1.

00:04:29.750 --> 00:04:31.584
And we proved this in
the calculus playlist,

00:04:31.584 --> 00:04:33.916
and if none of this makes any
sense to you, don't worry.

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We're only in
geometry right now.

00:04:35.520 --> 00:04:37.170
But he gets to
have all the-- This

00:04:37.170 --> 00:04:40.440
is a magical thing because e
comes out of compound interest,

00:04:40.440 --> 00:04:44.040
and growth and decay,
exponential growth and decay.

00:04:44.040 --> 00:04:46.710
i squared is negative
1-- seems a very bizarre

00:04:46.710 --> 00:04:47.820
imaginary number.

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Pi is a ratio of
the circumference

00:04:50.960 --> 00:04:52.390
of a circle to its diameter.

00:04:52.390 --> 00:04:54.270
And negative 1 is,
well, negative 1.

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And so Euler showed that
there's a lot of reason

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to believe that these
four numbers that

00:04:59.650 --> 00:05:02.790
come from all of these different
weird realities of the world

00:05:02.790 --> 00:05:05.400
are connected in
this very tight way.

00:05:05.400 --> 00:05:08.150
And he also showed that
these special points

00:05:08.150 --> 00:05:10.750
are all on the same
line, which tells us

00:05:10.750 --> 00:05:13.820
something kind of crazy and
mystical about our reality.

00:05:13.820 --> 00:05:16.130
And if that is not
enough for you,

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if you were to take the
midpoint on the Euler

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line between the orthocenter--
I'm not going to prove it here,

00:05:21.184 --> 00:05:23.600
I'm just going to give you a
little bit of a tidbit of it.

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If you take the midpoint
between the orthocenter

00:05:26.290 --> 00:05:29.060
and the circumcenter--
so let's take this.

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So I'm going to try
to look at it roughly.

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So it looks like it's
right about there.

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So this length is going
to be equal to this length

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right over here.

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This point that sits
on the Euler line

00:05:39.366 --> 00:05:41.240
is going to be the center
of something called

00:05:41.240 --> 00:05:44.430
the nine-point circle,
which intersects

00:05:44.430 --> 00:05:46.650
this triangle at nine points.

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And we'll see this kind of
nine interesting points.

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So let me label that as well.

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So it's cool enough that
these three special points are

00:05:54.121 --> 00:05:56.620
on the Euler line, but there's
actually four special points.

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And actually, there's
a few more than that

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are kind of interesting.

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So that orange point
right over there

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is the center of the
nine-point circle.

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And maybe I'll do another video,
another video just on that.

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So anyway, hopefully you found
that kind of interesting.


Review of triangle properties

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:01.820
What I want to do
in this video is

00:00:01.820 --> 00:00:04.750
review all of the neat
and bizarre things

00:00:04.750 --> 00:00:07.430
that we've learned
about triangles.

00:00:07.430 --> 00:00:09.470
So first, we learned--
so let me just draw

00:00:09.470 --> 00:00:11.840
a bunch of triangles
for ourselves.

00:00:11.840 --> 00:00:14.140
So let's have a triangle
right over there.

00:00:14.140 --> 00:00:15.640
The first thing
that we talked about

00:00:15.640 --> 00:00:19.710
is the perpendicular bisectors
of the sides of the triangles.

00:00:19.710 --> 00:00:23.880
So if we take, so let's bisect
this side right over here.

00:00:23.880 --> 00:00:26.379
And let's draw a
perpendicular line to it.

00:00:26.379 --> 00:00:28.670
So this line right over here
would be the perpendicular

00:00:28.670 --> 00:00:31.430
bisector of this
side right over here.

00:00:31.430 --> 00:00:33.820
So it's bisecting and
it is perpendicular.

00:00:33.820 --> 00:00:36.510
Let's draw another perpendicular
bisector right over here.

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So we're going to-- this is
the midpoint of that side.

00:00:39.350 --> 00:00:41.410
Let's draw a perpendicular.

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So this is perpendicular
and this length

00:00:43.270 --> 00:00:48.390
is equal to this length, and
then let's do one over here.

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This looks like the midpoint
of that side right over there.

00:00:51.270 --> 00:00:53.710
And then we will
draw a perpendicular.

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It's perpendicular, and
we know that this length

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is equal to this
length right over here.

00:00:59.320 --> 00:01:03.230
And what we learned is where
all of these perpendicular

00:01:03.230 --> 00:01:05.630
bisectors intersect, and
what's neat about this,

00:01:05.630 --> 00:01:08.004
and frankly all the things
that we're going to talk about

00:01:08.004 --> 00:01:10.990
in this video, is that they do
intersect in one unique point.

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That one unique point is
equidistant from the vertices

00:01:15.320 --> 00:01:16.410
of this triangle.

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So this distance, is going to
be equal to this distance, which

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is going to be equal
to that distance.

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And because it's
equidistant to the vertices,

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you could draw a circle
of that radius that

00:01:27.860 --> 00:01:29.035
goes through the vertices.

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And that's why we call
this right over here,

00:01:39.390 --> 00:01:42.594
that point, that intersection
of the perpendicular bisectors.

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So let me write
this down, just so

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that we can keep
track of things.

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Perpendicular bisectors.

00:01:51.510 --> 00:01:56.910
We call this point right
over here our circumcenter,

00:01:56.910 --> 00:02:01.500
because it is the center of
our circumcircle, a circle that

00:02:01.500 --> 00:02:04.264
can be circumscribed
about this triangle.

00:02:04.264 --> 00:02:05.430
So this is our circumcircle.

00:02:10.250 --> 00:02:13.210
And the radius of the
circumcircle, the distance

00:02:13.210 --> 00:02:17.660
between the circumcenter and the
vertices is the circumradius.

00:02:17.660 --> 00:02:19.327
So that was the
perpendicular bisectors.

00:02:19.327 --> 00:02:21.410
Now the next thing we
learned, and the whole point

00:02:21.410 --> 00:02:22.974
of this video is
just to make sure

00:02:22.974 --> 00:02:24.890
that we can differentiate
between these things

00:02:24.890 --> 00:02:26.370
and not get too confused.

00:02:26.370 --> 00:02:30.210
So let me draw another arbitrary
triangle right over here.

00:02:30.210 --> 00:02:32.090
The next thing we
thought about is well,

00:02:32.090 --> 00:02:35.084
what about if we were
to bisect the angles?

00:02:35.084 --> 00:02:36.750
So we're not talking
about perpendicular

00:02:36.750 --> 00:02:38.330
bisecting the
sides, but we're now

00:02:38.330 --> 00:02:41.760
talking about bisecting
the angles themselves.

00:02:41.760 --> 00:02:44.830
So we could bisect this
angle right over here.

00:02:44.830 --> 00:02:47.490
Let me draw my best
attempt to draw it.

00:02:47.490 --> 00:02:50.540
And so this angle is going
to be equal to that angle.

00:02:50.540 --> 00:02:54.190
We could bisect this
angle right over here.

00:02:54.190 --> 00:02:57.530
We could bisect-- I could do
a better version than that.

00:02:57.530 --> 00:03:01.440
So that looks--
well, one more try.

00:03:01.440 --> 00:03:03.880
So I could bisect it like that.

00:03:03.880 --> 00:03:06.210
And them if I'm
bisecting it this angle

00:03:06.210 --> 00:03:08.310
is going to be
equal to that angle.

00:03:08.310 --> 00:03:13.920
And then if I
bisect this one, we

00:03:13.920 --> 00:03:16.800
know that this
angle is going to be

00:03:16.800 --> 00:03:19.280
equal to that angle over there.

00:03:19.280 --> 00:03:21.530
And once again, we have
proven to ourselves

00:03:21.530 --> 00:03:25.110
that they all intersect
in a unique point.

00:03:25.110 --> 00:03:28.220
And this point, instead of being
equidistant from the vertices,

00:03:28.220 --> 00:03:32.636
this point is equidistant from
the sides of the triangle.

00:03:32.636 --> 00:03:34.010
So if you dropped
a perpendicular

00:03:34.010 --> 00:03:35.000
to each of the sides.

00:03:35.000 --> 00:03:38.520
So this distance is going to be
equal to that distance, which

00:03:38.520 --> 00:03:40.540
is going to be equal
to that distance.

00:03:40.540 --> 00:03:42.960
And because of that,
we can draw a circle

00:03:42.960 --> 00:03:45.570
that is tangent to the
sides that has this radius.

00:03:45.570 --> 00:03:49.990
So we could draw a circle
that looks like this.

00:03:49.990 --> 00:03:51.590
And we call this
circle, because it's

00:03:51.590 --> 00:03:53.860
kind of inside the triangle,
we call it an incircle.

00:03:56.380 --> 00:03:58.530
And this point,
we can call, which

00:03:58.530 --> 00:04:02.690
is the intersection of
these angle bisectors,

00:04:02.690 --> 00:04:07.370
we can call this the inradius.

00:04:07.370 --> 00:04:10.420
Now the other thing we
learned about angle bisectors,

00:04:10.420 --> 00:04:12.280
and this we just
have to draw one.

00:04:12.280 --> 00:04:15.560
So let me just draw another
triangle right over here.

00:04:15.560 --> 00:04:17.560
And let me draw
an angle bisector.

00:04:17.560 --> 00:04:19.029
So I'm going to
bisect this angle.

00:04:19.029 --> 00:04:21.089
So this angle is
equal to that angle.

00:04:21.089 --> 00:04:23.130
And let me label
some points here.

00:04:23.130 --> 00:04:26.200
So let's say that this
is-- change the colors.

00:04:26.200 --> 00:04:31.470
Let's say that is A,
this is B, this is C,

00:04:31.470 --> 00:04:34.810
and this is D. We learned
that if AC is really the angle

00:04:34.810 --> 00:04:43.500
bisector of angle BAD, that the
ratio between-- that AB over BC

00:04:43.500 --> 00:04:46.936
is going to be equal to
the ratio of AD to DC.

00:04:49.660 --> 00:04:53.050
Sometimes this is called
the angle bisector theorem.

00:04:53.050 --> 00:04:54.660
So that's neat.

00:04:54.660 --> 00:04:56.860
So the next thing we
learned is-- let's

00:04:56.860 --> 00:04:58.230
draw another triangle here.

00:04:58.230 --> 00:05:00.190
This is just to be an
overview of everything

00:05:00.190 --> 00:05:02.430
we've been covering in
the last few videos.

00:05:02.430 --> 00:05:04.640
So let me draw
another triangle here.

00:05:04.640 --> 00:05:07.090
So now instead of drawing the
perpendicular bisector-- so

00:05:07.090 --> 00:05:07.880
let me label everything.

00:05:07.880 --> 00:05:08.965
This was angle bisectors.

00:05:13.180 --> 00:05:15.775
And now what I'm going to
think about are the medians.

00:05:18.930 --> 00:05:22.200
So the perpendicular bisectors
were from the midpoint,

00:05:22.200 --> 00:05:24.890
were lines that
bisect the sides,

00:05:24.890 --> 00:05:27.570
and they are perpendicular,
but don't necessarily

00:05:27.570 --> 00:05:28.720
go through the vertices.

00:05:28.720 --> 00:05:30.900
When we talk about
medians, we are

00:05:30.900 --> 00:05:34.760
talking about points
that bisect the sides,

00:05:34.760 --> 00:05:37.060
but they go to the
vertices, and they're not

00:05:37.060 --> 00:05:39.230
necessarily perpendicular.

00:05:39.230 --> 00:05:41.250
So let's draw some medians here.

00:05:41.250 --> 00:05:43.210
So let's say this is the
midpoint of that side

00:05:43.210 --> 00:05:44.530
right over there.

00:05:44.530 --> 00:05:46.306
So we could draw a
median like that.

00:05:46.306 --> 00:05:47.930
No, this is going
through the vertices,

00:05:47.930 --> 00:05:51.270
these did not necessarily
go through the vertices.

00:05:51.270 --> 00:05:54.440
This right over here is not
necessarily perpendicular.

00:05:54.440 --> 00:05:55.970
But we do know
that this length is

00:05:55.970 --> 00:05:57.940
equal to that length
right over there.

00:05:57.940 --> 00:06:00.770
Let me draw a couple of more
medians right over here.

00:06:00.770 --> 00:06:04.750
So this, the midpoint looks
like it's right about here.

00:06:07.770 --> 00:06:09.630
So this length is
equal to that length,

00:06:09.630 --> 00:06:11.400
and notice it goes
through the vertex,

00:06:11.400 --> 00:06:13.530
but it's not necessarily
perpendicular.

00:06:13.530 --> 00:06:16.160
And then this one--
see the midpoint

00:06:16.160 --> 00:06:20.437
looks like it's
right about there.

00:06:20.437 --> 00:06:22.270
And once again, all of
these are concurrent.

00:06:22.270 --> 00:06:24.400
They all intersect at one
point right over here.

00:06:24.400 --> 00:06:26.150
And so this length
right over here

00:06:26.150 --> 00:06:27.840
is equal to this
length right over here.

00:06:27.840 --> 00:06:30.010
There's a bunch of neat
things about medians.

00:06:30.010 --> 00:06:33.950
When you draw the three medians
like this, that unique point

00:06:33.950 --> 00:06:37.987
where they intersect, we
called it the centroid.

00:06:37.987 --> 00:06:39.820
And as I mentioned, and
you might learn this

00:06:39.820 --> 00:06:43.845
later on in physics, is if
this was a uniform triangle,

00:06:43.845 --> 00:06:45.720
if it had a uniform
density, and if you

00:06:45.720 --> 00:06:47.900
were to throw it or
rotate it in the air,

00:06:47.900 --> 00:06:50.440
it would rotate around
its centroid, which

00:06:50.440 --> 00:06:52.222
would essentially be
its center of mass.

00:06:52.222 --> 00:06:54.680
It would rotate around that as
it's flying through the air.

00:06:54.680 --> 00:06:57.310
If it had some type of
rotational, or I guess

00:06:57.310 --> 00:06:59.280
you could say angular momentum.

00:06:59.280 --> 00:07:02.760
But the neat thing about this
is it also divides this triangle

00:07:02.760 --> 00:07:05.990
into six triangles
of equal area.

00:07:05.990 --> 00:07:09.250
So this triangle has the
same area as that triangle.

00:07:09.250 --> 00:07:11.560
We proved this in
several videos ago.

00:07:11.560 --> 00:07:16.590
Each of these six triangles
all have the same area.

00:07:16.590 --> 00:07:19.380
The other thing that we
learned about medians

00:07:19.380 --> 00:07:23.940
is that where the centroid
sits on each of the medians

00:07:23.940 --> 00:07:26.010
is 2/3 along the median.

00:07:26.010 --> 00:07:29.510
So the ratio of this side, of
this length to this length,

00:07:29.510 --> 00:07:30.950
is 2 to 1.

00:07:30.950 --> 00:07:33.520
Or this is 2/3 along
the way of the median.

00:07:33.520 --> 00:07:36.550
This is 2/3 of the median,
this is 1/3 of the median.

00:07:36.550 --> 00:07:38.980
So the ratio is 2 to 1.

00:07:38.980 --> 00:07:40.540
Another related
thing we learned,

00:07:40.540 --> 00:07:42.600
this wasn't really
necessarily about medians,

00:07:42.600 --> 00:07:47.180
but it's a related concept, was
the idea of a medial triangle.

00:07:47.180 --> 00:07:49.370
A medial triangle
like this, where

00:07:49.370 --> 00:07:54.260
you take the midpoint
of each side,

00:07:54.260 --> 00:07:56.040
and you draw a
triangle that connects

00:07:56.040 --> 00:07:57.770
the midpoints of each side.

00:07:57.770 --> 00:08:01.400
We call this triangle
a medial triangle.

00:08:01.400 --> 00:08:04.160
And we proved to
ourselves that when

00:08:04.160 --> 00:08:07.130
you draw a medial triangle,
it separates this triangle

00:08:07.130 --> 00:08:10.570
into four triangles that
not only have equal area,

00:08:10.570 --> 00:08:15.110
but the four triangles here are
actually congruent triangles.

00:08:15.110 --> 00:08:16.860
And not only are they
congruent, but we've

00:08:16.860 --> 00:08:20.090
shown that this side is
parallel to this side.

00:08:20.090 --> 00:08:22.050
That-- let me use some
more colors here--

00:08:22.050 --> 00:08:24.390
this side is parallel.

00:08:24.390 --> 00:08:26.980
Actually, I shouldn't
draw two arrows like that.

00:08:26.980 --> 00:08:28.770
That side is parallel
to that side.

00:08:28.770 --> 00:08:31.240
This side is parallel
to this side.

00:08:31.240 --> 00:08:35.179
And then you have this side
is parallel to this side right

00:08:35.179 --> 00:08:35.679
here.

00:08:35.679 --> 00:08:37.530
And this length is
1/2 of that length,

00:08:37.530 --> 00:08:38.947
this length is 1/2
of that length,

00:08:38.947 --> 00:08:40.363
this length is 1/2
of that length.

00:08:40.363 --> 00:08:42.080
And it really just
comes out of the fact

00:08:42.080 --> 00:08:45.700
that these are four
congruent triangles.

00:08:45.700 --> 00:08:47.460
And then the last thing
that we touched on

00:08:47.460 --> 00:08:49.330
is drawing altitudes
of a triangle.

00:08:49.330 --> 00:08:51.290
So there's medians,
medial triangles,

00:08:51.290 --> 00:08:54.460
and I'll draw one last
triangle over here.

00:08:54.460 --> 00:08:56.574
And here, I'm going to go
from each of the vertex,

00:08:56.574 --> 00:08:58.990
and I'm not going to go to the
midpoint of the other side.

00:08:58.990 --> 00:09:00.364
I'm going to drop
a perpendicular

00:09:00.364 --> 00:09:02.460
to the other side.

00:09:02.460 --> 00:09:04.160
So here I will drop
a perpendicular,

00:09:04.160 --> 00:09:07.640
but this isn't necessarily
bisecting the other side.

00:09:07.640 --> 00:09:09.440
Once again, going to
drop a perpendicular

00:09:09.440 --> 00:09:11.780
but not necessarily
bisecting the other side.

00:09:11.780 --> 00:09:13.830
And then drop a
perpendicular but not

00:09:13.830 --> 00:09:16.120
necessarily bisecting
the other side.

00:09:16.120 --> 00:09:17.780
And we've also
proven to ourselves--

00:09:17.780 --> 00:09:19.571
so these are the
altitudes of the triangle.

00:09:22.260 --> 00:09:28.155
And these also intersect
in a unique point.

00:09:28.155 --> 00:09:29.930
And I want to be clear,
this unique point

00:09:29.930 --> 00:09:33.452
does not necessarily have to
be inside of the triangles.

00:09:33.452 --> 00:09:35.910
And the same thing was true of
the perpendicular bisectors.

00:09:35.910 --> 00:09:38.570
It actually could be
outside of the triangle.

00:09:38.570 --> 00:09:43.420
And this unique point
we call an orthocenter.

00:09:43.420 --> 00:09:44.484
So I'll leave you there.

00:09:44.484 --> 00:09:46.150
And hopefully this
was useful, because I

00:09:46.150 --> 00:09:47.940
know it can get confusing.

00:09:47.940 --> 00:09:50.519
How's a median different
than a circumcenter, which

00:09:50.519 --> 00:09:52.560
is different than an
orthocenter, or an inradius,

00:09:52.560 --> 00:09:53.851
or any of these type of things?

00:09:53.851 --> 00:09:56.820
So hopefully this clarified
things a little bit.


Derivative intuition module

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:03.360
This right here is our
derivative intuition module.

00:00:03.360 --> 00:00:06.210
And it was a module that was
contributed by Ben Eater,

00:00:06.210 --> 00:00:08.862
and it's a pretty interesting
module, one, it's graphical

00:00:08.862 --> 00:00:09.570
and all the rest.

00:00:09.570 --> 00:00:11.280
But what's really
cool about it is

00:00:11.280 --> 00:00:16.219
it really helps to conceptualize
what a derivative is all about.

00:00:16.219 --> 00:00:18.110
And all it really
is is the slope

00:00:18.110 --> 00:00:20.170
of the tangent
line at any point.

00:00:20.170 --> 00:00:22.800
So in this exercise
right over here,

00:00:22.800 --> 00:00:26.020
they've graphed f of x is
equal to 2x to the third power,

00:00:26.020 --> 00:00:29.580
so this is 2x to the third
power right over here.

00:00:29.580 --> 00:00:34.150
And what we need to do is move
these orange dots up and down,

00:00:34.150 --> 00:00:37.220
so these orange dots
right now show a line.

00:00:37.220 --> 00:00:39.280
So when I highlight
on that orange dot,

00:00:39.280 --> 00:00:41.770
you saw this line down
here became orange.

00:00:41.770 --> 00:00:43.760
And right now that
orange dot is 0

00:00:43.760 --> 00:00:47.740
because the slope of this
line right over here is 0.

00:00:47.740 --> 00:00:50.620
But we see that that line right
now is definitely not tangent,

00:00:50.620 --> 00:00:53.750
or the slope at that point
in our curve 2x to the third,

00:00:53.750 --> 00:00:55.180
is definitely not 0.

00:00:55.180 --> 00:00:59.020
So what we need to do
is we need to move this,

00:00:59.020 --> 00:01:02.860
so it becomes pretty close, as
close as we can kind of eyeball

00:01:02.860 --> 00:01:06.150
it, to the slope of
the tangent line.

00:01:06.150 --> 00:01:09.870
And so that looks about-- right
about there looks about right.

00:01:09.870 --> 00:01:12.170
That looks pretty
close to the slope

00:01:12.170 --> 00:01:14.300
of the curve at that point.

00:01:14.300 --> 00:01:16.840
And you could even see
while I do it, up here,

00:01:16.840 --> 00:01:20.210
you see DDX of f of negative 2.

00:01:20.210 --> 00:01:22.772
So the derivative at f
is equal to negative 2.

00:01:22.772 --> 00:01:24.730
So that's just saying
that when the function is

00:01:24.730 --> 00:01:27.350
equal to negative 2, what
is the derivative there?

00:01:27.350 --> 00:01:28.150
That's the slope.

00:01:28.150 --> 00:01:29.780
So by putting it
all the way up here,

00:01:29.780 --> 00:01:33.280
I'm saying that the slope is
equal to 24 at that point.

00:01:33.280 --> 00:01:36.310
So that's a pretty good
approximation just eyeballing

00:01:36.310 --> 00:01:37.400
it like that.

00:01:37.400 --> 00:01:39.930
Now, let's do this
point right over here.

00:01:39.930 --> 00:01:43.050
So it looks like the slope
is decreasing a little bit,

00:01:43.050 --> 00:01:44.800
but it's still very positive.

00:01:44.800 --> 00:01:49.180
So the slope looks maybe it's
close to around 14 or 15.

00:01:49.180 --> 00:01:51.250
And then over here,
once again, our slope

00:01:51.250 --> 00:01:53.680
has decreased bit it's
still reasonably positive.

00:01:53.680 --> 00:01:55.860
The slope looks
like it's about 5,

00:01:55.860 --> 00:01:57.482
so the slope is a decreasing.

00:01:57.482 --> 00:01:58.940
Right over here
the slope does look

00:01:58.940 --> 00:02:01.550
like it's about 0
right over here.

00:02:01.550 --> 00:02:03.600
This looks like an
inflection point actually.

00:02:03.600 --> 00:02:05.080
So our slope is 0 right there.

00:02:05.080 --> 00:02:06.900
And you see the
derivative at f is

00:02:06.900 --> 00:02:09.940
0 is 0, which just means
this is fancy notation

00:02:09.940 --> 00:02:11.690
to say that the slope
of the tangent line,

00:02:11.690 --> 00:02:14.860
or you could say the slope of
the curve at that point, is 0.

00:02:14.860 --> 00:02:17.600
It's flat there, and you
see it really is flat.

00:02:17.600 --> 00:02:19.580
And then you go
over here, and we

00:02:19.580 --> 00:02:22.890
want to find the
derivative at f of 1.

00:02:22.890 --> 00:02:27.660
And so once again, it looks like
the slope is now increasing.

00:02:30.152 --> 00:02:31.110
So this is interesting.

00:02:31.110 --> 00:02:33.780
Up here, the slope
was very positive,

00:02:33.780 --> 00:02:35.870
but it was decreasing
as we went here.

00:02:35.870 --> 00:02:37.640
Notice, it's getting
flatter and flatter.

00:02:37.640 --> 00:02:41.400
Then it goes to 0, and now the
slope starts increasing again.

00:02:41.400 --> 00:02:43.490
So at this point, we
want to make it tangent.

00:02:43.490 --> 00:02:45.190
And so now notice
the slope has gone

00:02:45.190 --> 00:02:47.770
up relative to
when we were at 1.

00:02:47.770 --> 00:02:50.450
So that looks like the
slope right over there.

00:02:50.450 --> 00:02:52.380
And then we do the
slope over there.

00:02:52.380 --> 00:02:55.100
And if we get
pretty close, it'll

00:02:55.100 --> 00:02:57.720
actually draw the curve
of the derivative.

00:02:57.720 --> 00:02:58.670
So there you go.

00:02:58.670 --> 00:02:59.620
And so what's neat
about here, we

00:02:59.620 --> 00:03:02.040
didn't find the derivative
at every point in the curve,

00:03:02.040 --> 00:03:04.320
we found it at just add
the special points that we

00:03:04.320 --> 00:03:06.110
were able to move
these orange dots for.

00:03:06.110 --> 00:03:08.610
But since we got the
orange dots close enough

00:03:08.610 --> 00:03:11.540
to the actual derivatives,
just by eyeballing it,

00:03:11.540 --> 00:03:14.750
just trying to find the--
trying to look at what

00:03:14.750 --> 00:03:16.799
the slope of this
tangent line is, it said,

00:03:16.799 --> 00:03:18.340
OK, you've done it
right, and it drew

00:03:18.340 --> 00:03:20.024
the whole slope
of the derivative.

00:03:20.024 --> 00:03:22.440
And what this entire derivative
is saying is at any point,

00:03:22.440 --> 00:03:24.023
this is the slope
of the tangent line.

00:03:24.023 --> 00:03:28.050
So even though we didn't move
a dot around at negative 0.5,

00:03:28.050 --> 00:03:31.060
if you go right up here,
that looks like it's around--

00:03:31.060 --> 00:03:34.250
this is 2 and 1/2, so this
would be a little bit over 1.

00:03:34.250 --> 00:03:36.890
It looks like that would be
the slope of the tangent line

00:03:36.890 --> 00:03:37.750
right over there.

00:03:37.750 --> 00:03:42.202
Or if we went all the
way to negative 2.25,

00:03:42.202 --> 00:03:43.660
the slope of the
tangent line looks

00:03:43.660 --> 00:03:47.262
like it's approximately
30 right over there.

00:03:47.262 --> 00:03:48.720
So once you have
the curve, you now

00:03:48.720 --> 00:03:50.080
have the slope of
the tangent line,

00:03:50.080 --> 00:03:51.538
or you have the
slope of the curve,

00:03:51.538 --> 00:03:53.690
at every point that
we see over here.

00:03:53.690 --> 00:03:57.341
So this is just, I thought,
a pretty neat exercise.


Proof: Triangle altitudes are concurrent (orthocenter)

WEBVTT
Kind: captions
Language: en

00:00:00.374 --> 00:00:01.790
What I want to do
in this video is

00:00:01.790 --> 00:00:04.692
to show that if we start
with any arbitrary triangle--

00:00:04.692 --> 00:00:07.150
and this will be the arbitrary
triangle that we're starting

00:00:07.150 --> 00:00:10.010
with-- that we can
always make this

00:00:10.010 --> 00:00:13.265
the medial triangle
of a larger triangle.

00:00:13.265 --> 00:00:15.090
And when we say the
medial triangle,

00:00:15.090 --> 00:00:18.410
we mean that each of the
vertices of this triangle

00:00:18.410 --> 00:00:21.887
will be the midpoint of the
sides of a larger triangle.

00:00:21.887 --> 00:00:24.220
And I wanted to show that you
can always construct that.

00:00:24.220 --> 00:00:25.730
If you start with
this triangle, you

00:00:25.730 --> 00:00:29.360
can always have this be the
medial triangle of a larger

00:00:29.360 --> 00:00:30.380
triangle.

00:00:30.380 --> 00:00:33.560
So to do that, let's
draw a line that

00:00:33.560 --> 00:00:35.870
goes through this
point right over here,

00:00:35.870 --> 00:00:39.010
but that's parallel to
this line down here.

00:00:39.010 --> 00:00:44.470
So this line and this line up
here are going to be parallel.

00:00:44.470 --> 00:00:46.302
So just like that.

00:00:46.302 --> 00:00:47.760
And immediately we
can start to say

00:00:47.760 --> 00:00:49.940
some interesting things
about the angles.

00:00:49.940 --> 00:00:53.082
So if we have a transversal
right over here,

00:00:53.082 --> 00:00:55.290
we could view this side as
a transversal of these two

00:00:55.290 --> 00:00:58.160
parallel lines, or of
this line in the segment.

00:00:58.160 --> 00:01:01.310
We know that alternate
interior angles are congruent.

00:01:01.310 --> 00:01:04.250
So that angle is going to
be congruent to that angle.

00:01:04.250 --> 00:01:07.030
And we also know that
this angle in blue,

00:01:07.030 --> 00:01:10.760
is going to be congruent to
that angle right over there.

00:01:10.760 --> 00:01:12.910
Now, let's do that for
the other two sides.

00:01:12.910 --> 00:01:16.710
So let's create a
line that is parallel

00:01:16.710 --> 00:01:19.020
to this side of the
triangle, but that

00:01:19.020 --> 00:01:22.090
goes through this
point right over here.

00:01:22.090 --> 00:01:27.090
So let me draw it
as well as possible.

00:01:27.090 --> 00:01:29.432
And so these two characters
are going to be parallel,

00:01:29.432 --> 00:01:31.640
and you could always construct
a line that's parallel

00:01:31.640 --> 00:01:35.510
to another line that goes to a
point that's not on that line.

00:01:35.510 --> 00:01:39.240
And so once again, we can use
alternate interior angles.

00:01:39.240 --> 00:01:42.520
We know that if this angle
right over here-- let's

00:01:42.520 --> 00:01:45.940
say we have this orange angle--
it's alternate interior angle

00:01:45.940 --> 00:01:48.640
is this angle right over there.

00:01:48.640 --> 00:01:50.380
We also have
corresponding angles.

00:01:50.380 --> 00:01:55.060
This blue angle corresponds
to this angle right over here.

00:01:55.060 --> 00:01:59.070
So it will correspond to
that angle right over there.

00:01:59.070 --> 00:02:01.870
And now let's draw another
line that is parallel

00:02:01.870 --> 00:02:05.070
to this line right
over here, but it

00:02:05.070 --> 00:02:06.580
goes through this vertex.

00:02:06.580 --> 00:02:09.470
It goes through the vertex
that's opposite that line.

00:02:09.470 --> 00:02:11.650
And so let me just draw it.

00:02:11.650 --> 00:02:15.330
And you can always construct
these parallel lines just

00:02:15.330 --> 00:02:16.360
like that.

00:02:16.360 --> 00:02:17.500
And let's see what happens.

00:02:17.500 --> 00:02:19.333
So once again, these
two lines are parallel.

00:02:19.333 --> 00:02:21.650
So you could view this
green line as a transversal.

00:02:21.650 --> 00:02:23.660
If this green line
is a transversal,

00:02:23.660 --> 00:02:29.420
this corresponding angle is
this angle right over here.

00:02:29.420 --> 00:02:32.510
If we view this green line
as a transversal of both

00:02:32.510 --> 00:02:35.930
of these pink lines,
then this angle

00:02:35.930 --> 00:02:39.470
corresponds to this
angle right over here.

00:02:39.470 --> 00:02:42.970
If we view this yellow line
as a transversal of both

00:02:42.970 --> 00:02:47.080
of these pink lines-- actually,
let's look at it this way.

00:02:47.080 --> 00:02:50.270
View the pink line as a
transversal of these two

00:02:50.270 --> 00:02:53.970
yellow lines, then we know
that this angle corresponds

00:02:53.970 --> 00:02:56.300
to this angle right over here.

00:02:56.300 --> 00:02:59.300
And if you view this yellow line
as a transversal of these two

00:02:59.300 --> 00:03:02.420
pink lines, then this
angle corresponds

00:03:02.420 --> 00:03:06.500
to this angle right over here.

00:03:06.500 --> 00:03:11.390
And then the last thing
we need to think about

00:03:11.390 --> 00:03:18.030
is if we think about the
two green parallel lines

00:03:18.030 --> 00:03:22.470
and you view this yellow
line as a transversal,

00:03:22.470 --> 00:03:27.460
then this corresponding angle
in orange is right over here.

00:03:27.460 --> 00:03:30.800
This corresponds to that
angle, because this yellow line

00:03:30.800 --> 00:03:33.900
is a transversal on both
of these green lines.

00:03:33.900 --> 00:03:37.600
So what I've just shown starting
with this inner triangle

00:03:37.600 --> 00:03:40.740
right over here is that if I
construct these parallel lines

00:03:40.740 --> 00:03:43.934
in this way, that I
now have four triangles

00:03:43.934 --> 00:03:45.850
if I include the original
one, and they're all

00:03:45.850 --> 00:03:47.210
going to be similar
to each other.

00:03:47.210 --> 00:03:48.240
And we know that
they're all similar

00:03:48.240 --> 00:03:50.073
because they all have
the exact same angles.

00:03:50.073 --> 00:03:52.050
You just need two angles
to prove similarity.

00:03:52.050 --> 00:03:55.732
But all four of these triangles
have the exact three angles.

00:03:55.732 --> 00:03:58.190
Now, the other thing we can
show is that they're congruent.

00:03:58.190 --> 00:04:00.260
So all of these
four are similar.

00:04:00.260 --> 00:04:01.780
And we also know
they're congruent.

00:04:01.780 --> 00:04:06.270
For example, this side
right over here in yellow

00:04:06.270 --> 00:04:08.490
is the side in this
triangle, between the orange

00:04:08.490 --> 00:04:11.240
and the green side, is the
side between the orange

00:04:11.240 --> 00:04:13.920
and the green side on this
triangle right over here.

00:04:13.920 --> 00:04:18.720
So these two-- we have an
angle, a side, and an angle.

00:04:18.720 --> 00:04:20.000
Angle-side-angle congruency.

00:04:20.000 --> 00:04:23.430
So these two are going to
be congruent to each other.

00:04:23.430 --> 00:04:26.600
Then over here, on
this inner triangle,

00:04:26.600 --> 00:04:28.230
our original triangle,
the side that's

00:04:28.230 --> 00:04:30.680
between the orange
and the blue side

00:04:30.680 --> 00:04:33.450
is going to be congruent to
the side between the orange

00:04:33.450 --> 00:04:35.400
and the blue side
on that triangle.

00:04:35.400 --> 00:04:37.720
Once again, we have
angle-side-angle congruency.

00:04:37.720 --> 00:04:40.450
So this is congruent to this,
which is congruent to that.

00:04:40.450 --> 00:04:42.920
All of these are
going to be congruent.

00:04:42.920 --> 00:04:49.350
And by the same exact
argument, this middle triangle

00:04:49.350 --> 00:04:52.340
is going to be congruent
to this bottom triangle.

00:04:52.340 --> 00:04:56.690
You have an angle, blue angle,
purple side, green angle.

00:04:56.690 --> 00:04:59.260
Blue angle, purple
side, green angle.

00:04:59.260 --> 00:05:00.960
They're congruent to each other.

00:05:00.960 --> 00:05:02.390
So you have all
of these triangles

00:05:02.390 --> 00:05:04.440
are congruent to each other.

00:05:04.440 --> 00:05:06.980
So their corresponding
sides are equal.

00:05:06.980 --> 00:05:10.400
So if you look at this
triangle over here,

00:05:10.400 --> 00:05:15.540
we know that the side
between the blue angle

00:05:15.540 --> 00:05:19.070
and the green angle
is going to be

00:05:19.070 --> 00:05:21.510
equal to this angle
right over here.

00:05:21.510 --> 00:05:24.032
Sorry, equal to this length.

00:05:24.032 --> 00:05:25.740
So it's going to be
equal to this length.

00:05:25.740 --> 00:05:27.230
Between the blue
and the green we

00:05:27.230 --> 00:05:29.040
have this length, between
the blue and the green

00:05:29.040 --> 00:05:31.290
we have that length, between
the blue and the green we

00:05:31.290 --> 00:05:33.320
have that length
right over there.

00:05:33.320 --> 00:05:35.920
So you immediately see that
this point-- and let me label it

00:05:35.920 --> 00:05:37.900
now, maybe I should've
labeled it before.

00:05:37.900 --> 00:05:39.770
If we call that point
A, we see that A

00:05:39.770 --> 00:05:44.000
is the midpoint of--
let's call this point B,

00:05:44.000 --> 00:05:46.550
and call this point
C right over here.

00:05:46.550 --> 00:05:49.364
So A is the midpoint of BC.

00:05:49.364 --> 00:05:50.280
So that's fair enough.

00:05:50.280 --> 00:05:52.500
So I was able to
construct it in that way.

00:05:52.500 --> 00:05:54.540
Now let's look at
the other sides.

00:05:54.540 --> 00:05:56.990
So this green side
on all the triangles

00:05:56.990 --> 00:06:00.620
is the side between the
blue and the orange angle.

00:06:00.620 --> 00:06:02.560
So between the blue
and the orange angle,

00:06:02.560 --> 00:06:05.450
you have the green side, between
the blue and the orange angle

00:06:05.450 --> 00:06:07.310
you have the green side.

00:06:07.310 --> 00:06:10.250
So once again, this length
is equal to this length.

00:06:10.250 --> 00:06:13.080
And so if we call this
point over here D,

00:06:13.080 --> 00:06:15.170
and maybe this
point over here E,

00:06:15.170 --> 00:06:18.560
you see that D is
the midpoint of BE.

00:06:18.560 --> 00:06:21.340
And then finally,
the yellow side

00:06:21.340 --> 00:06:22.840
is between the green
and the orange.

00:06:22.840 --> 00:06:24.850
So between the green and the
orange, we have a yellow side.

00:06:24.850 --> 00:06:27.183
Between the green and the
orange you have a yellow side.

00:06:27.183 --> 00:06:28.800
All of these triangles
are congruent.

00:06:28.800 --> 00:06:30.810
So once again, let
me call this F.

00:06:30.810 --> 00:06:35.450
We see that F is
the midpoint of EC.

00:06:35.450 --> 00:06:37.730
So we've done what
we wanted to do.

00:06:37.730 --> 00:06:39.940
We've shown that if you
start with any arbitrary

00:06:39.940 --> 00:06:54.880
triangle, triangle ADF, we can
construct a triangle BCE so

00:06:54.880 --> 00:07:05.750
that ADF is triangle
BCE's medial triangle.

00:07:05.750 --> 00:07:10.890
And all that means is
that the vertices of ADF

00:07:10.890 --> 00:07:14.190
sit on the midpoints of BCE.

00:07:14.190 --> 00:07:16.680
So you might say Sal, that
by itself is interesting,

00:07:16.680 --> 00:07:18.311
but what's the
whole point of this?

00:07:18.311 --> 00:07:19.810
The whole point of
this is actually,

00:07:19.810 --> 00:07:23.310
I wanted to use this fact that
if you give me any triangle,

00:07:23.310 --> 00:07:25.890
I can make it the medial
triangle of the larger one

00:07:25.890 --> 00:07:29.830
to prove that the altitudes of
this triangle are concurrent.

00:07:29.830 --> 00:07:32.730
And to see that, let me
first draw the altitudes.

00:07:32.730 --> 00:07:35.450
So an altitude from
vertex A looks like this.

00:07:35.450 --> 00:07:37.810
It starts at the vertex,
goes to the opposite side,

00:07:37.810 --> 00:07:40.180
and is perpendicular
to the opposite side.

00:07:40.180 --> 00:07:42.620
If I draw an altitude
from vertex D,

00:07:42.620 --> 00:07:44.560
it would look like this.

00:07:44.560 --> 00:07:47.080
And if I draw an
altitude from vertex F,

00:07:47.080 --> 00:07:49.560
it will look like this.

00:07:49.560 --> 00:07:51.620
And what I did, this
whole set up of this video

00:07:51.620 --> 00:07:57.450
is to show, to prove that these
will always be concurrent.

00:07:57.450 --> 00:07:59.880
And you might say,
wait how do we

00:07:59.880 --> 00:08:01.970
know that they are concurrent?

00:08:01.970 --> 00:08:03.720
Well all you have to
do is think about how

00:08:03.720 --> 00:08:06.610
they interact with
the larger triangle.

00:08:06.610 --> 00:08:09.790
What are these altitudes
to the larger triangle?

00:08:09.790 --> 00:08:12.620
Well, this yellow altitude
to the larger triangle.

00:08:12.620 --> 00:08:16.170
Remember, these two yellow
lines, line AD and line CE

00:08:16.170 --> 00:08:17.480
are parallel.

00:08:17.480 --> 00:08:22.070
So if this is a 90-degree angle,
so its alternate interior angle

00:08:22.070 --> 00:08:23.950
is also going to be 90 degrees.

00:08:23.950 --> 00:08:27.120
So this right over here
is perpendicular to CE,

00:08:27.120 --> 00:08:29.890
and it bisects CE,
because we know

00:08:29.890 --> 00:08:32.510
that ADE is the medial triangle.

00:08:32.510 --> 00:08:33.919
This is the midpoint.

00:08:33.919 --> 00:08:36.850
So this right over here
is perpendicular bisector.

00:08:36.850 --> 00:08:42.620
This is a perpendicular bisector
for the larger triangle,

00:08:42.620 --> 00:08:45.620
for triangle BCE.

00:08:45.620 --> 00:08:47.410
So this altitude
for the smaller one

00:08:47.410 --> 00:08:49.550
is a perpendicular bisector
for the larger one.

00:08:49.550 --> 00:08:51.750
We can do that for all of them.

00:08:51.750 --> 00:08:53.972
If this angle right
over here is 90 degrees,

00:08:53.972 --> 00:08:55.430
then this angle
right over there is

00:08:55.430 --> 00:08:58.013
going to be 90 degrees, because
this line is parallel to this,

00:08:58.013 --> 00:09:02.880
this is a transversal, alternate
interior angles are the same.

00:09:02.880 --> 00:09:05.890
So this line right over here,
this altitude of the smaller

00:09:05.890 --> 00:09:09.390
triangle, it bisects right at
the midpoint of the larger one,

00:09:09.390 --> 00:09:12.520
on this side, and it's also
a perpendicular bisector.

00:09:12.520 --> 00:09:15.702
So it's a perpendicular
bisector of the larger triangle.

00:09:15.702 --> 00:09:17.160
And then finally,
the same thing is

00:09:17.160 --> 00:09:19.750
true of this altitude
right over here.

00:09:19.750 --> 00:09:21.820
It bisects this
side of the larger

00:09:21.820 --> 00:09:24.374
triangle at a 90-degree angle.

00:09:24.374 --> 00:09:26.540
We know that because these
two magenta lines the way

00:09:26.540 --> 00:09:28.420
we constructed the
larger triangle,

00:09:28.420 --> 00:09:29.830
they're going to be parallel.

00:09:29.830 --> 00:09:32.550
So once again, this is a
perpendicular bisector.

00:09:32.550 --> 00:09:36.500
So this whole reason, if you
just give me any triangle,

00:09:36.500 --> 00:09:38.220
I can take its
altitudes and I know

00:09:38.220 --> 00:09:40.989
that its altitude are going
to intersect in one point.

00:09:40.989 --> 00:09:42.280
They're going to be concurrent.

00:09:42.280 --> 00:09:45.310
Because for any triangle, I
can make it the medial triangle

00:09:45.310 --> 00:09:47.520
of a larger one, and
then it's altitudes

00:09:47.520 --> 00:09:50.040
will be the perpendicular
bisector for the larger

00:09:50.040 --> 00:09:50.690
triangle.

00:09:50.690 --> 00:09:53.410
And we already know that
the perpendicular bisectors

00:09:53.410 --> 00:09:55.380
for any triangle are concurrent.

00:09:55.380 --> 00:10:00.700
They do intersect in
exactly one point.


Median centroid right triangle example

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.740
So we're told that
AE is equal to 12.

00:00:03.740 --> 00:00:05.470
That's this side
right over here.

00:00:05.470 --> 00:00:07.865
And EC is equal to 18.

00:00:11.054 --> 00:00:13.470
And then they've drawn a bunch
of the medians here for us.

00:00:13.470 --> 00:00:16.350
So we know that they are medians
because, when they intersect

00:00:16.350 --> 00:00:19.100
the opposite side, they're
telling us that this length is

00:00:19.100 --> 00:00:21.900
equal to this length, or
that ED is equal to DC,

00:00:21.900 --> 00:00:25.070
CB is equal to BA,
AF is equal to FE,

00:00:25.070 --> 00:00:27.735
or that F, B, and
D are the midpoints

00:00:27.735 --> 00:00:30.560
and that G, then, would
be the centroid where

00:00:30.560 --> 00:00:32.310
the medians intersect.

00:00:32.310 --> 00:00:37.710
And so the first thing they ask
us is, what is the area of BGC?

00:00:37.710 --> 00:00:39.460
So BGC right here.

00:00:39.460 --> 00:00:42.280
That is this triangle
right over there.

00:00:42.280 --> 00:00:43.780
And to figure out
that area, we just

00:00:43.780 --> 00:00:47.850
have to remind ourselves that
the three medians of a triangle

00:00:47.850 --> 00:00:54.702
divide a triangle into six
triangles that have equal area.

00:00:54.702 --> 00:00:57.160
So if we know the area of the
entire triangle-- and I think

00:00:57.160 --> 00:00:58.410
we can figure this out.

00:00:58.410 --> 00:00:59.610
This is a right triangle.

00:00:59.610 --> 00:01:00.770
They're telling us that.

00:01:00.770 --> 00:01:04.662
AE-- this entire distance right
over here-- is going to be 12.

00:01:04.662 --> 00:01:05.745
So this is going to be 12.

00:01:05.745 --> 00:01:07.560
Let me make sure I
have enough space.

00:01:07.560 --> 00:01:10.450
This entire distance
right over here is 18.

00:01:10.450 --> 00:01:11.800
They tell us that.

00:01:11.800 --> 00:01:18.530
So the area of
AEC is going to be

00:01:18.530 --> 00:01:21.480
equal to 1/2 times
the base-- which

00:01:21.480 --> 00:01:25.950
is 18-- times the height--
which is 12-- which

00:01:25.950 --> 00:01:29.520
is equal to 9 times
12, which is 108.

00:01:29.520 --> 00:01:33.340
That's the area of this entire
right triangle, triangle AEC.

00:01:33.340 --> 00:01:38.550
If we want the area of BGC or
any of these smaller of the six

00:01:38.550 --> 00:01:40.830
triangles-- if we ignore
this little altitude

00:01:40.830 --> 00:01:44.060
right over here, the ones that
are bounded by the medians--

00:01:44.060 --> 00:01:46.005
then we just have
to divide this by 6.

00:01:46.005 --> 00:01:47.380
Because they all
have equal area.

00:01:47.380 --> 00:01:49.160
We've proven that
in a previous video.

00:01:49.160 --> 00:01:55.480
So the area of BGC is
equal to the area of AEC,

00:01:55.480 --> 00:02:01.120
the entire triangle, divided by
6, which is 108 divided by 6.

00:02:01.120 --> 00:02:01.950
Which is what?

00:02:01.950 --> 00:02:03.765
It's 60-- let's see.

00:02:03.765 --> 00:02:05.880
You get 10 and then 48.

00:02:05.880 --> 00:02:08.070
Looks like it would be 18.

00:02:08.070 --> 00:02:08.735
It would be 18.

00:02:08.735 --> 00:02:10.110
And that's right
because it would

00:02:10.110 --> 00:02:14.239
be-- 108 is the same
thing as 18 times 6.

00:02:14.239 --> 00:02:15.280
So we did our first part.

00:02:15.280 --> 00:02:17.400
The area of that right
over there is 18.

00:02:17.400 --> 00:02:18.900
And if we wanted,
we could say, hey,

00:02:18.900 --> 00:02:20.760
the area of any of
these triangles--

00:02:20.760 --> 00:02:23.300
the ones that are bounded
by the medians-- this

00:02:23.300 --> 00:02:24.250
is going to be 18.

00:02:24.250 --> 00:02:25.520
This is going to be 18.

00:02:25.520 --> 00:02:29.310
This entire FGE triangle
is going to be 18,

00:02:29.310 --> 00:02:31.900
but we did this first
part right over there.

00:02:31.900 --> 00:02:37.240
Now they ask us, what
is the length of AG?

00:02:37.240 --> 00:02:40.090
So AG is the distance.

00:02:40.090 --> 00:02:43.520
It's the longer part of
this median right over here.

00:02:43.520 --> 00:02:45.365
And to figure out
what AG is, we just

00:02:45.365 --> 00:02:49.300
have to remind ourselves
that the centroid is always

00:02:49.300 --> 00:02:51.570
2/3 along the way
of the medians,

00:02:51.570 --> 00:02:54.230
or it divides the
median into two segments

00:02:54.230 --> 00:02:57.100
that have a ratio of 2 to 1.

00:02:57.100 --> 00:02:59.200
So if we know the entire
length of this median,

00:02:59.200 --> 00:03:00.880
we could just take 2/3 of that.

00:03:00.880 --> 00:03:03.430
And that'll give us
the length of AG.

00:03:03.430 --> 00:03:06.200
And lucky for us, this
is a right triangle.

00:03:06.200 --> 00:03:09.670
And we know that F and
D are the midpoints.

00:03:09.670 --> 00:03:11.930
So for example, we
know this AE is 12.

00:03:11.930 --> 00:03:12.700
That was given.

00:03:12.700 --> 00:03:16.150
We know that ED is
half of this 18.

00:03:16.150 --> 00:03:19.450
So ED right over here--
I'll do this in a new color.

00:03:19.450 --> 00:03:21.150
ED is going to be 9.

00:03:23.719 --> 00:03:25.760
So then we could just use
the Pythagorean theorem

00:03:25.760 --> 00:03:27.530
to figure out what AD is.

00:03:27.530 --> 00:03:32.000
AD is the hypotenuse
of this right triangle.

00:03:32.000 --> 00:03:35.920
So we're looking at
triangle AED right now.

00:03:35.920 --> 00:03:36.920
Let me write this down.

00:03:36.920 --> 00:03:44.310
We know that 12
squared plus 9 squared

00:03:44.310 --> 00:03:46.513
is going to be
equal to AD squared.

00:03:53.680 --> 00:03:56.210
12 squared is 144.

00:03:56.210 --> 00:03:58.200
144 plus 81.

00:04:01.800 --> 00:04:04.290
And so this is going to
be equal to AD squared.

00:04:04.290 --> 00:04:05.360
So this is what?

00:04:05.360 --> 00:04:07.490
This is 225.

00:04:07.490 --> 00:04:11.350
So we have 225 is
equal to AD squared.

00:04:14.240 --> 00:04:18.560
And 225, you may or may not
recognize, is 15 squared.

00:04:18.560 --> 00:04:21.489
So AD is equal to 15.

00:04:21.489 --> 00:04:23.780
You want to take the principal
root, the positive root,

00:04:23.780 --> 00:04:26.570
because we're talking about
distances or lengths of sides.

00:04:26.570 --> 00:04:28.140
We don't care about
the negatives.

00:04:28.140 --> 00:04:29.700
So AD is equal to 15.

00:04:29.700 --> 00:04:31.500
So this whole thing
right over here

00:04:31.500 --> 00:04:33.620
is going to be equal to 15.

00:04:33.620 --> 00:04:36.253
And AG is going to be 2/3 of AD.

00:04:38.790 --> 00:04:40.310
We proved that in
a previous video,

00:04:40.310 --> 00:04:43.312
that the centroid is
2/3 along the way of any

00:04:43.312 --> 00:04:44.020
of these medians.

00:04:44.020 --> 00:04:45.769
And we could do it for
any of the medians.

00:04:45.769 --> 00:04:52.680
So it's equal to 2/3 times
15, which is equal to 10.

00:04:52.680 --> 00:05:00.070
So AG right over
here is equal to 10.

00:05:00.070 --> 00:05:01.340
So we did the second part.

00:05:01.340 --> 00:05:04.397
Now, this third part,
what is the area of FGH?

00:05:08.380 --> 00:05:14.210
So let me color it in-- FGH.

00:05:14.210 --> 00:05:18.970
So if we knew this length-- if
we knew HG and if we knew FH--

00:05:18.970 --> 00:05:21.540
we could easily figure
out what that area is.

00:05:21.540 --> 00:05:23.230
And there's actually
multiple ways

00:05:23.230 --> 00:05:26.550
of figuring out either
one of those things.

00:05:26.550 --> 00:05:31.430
So one way that we can think
about finding what HG is

00:05:31.430 --> 00:05:36.150
is to remind
ourselves that HG is

00:05:36.150 --> 00:05:42.640
the altitude of either
triangle FGE or triangle AFG.

00:05:42.640 --> 00:05:44.850
And both of them
have a base of 6.

00:05:47.680 --> 00:05:50.680
So this is 6, and
this is 6 over here.

00:05:50.680 --> 00:05:53.360
And then they have a
height equal to GH.

00:05:53.360 --> 00:05:54.810
And we know what the area is.

00:05:54.810 --> 00:05:59.070
We know that the area
is already equal to 18.

00:05:59.070 --> 00:06:01.520
So let's take this
triangle up here.

00:06:01.520 --> 00:06:05.920
So we're talking about
the area of triangle AFG.

00:06:05.920 --> 00:06:09.480
So we know it's 1/2
times its base, which

00:06:09.480 --> 00:06:14.660
is 6, times its height,
which is GH-- times GH.

00:06:14.660 --> 00:06:16.250
That's just 1/2
base times height

00:06:16.250 --> 00:06:18.370
is equal to the area
of this triangle, which

00:06:18.370 --> 00:06:22.230
is going to be equal to 18.

00:06:22.230 --> 00:06:24.580
And so then, we just have
to tell ourselves, well,

00:06:24.580 --> 00:06:28.860
this is-- 3 times
GH is equal to 18.

00:06:28.860 --> 00:06:35.300
If we divide both sides of
this by 3, GH is equal to 6.

00:06:35.300 --> 00:06:37.900
So that is one way to do it.

00:06:37.900 --> 00:06:39.960
GH is equal to 6.

00:06:39.960 --> 00:06:42.020
You could have also made
the similarity argument.

00:06:42.020 --> 00:06:44.720
Then you could have said,
look, this triangle up here

00:06:44.720 --> 00:06:49.230
is similar to this larger
triangle over here.

00:06:49.230 --> 00:06:52.420
This hypotenuse is 2/3 of the
length of this entire thing.

00:06:52.420 --> 00:06:54.795
So this is going to
be 2/3 of this 9.

00:06:54.795 --> 00:06:57.170
So that's another way that
you could have gotten 6 there.

00:06:57.170 --> 00:06:59.470
But either way, we
got this length.

00:06:59.470 --> 00:07:01.860
Now we just have to
figure out what FH is.

00:07:01.860 --> 00:07:09.190
And we could figure out what
FH is if we know what AH is.

00:07:09.190 --> 00:07:11.220
Because we know A to F is 6.

00:07:11.220 --> 00:07:15.810
So FH is going to
be AH minus AF.

00:07:15.810 --> 00:07:17.190
So let's figure out what AH is.

00:07:17.190 --> 00:07:19.456
Well, once again, we can
make a similarity argument.

00:07:19.456 --> 00:07:20.830
And if we want to
do it formally,

00:07:20.830 --> 00:07:26.330
we see that both this larger
right triangle and the smaller

00:07:26.330 --> 00:07:29.380
right triangle, both have
a 90-degree angle there.

00:07:29.380 --> 00:07:31.630
They both have this
angle in common.

00:07:31.630 --> 00:07:33.190
So they have two
angles in common.

00:07:33.190 --> 00:07:36.020
They are definitely
similar triangles.

00:07:36.020 --> 00:07:40.050
And so we know the ratio of
AH-- let me do it in orange.

00:07:40.050 --> 00:07:44.590
We know that the ratio
of AH to AE-- which

00:07:44.590 --> 00:07:49.330
is 12-- is equal to
the ratio of AG-- which

00:07:49.330 --> 00:07:55.230
is 10-- to the ratio of AD--
which we already figured out

00:07:55.230 --> 00:07:57.170
was 15.

00:07:57.170 --> 00:08:00.022
So one way to think about it
is AH is going to be 2/3 of 12.

00:08:00.022 --> 00:08:01.980
Or we can just work
through the math just using

00:08:01.980 --> 00:08:03.560
the similar triangles.

00:08:03.560 --> 00:08:06.410
So this right-hand side
over here is just 2/3.

00:08:06.410 --> 00:08:09.460
And so AH-- multiplying
both sides by 12--

00:08:09.460 --> 00:08:15.700
is equal to 2/3 times
12, which is just 8.

00:08:15.700 --> 00:08:17.970
So AH here is 8.

00:08:17.970 --> 00:08:19.640
AH is 8.

00:08:19.640 --> 00:08:21.860
AF is 6.

00:08:21.860 --> 00:08:27.129
So FH right over here
is going to be 2.

00:08:27.129 --> 00:08:28.670
And so now we have
enough information

00:08:28.670 --> 00:08:32.409
to figure out the area of FHG.

00:08:32.409 --> 00:08:33.669
So let me write it over here.

00:08:33.669 --> 00:08:37.270
It's going to be
1/2 times the base.

00:08:37.270 --> 00:08:39.360
I'll just use FH
as the base here,

00:08:39.360 --> 00:08:41.490
although I could
do it either way.

00:08:41.490 --> 00:08:43.299
Well, I'll use FH as the base.

00:08:43.299 --> 00:08:49.490
1/2 times 2 times the height--
times 6-- which is equal to 6.

00:08:49.490 --> 00:08:51.329
And we are done.

00:08:51.329 --> 00:08:52.370
And you could keep going.

00:08:52.370 --> 00:08:53.880
You could figure out the
length of pretty much

00:08:53.880 --> 00:08:56.280
all of these segments here
using some of these techniques

00:08:56.280 --> 00:08:57.290
or any of these areas.

00:08:57.290 --> 00:09:00.370
Well, we've actually
figured out most of them.


Proving that the centroid is 2-3rds along the median

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:03.300
I've drawn an arbitrary
triangle right over here,

00:00:03.300 --> 00:00:08.210
and I've also drawn its three
medians: median EB, median FC,

00:00:08.210 --> 00:00:09.990
and median AD.

00:00:09.990 --> 00:00:13.400
And we know that where the three
medians intersect at point G

00:00:13.400 --> 00:00:16.309
right over here, we
call that the centroid.

00:00:16.309 --> 00:00:19.710
What I want to do in this
video is prove to you

00:00:19.710 --> 00:00:24.520
that the centroid is exactly 2/3
along the way of each median.

00:00:24.520 --> 00:00:26.130
Or another way to
think about it,

00:00:26.130 --> 00:00:27.804
we can pick any one
of these medians,

00:00:27.804 --> 00:00:28.970
and let's say let's pick EB.

00:00:28.970 --> 00:00:30.860
What I want to do
is I want to prove

00:00:30.860 --> 00:00:38.230
that EG is equal to 2 times GB.

00:00:38.230 --> 00:00:41.900
So whatever distance this is
it's twice this distance there.

00:00:41.900 --> 00:00:44.290
Or another way to
think about it is EG

00:00:44.290 --> 00:00:48.090
is 2/3 along the way of EB.

00:00:48.090 --> 00:00:50.240
And the logic that I'm
using to prove this you

00:00:50.240 --> 00:00:52.060
can use for any of
the medians to show

00:00:52.060 --> 00:00:55.780
that the centroid is exactly
2/3 along the way of any median,

00:00:55.780 --> 00:00:57.290
or divides it into
a segment that's

00:00:57.290 --> 00:01:00.320
twice as long as
the other segment.

00:01:00.320 --> 00:01:07.570
And to do that, let's focus--
I want to focus on triangle ABE

00:01:07.570 --> 00:01:08.240
right over here.

00:01:08.240 --> 00:01:11.996
And I'm going to draw this
median as essentially the base.

00:01:11.996 --> 00:01:13.120
So let me draw it that way.

00:01:13.120 --> 00:01:15.920
I'm going to try to
color code it similarly.

00:01:15.920 --> 00:01:21.110
So we draw it a little
bit flatter than that.

00:01:21.110 --> 00:01:22.480
So it's like that.

00:01:22.480 --> 00:01:25.182
And then we have the
two yellow sides,

00:01:25.182 --> 00:01:28.070
so it looks something like this.

00:01:28.070 --> 00:01:30.250
It looks something like that.

00:01:30.250 --> 00:01:33.520
And then we have the
centroid, right over here

00:01:33.520 --> 00:01:36.830
at G. That is our
centroid, and then we

00:01:36.830 --> 00:01:40.780
have this magenta
line going to A.

00:01:40.780 --> 00:01:43.000
Let me draw it a little
bit neater than that.

00:01:43.000 --> 00:01:45.100
We have that line
going to A, and then we

00:01:45.100 --> 00:01:47.700
have this blue line going
to F right over here.

00:01:52.340 --> 00:01:55.650
And let me label all the points.

00:01:55.650 --> 00:01:57.040
Go back to the orange color.

00:01:57.040 --> 00:02:00.660
So this is going to be
E, this is going to be B,

00:02:00.660 --> 00:02:05.046
this is going to be A, this is
going to be F right over here.

00:02:05.046 --> 00:02:07.420
And just to make sure we have
all the same markings, that

00:02:07.420 --> 00:02:11.170
little marking there is that
marking, these two markings,

00:02:11.170 --> 00:02:14.120
these two markings are on
this side right over there.

00:02:14.120 --> 00:02:17.070
And the whole way that I'm
going to prove that EG is twice

00:02:17.070 --> 00:02:19.690
as long as GB is just
refer to the result

00:02:19.690 --> 00:02:21.380
that we did, I think,
a couple of videos

00:02:21.380 --> 00:02:27.080
ago that the medians divide
this triangle into six smaller

00:02:27.080 --> 00:02:30.140
triangles that all
have equal area.

00:02:30.140 --> 00:02:32.617
So another way to think about
it is each of these three

00:02:32.617 --> 00:02:33.950
small triangles have equal area.

00:02:33.950 --> 00:02:37.190
These are three of the total
of six smaller triangles.

00:02:37.190 --> 00:02:39.530
So these three all
have equal area.

00:02:39.530 --> 00:02:42.480
So let's think about this
triangle right over here.

00:02:42.480 --> 00:02:46.330
Let's think about this
triangle, triangle AGB.

00:02:46.330 --> 00:02:48.590
This is triangle AGB
right over there.

00:02:48.590 --> 00:02:49.920
Those are the same triangles.

00:02:49.920 --> 00:02:56.020
And let's compare that to
triangle EAG right over here.

00:02:56.020 --> 00:02:57.910
Let's compare it
to this triangle,

00:02:57.910 --> 00:03:00.300
which is this triangle
right over here

00:03:00.300 --> 00:03:02.410
on the original drawing.

00:03:02.410 --> 00:03:06.580
Now, they both have
the exact same height.

00:03:06.580 --> 00:03:11.880
If you view EG as their base--
or I the guess their shared

00:03:11.880 --> 00:03:14.440
base, they don't have
the exact same base.

00:03:14.440 --> 00:03:17.690
The smaller triangle
has the base E-- sorry,

00:03:17.690 --> 00:03:20.720
the smaller orange triangle
has GB as it's base.

00:03:20.720 --> 00:03:24.260
The larger blue triangle
has EG as its base,

00:03:24.260 --> 00:03:27.310
but they definitely both have
the same height, or altitude,

00:03:27.310 --> 00:03:29.140
when you draw it this way.

00:03:29.140 --> 00:03:33.610
So their height, in both
cases, is this right over here.

00:03:33.610 --> 00:03:37.660
Now the other thing
that we do know

00:03:37.660 --> 00:03:41.010
is that this blue
triangle EAG has

00:03:41.010 --> 00:03:43.260
twice the area of
the orange triangle.

00:03:43.260 --> 00:03:44.310
How do we know that?

00:03:44.310 --> 00:03:46.300
Because it's got two of
these triangles in it.

00:03:46.300 --> 00:03:49.060
So one way to think about it
is if this orange triangle has

00:03:49.060 --> 00:03:51.144
area x, actually let
me call it a-- well

00:03:51.144 --> 00:03:53.060
I already used a, so I'm
going to call it area

00:03:53.060 --> 00:03:57.090
x-- then each of these
blue triangles have area x.

00:03:57.090 --> 00:04:05.520
Or you could say this entire
blue region has area 2x.

00:04:05.520 --> 00:04:08.210
So if you look at this blue
triangle right over here,

00:04:08.210 --> 00:04:13.880
we know that 1/2 times base
times height is equal to area.

00:04:13.880 --> 00:04:17.670
So we get 1/2-- the base is EG.

00:04:17.670 --> 00:04:19.839
1/2-- I'll do that
in the green color--

00:04:19.839 --> 00:04:27.940
1/2 EG times height times
this yellow height is

00:04:27.940 --> 00:04:29.200
going to be equal to 2x.

00:04:32.720 --> 00:04:36.170
I'm just applying the formula
for area of a triangle.

00:04:36.170 --> 00:04:39.320
1/2 base times height
is equal to area.

00:04:39.320 --> 00:04:40.720
This is our area.

00:04:40.720 --> 00:04:43.020
Now, let's do the same thing
for this orange triangle.

00:04:43.020 --> 00:04:47.120
1/2-- let me scroll over a
little bit to the right--

00:04:47.120 --> 00:04:56.060
we have 1/2 GB times
the yellow height is

00:04:56.060 --> 00:04:57.585
going to be equal to x.

00:05:01.610 --> 00:05:02.730
Well we can substitute it.

00:05:02.730 --> 00:05:06.520
If this is equal to x, we can
place this entire expression

00:05:06.520 --> 00:05:07.880
right over here for x.

00:05:07.880 --> 00:05:08.770
So let's do that.

00:05:08.770 --> 00:05:11.730
We get 1/2-- and
you might already

00:05:11.730 --> 00:05:15.520
see where this is going, but
I won't skip any steps here--

00:05:15.520 --> 00:05:28.982
we get 1/2 times EG times
h is equal to 2 times x.

00:05:28.982 --> 00:05:30.940
But instead of x, I'm
going to write this here.

00:05:30.940 --> 00:05:38.370
Is equal to 2 times 1/2
times GB times this length--

00:05:38.370 --> 00:05:43.570
times the base of the
smaller triangle times h.

00:05:43.570 --> 00:05:45.950
And now we can
just simplify this.

00:05:45.950 --> 00:05:50.620
We have 2 times 1/2
is just going to be 1.

00:05:50.620 --> 00:05:53.950
You can divide both
sides by h, and we

00:05:53.950 --> 00:05:57.190
are left with 1/2
EG is equal to GB.

00:05:57.190 --> 00:06:00.329
Or we could write EG
over 2 is equal to-- let

00:06:00.329 --> 00:06:01.870
me do it in the same
color since I've

00:06:01.870 --> 00:06:04.840
gone this far with
the same colors.

00:06:04.840 --> 00:06:12.855
So we can write 1/2
EG is equal to GB.

00:06:16.290 --> 00:06:17.420
And we're done.

00:06:17.420 --> 00:06:20.116
This is essentially saying
that GB is half of EG.

00:06:23.670 --> 00:06:26.430
So for example, if EG is
2, this is going to be 1.

00:06:26.430 --> 00:06:28.610
If EG is 4, this
is going to be 2.

00:06:28.610 --> 00:06:30.111
So we've actually
proven our result.

00:06:30.111 --> 00:06:31.693
Well actually, let's
go back to-- this

00:06:31.693 --> 00:06:33.110
is the result we
wanted to prove.

00:06:33.110 --> 00:06:35.800
To get to there we just multiply
both sides of this equation

00:06:35.800 --> 00:06:36.410
by 2.

00:06:36.410 --> 00:06:39.960
You multiply this, the
left-hand side by 2, you get EG.

00:06:39.960 --> 00:06:42.880
You multiply the right-hand
side by 2, you get GB.

00:06:42.880 --> 00:06:46.800
So we've proven that
EG is twice GB .

00:06:46.800 --> 00:06:49.580
And you can apply the same
logic to any of the medians

00:06:49.580 --> 00:06:51.310
to show that the
centroid is exactly

00:06:51.310 --> 00:06:54.741
2/3 along the way of the median.


Exploring medial triangles

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:02.630
So I've got an
arbitrary triangle here.

00:00:02.630 --> 00:00:04.830
We'll call it triangle ABC.

00:00:04.830 --> 00:00:07.130
And what I want to do
is look at the midpoints

00:00:07.130 --> 00:00:08.840
of each of the sides of ABC.

00:00:08.840 --> 00:00:12.050
So this is the midpoint of
one of the sides, of side BC.

00:00:12.050 --> 00:00:17.160
Let's call that point D. Let's
call this midpoint E. And let's

00:00:17.160 --> 00:00:20.720
call this midpoint
right over here F.

00:00:20.720 --> 00:00:22.320
And since it's the
midpoint, we know

00:00:22.320 --> 00:00:25.380
that the distance between BD
is equal to the distance from D

00:00:25.380 --> 00:00:28.210
to C. So this distance is
equal to this distance.

00:00:28.210 --> 00:00:31.380
We know that AE is equal
to EC, so this distance

00:00:31.380 --> 00:00:32.970
is equal to that distance.

00:00:32.970 --> 00:00:35.620
And we know that
AF is equal to FB,

00:00:35.620 --> 00:00:39.150
so this distance is
equal to this distance.

00:00:39.150 --> 00:00:41.399
Instead of drawing medians
going from these midpoints

00:00:41.399 --> 00:00:42.940
to the vertices,
what I want to do is

00:00:42.940 --> 00:00:46.090
I want to connect these
midpoints and see what happens.

00:00:46.090 --> 00:00:48.830
So if I connect them, I
clearly have three points.

00:00:48.830 --> 00:00:52.970
So if you connect three
non-linear points like this,

00:00:52.970 --> 00:00:55.710
you will get another triangle.

00:00:55.710 --> 00:00:59.470
And this triangle that's formed
from the midpoints of the sides

00:00:59.470 --> 00:01:02.623
of this larger triangle-- we
call this a medial triangle.

00:01:06.800 --> 00:01:08.880
And that's all nice
and cute by itself.

00:01:08.880 --> 00:01:10.060
But what we're going
to see in this video

00:01:10.060 --> 00:01:11.590
is that the medial
triangle actually

00:01:11.590 --> 00:01:13.800
has some very neat properties.

00:01:13.800 --> 00:01:15.210
What we're actually
going to show

00:01:15.210 --> 00:01:19.290
is that it divides any triangle
into four smaller triangles

00:01:19.290 --> 00:01:21.220
that are congruent
to each other,

00:01:21.220 --> 00:01:24.530
that all four of these triangles
are identical to each other.

00:01:24.530 --> 00:01:26.610
And they're all similar
to the larger triangle.

00:01:26.610 --> 00:01:28.430
And you could think
of them each as having

00:01:28.430 --> 00:01:31.320
1/4 of the area of
the larger triangle.

00:01:31.320 --> 00:01:33.650
So let's go about proving it.

00:01:33.650 --> 00:01:35.700
So first, let's focus
on this triangle down

00:01:35.700 --> 00:01:37.310
here, triangle CDE.

00:01:37.310 --> 00:01:40.200
And it looks similar
to the larger triangle,

00:01:40.200 --> 00:01:42.450
to triangle CBA.

00:01:42.450 --> 00:01:44.964
But let's prove it to ourselves.

00:01:44.964 --> 00:01:47.130
So one thing we can say is,
well, look, both of them

00:01:47.130 --> 00:01:49.570
share this angle
right over here.

00:01:49.570 --> 00:01:53.650
Both the larger triangle,
triangle CBA, has this angle.

00:01:53.650 --> 00:01:56.630
And the smaller triangle,
CDE, has this angle.

00:01:56.630 --> 00:01:58.460
So they definitely
share that angle.

00:01:58.460 --> 00:02:00.950
And then let's think about
the ratios of the sides.

00:02:00.950 --> 00:02:14.240
We know that the ratio of CD
to CB is equal to 1 over 2.

00:02:14.240 --> 00:02:18.350
This is 1/2 of this entire
side, is equal to 1 over 2.

00:02:18.350 --> 00:02:22.980
And that's the same thing
as the ratio of CE to CA.

00:02:22.980 --> 00:02:27.040
CE is exactly 1/2 of CA,
because E is the midpoint.

00:02:27.040 --> 00:02:30.880
It's equal to CE over CA.

00:02:30.880 --> 00:02:33.295
So we have an angle,
corresponding angles that

00:02:33.295 --> 00:02:35.150
are congruent, and
then the ratios

00:02:35.150 --> 00:02:37.840
of two corresponding sides
on either side of that angle

00:02:37.840 --> 00:02:38.480
are the same.

00:02:38.480 --> 00:02:45.010
CD over CB is 1/2, CE over CA
is 1/2, and the angle in between

00:02:45.010 --> 00:02:46.120
is congruent.

00:02:46.120 --> 00:03:03.460
So by SAS similarity, we
know that triangle CDE

00:03:03.460 --> 00:03:08.070
is similar to triangle CBA.

00:03:12.400 --> 00:03:14.860
And just from that, you can
get some interesting results.

00:03:14.860 --> 00:03:16.318
Because then we
know that the ratio

00:03:16.318 --> 00:03:20.230
of this side of the smaller
triangle to the longer triangle

00:03:20.230 --> 00:03:21.880
is also going to be 1/2.

00:03:21.880 --> 00:03:23.880
Because the other two
sides have a ratio of 1/2,

00:03:23.880 --> 00:03:25.710
and we're dealing with
similar triangles.

00:03:25.710 --> 00:03:27.860
So this is going
to be 1/2 of that.

00:03:27.860 --> 00:03:32.750
And we know 1/2 of AB is just
going to be the length of FA.

00:03:32.750 --> 00:03:34.960
So we know that this
length right over here

00:03:34.960 --> 00:03:37.911
is going to be the
same as FA or FB.

00:03:37.911 --> 00:03:39.910
And we get that straight
from similar triangles.

00:03:39.910 --> 00:03:49.290
Because these are similar,
we know that DE over BA

00:03:49.290 --> 00:03:51.730
has got to be equal
to these ratios,

00:03:51.730 --> 00:03:54.650
the other corresponding
sides, which is equal to 1/2.

00:03:54.650 --> 00:03:57.580
And so that's how we got
that right over there.

00:03:57.580 --> 00:04:01.395
Now let's think about
this triangle up here.

00:04:04.070 --> 00:04:05.187
We could call it BDF.

00:04:07.800 --> 00:04:09.610
So first of all, if
we compare triangle

00:04:09.610 --> 00:04:12.930
BDF to the larger
triangle, they both share

00:04:12.930 --> 00:04:15.780
this angle right
over here, angle ABC.

00:04:15.780 --> 00:04:17.810
They both have that
angle in common.

00:04:17.810 --> 00:04:20.279
And we're going to have
the exact same argument.

00:04:20.279 --> 00:04:21.714
You can just look
at this diagram.

00:04:21.714 --> 00:04:27.120
And you know that the ratio
of BA-- let me do it this way.

00:04:27.120 --> 00:04:33.950
The ratio of BF to
BA is equal to 1/2,

00:04:33.950 --> 00:04:36.725
which is also the
ratio of BD to BC.

00:04:40.820 --> 00:04:42.830
The ratio of this
to that is the same

00:04:42.830 --> 00:04:45.710
as the ratio of this
to that, which is 1/2.

00:04:45.710 --> 00:04:47.360
Because BD is 1/2 of
this whole length.

00:04:47.360 --> 00:04:49.500
BF is 1/2 of that whole length.

00:04:49.500 --> 00:04:51.860
And so you have
corresponding sides

00:04:51.860 --> 00:04:53.860
have the same ratio
on the two triangles,

00:04:53.860 --> 00:04:55.450
and they share an
angle in between.

00:04:55.450 --> 00:05:03.010
So once again, by
SAS similarity,

00:05:03.010 --> 00:05:07.980
we know that triangle--
I'll write it

00:05:07.980 --> 00:05:18.750
this way-- DBF is
similar to triangle CBA.

00:05:25.129 --> 00:05:27.420
And once again, we use this
exact same kind of argument

00:05:27.420 --> 00:05:28.435
that we did with this triangle.

00:05:28.435 --> 00:05:31.060
Well, if it's similar, the ratio
of all the corresponding sides

00:05:31.060 --> 00:05:32.070
have to be the same.

00:05:32.070 --> 00:05:33.380
And that ratio is 1/2.

00:05:33.380 --> 00:05:35.680
So the ratio of this
side to this side,

00:05:35.680 --> 00:05:41.420
the ratio of FD to
AC, has to be 1/2.

00:05:41.420 --> 00:05:44.100
Or FD has to be 1/2 of AC.

00:05:44.100 --> 00:05:46.750
And 1/2 of AC is just
the length of AE.

00:05:46.750 --> 00:05:50.710
So that is just going to be
that length right over there.

00:05:50.710 --> 00:05:52.700
I think you see
where this is going.

00:05:52.700 --> 00:05:55.470
And also, because it's similar,
all of the corresponding angles

00:05:55.470 --> 00:05:56.490
have to be the same.

00:05:56.490 --> 00:05:58.990
And we know that
the larger triangle

00:05:58.990 --> 00:06:00.870
has a yellow angle
right over there.

00:06:00.870 --> 00:06:03.766
So we'd have that yellow
angle right over here.

00:06:03.766 --> 00:06:05.140
And this triangle
right over here

00:06:05.140 --> 00:06:07.440
was also similar to
the larger triangle.

00:06:07.440 --> 00:06:09.760
So it will have that same
angle measure up here.

00:06:09.760 --> 00:06:12.660
We already showed that
in this first part.

00:06:12.660 --> 00:06:14.864
So now let's go to
this third triangle.

00:06:14.864 --> 00:06:16.030
I think you see the pattern.

00:06:16.030 --> 00:06:18.155
I'm sure you might be able
to just pause this video

00:06:18.155 --> 00:06:19.770
and prove it for yourself.

00:06:19.770 --> 00:06:25.800
But we see that the
ratio of AF over AB

00:06:25.800 --> 00:06:28.776
is going to be the
same as the ratio of AE

00:06:28.776 --> 00:06:33.380
over AC, which is equal to 1/2.

00:06:33.380 --> 00:06:36.970
So we have two corresponding
sides where the ratio is 1/2,

00:06:36.970 --> 00:06:38.960
from the smaller
to larger triangle.

00:06:38.960 --> 00:06:41.510
And they share a common angle.

00:06:41.510 --> 00:06:44.450
They share this angle in
between the two sides.

00:06:44.450 --> 00:06:47.470
So by SAS similarity--
this is getting repetitive

00:06:47.470 --> 00:06:59.563
now-- we know that triangle
EFA is similar to triangle CBA.

00:07:03.280 --> 00:07:05.460
And so the ratio of all
of the corresponding sides

00:07:05.460 --> 00:07:06.680
need to be 1/2.

00:07:06.680 --> 00:07:10.000
So the ratio of FE to
BC needs to be 1/2,

00:07:10.000 --> 00:07:13.630
or FE needs to be 1/2 of that,
which is just the length of BD.

00:07:13.630 --> 00:07:17.720
So this is just going to be
that length right over there.

00:07:17.720 --> 00:07:19.500
And you can also
say that since we've

00:07:19.500 --> 00:07:22.380
shown that this triangle, this
triangle, and this triangle--

00:07:22.380 --> 00:07:24.005
we haven't talked
about this middle one

00:07:24.005 --> 00:07:26.640
yet-- they're all similar
to the larger triangle.

00:07:26.640 --> 00:07:28.890
So they're also all going
to be similar to each other.

00:07:28.890 --> 00:07:32.080
So they're all going to have
the same corresponding angles.

00:07:32.080 --> 00:07:34.332
So if the larger triangle
had this yellow angle here,

00:07:34.332 --> 00:07:35.790
then all of the
triangles are going

00:07:35.790 --> 00:07:38.520
to have this yellow
angle right over there.

00:07:38.520 --> 00:07:40.620
And if the larger triangle
had this blue angle

00:07:40.620 --> 00:07:43.280
right over here, then in
the corresponding vertex,

00:07:43.280 --> 00:07:46.005
all of the triangles are
going to have that blue angle.

00:07:46.005 --> 00:07:47.880
All of the ones that
we've shown are similar.

00:07:47.880 --> 00:07:50.750
We haven't thought about this
middle triangle just yet.

00:07:50.750 --> 00:07:52.780
And of course, if this
is similar to the whole,

00:07:52.780 --> 00:07:56.310
it'll also have this
angle at this vertex

00:07:56.310 --> 00:07:58.820
right over here, because this
corresponds to that vertex,

00:07:58.820 --> 00:08:01.152
based on the similarity.

00:08:01.152 --> 00:08:02.068
So that's interesting.

00:08:02.068 --> 00:08:06.640
Now let's compare the
triangles to each other.

00:08:06.640 --> 00:08:09.240
We've now shown that
all of these triangles

00:08:09.240 --> 00:08:11.200
have the exact same three sides.

00:08:11.200 --> 00:08:14.520
Has this blue side-- or
actually, this one-mark side,

00:08:14.520 --> 00:08:16.760
this two-mark side, and
this three-mark side.

00:08:16.760 --> 00:08:18.220
One mark, two mark, three mark.

00:08:18.220 --> 00:08:19.910
One mark, two mark, three mark.

00:08:19.910 --> 00:08:22.670
And that even applies
to this middle triangle

00:08:22.670 --> 00:08:24.020
right over here.

00:08:24.020 --> 00:08:35.049
So by side-side-side
congruency, we now know--

00:08:35.049 --> 00:08:38.000
and we want to be careful to get
our corresponding sides right--

00:08:38.000 --> 00:08:45.220
we now know that triangle CDE
is congruent to triangle DBF.

00:08:49.707 --> 00:08:51.290
I want to get the
corresponding sides.

00:08:51.290 --> 00:08:51.980
I'm looking at the colors.

00:08:51.980 --> 00:08:54.420
I went from yellow to magenta
to blue, yellow, magenta,

00:08:54.420 --> 00:08:58.910
to blue, which is going to
be congruent to triangle

00:08:58.910 --> 00:09:04.921
EFA, which is going to be
congruent to this triangle

00:09:04.921 --> 00:09:05.420
in here.

00:09:05.420 --> 00:09:07.180
But we want to make
sure that we're

00:09:07.180 --> 00:09:09.480
getting the right
corresponding sides here.

00:09:09.480 --> 00:09:11.010
So to make sure we
do that, we just

00:09:11.010 --> 00:09:12.770
have to think about the angles.

00:09:12.770 --> 00:09:15.100
So we know-- and
this is interesting--

00:09:15.100 --> 00:09:18.330
that because the interior
angles of a triangle

00:09:18.330 --> 00:09:20.820
add up to 180 degrees,
we know this magenta

00:09:20.820 --> 00:09:24.600
angle plus this blue angle plus
this yellow angle equal 180.

00:09:24.600 --> 00:09:27.010
Here, we have the blue
angle and the magenta angle,

00:09:27.010 --> 00:09:29.610
and clearly they will
all add up to 180.

00:09:29.610 --> 00:09:31.600
So you must have the blue angle.

00:09:31.600 --> 00:09:33.850
The blue angle must
be right over here.

00:09:33.850 --> 00:09:37.300
Same argument-- yellow
angle and blue angle,

00:09:37.300 --> 00:09:39.510
we must have the magenta
angle right over here.

00:09:39.510 --> 00:09:40.690
They add up to 180.

00:09:40.690 --> 00:09:42.250
So this must be
the magenta angle.

00:09:42.250 --> 00:09:44.630
And then finally,
magenta and blue-- this

00:09:44.630 --> 00:09:47.192
must be the yellow
angle right over there.

00:09:47.192 --> 00:09:48.900
And so when we wrote
the congruency here,

00:09:48.900 --> 00:09:50.890
we started at CDE.

00:09:50.890 --> 00:09:53.710
We went yellow, magenta, blue.

00:09:53.710 --> 00:09:57.190
So over here, we're going
to go yellow, magenta, blue.

00:09:57.190 --> 00:09:59.544
So it's going to be
congruent to triangle FED.

00:10:05.440 --> 00:10:06.620
And so that's pretty cool.

00:10:06.620 --> 00:10:09.472
We just showed that all
three, that this triangle,

00:10:09.472 --> 00:10:11.430
this triangle, this
triangle, and that triangle

00:10:11.430 --> 00:10:12.530
are congruent.

00:10:12.530 --> 00:10:14.830
And also, we can look
at the corresponding--

00:10:14.830 --> 00:10:17.680
and that they all have
ratios relative to-- they're

00:10:17.680 --> 00:10:20.520
all similar to the larger
triangle, to triangle ABC.

00:10:20.520 --> 00:10:23.430
And that the ratio between
the sides is 1 to 2.

00:10:23.430 --> 00:10:25.940
And also, because we've looked
at corresponding angles,

00:10:25.940 --> 00:10:30.520
we see, for example,
that this angle

00:10:30.520 --> 00:10:31.770
is the same as that angle.

00:10:31.770 --> 00:10:33.860
So if you viewed DC
or if you viewed BC

00:10:33.860 --> 00:10:35.710
as a transversal,
all of a sudden

00:10:35.710 --> 00:10:40.440
it becomes pretty clear that FD
is going to be parallel to AC,

00:10:40.440 --> 00:10:43.130
because the corresponding
angles are congruent.

00:10:43.130 --> 00:10:46.900
So this is going to be parallel
to that right over there.

00:10:46.900 --> 00:10:49.060
And then you could use
that same exact argument

00:10:49.060 --> 00:10:51.780
to say, well, then this
side, because once again,

00:10:51.780 --> 00:10:56.080
corresponding angles
here and here-- you

00:10:56.080 --> 00:10:58.490
could say that
this is going to be

00:10:58.490 --> 00:11:01.750
parallel to that
right over there.

00:11:01.750 --> 00:11:06.360
And then finally, you make
the same argument over here.

00:11:06.360 --> 00:11:08.820
I want to make sure I get the
right corresponding angles.

00:11:08.820 --> 00:11:12.060
You have this line
and this line.

00:11:12.060 --> 00:11:13.850
And this angle
corresponds to that angle.

00:11:13.850 --> 00:11:14.980
They're the same.

00:11:14.980 --> 00:11:21.410
So this DE must
be parallel to BA.

00:11:21.410 --> 00:11:25.280
So that's another neat property
of this medial triangle,

00:11:25.280 --> 00:11:27.409
[? I thought. ?]
All of these things

00:11:27.409 --> 00:11:29.950
just jump out when you just try
to do something fairly simple

00:11:29.950 --> 00:11:32.180
with a triangle.


Medians divide into smaller triangles of equal area

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:05.350
But what I think about now is the if we're given some triangle,

00:00:05.360 --> 00:00:06.720
and here we have triangle ABC

00:00:06.730 --> 00:00:08.910
What are the medians of the triangle

00:00:08.920 --> 00:00:10.670
and how do they relate to each other

00:00:10.680 --> 00:00:12.470
and do they have some interesting properties,

00:00:12.480 --> 00:00:13.960
and you might guess that they do

00:00:13.970 --> 00:00:16.860
So the median is, if we start at one of the vertices

00:00:16.870 --> 00:00:18.610
so let's start at this one right over here

00:00:18.620 --> 00:00:21.110
And then we bisect the opposite sides

00:00:21.120 --> 00:00:23.660
So this right here would be a median

00:00:23.670 --> 00:00:26.970
And so it started there, and it bisects this side right over here

00:00:26.980 --> 00:00:29.240
So the length from B to let's call this D

00:00:29.250 --> 00:00:32.160
is equal to the length from D to C

00:00:32.170 --> 00:00:33.790
Now, let's do that with every side

00:00:33.800 --> 00:00:36.960
I can draw a median over here, like that

00:00:36.970 --> 00:00:39.310
So, let's call this point E

00:00:39.320 --> 00:00:40.840
So the length from A to E

00:00:40.850 --> 00:00:44.240
is going to be equal to the length from E to C

00:00:44.250 --> 00:00:45.860
Although it's kind of a little lopsided,

00:00:45.870 --> 00:00:47.160
but that gets it pretty close

00:00:47.170 --> 00:00:48.960
And then we're gonna draw another median

00:00:48.970 --> 00:00:50.430
And I'm not gonna prove it in this video,

00:00:50.440 --> 00:00:53.850
but all the medians, and this is another neat thing, that, you know,

00:00:53.860 --> 00:00:56.610
when you have three lines that will always intersect at one point,

00:00:56.620 --> 00:00:59.450
that all the medians do intersect at one point

00:00:59.460 --> 00:01:00.890
They all are concurrent

00:01:00.900 --> 00:01:02.990
They all hit one common point in the center

00:01:03.000 --> 00:01:05.070
So, let's draw it that way

00:01:05.080 --> 00:01:07.290
I'm not gonna prove it in this video, so this length,

00:01:07.300 --> 00:01:09.680
let's call this point right over here F

00:01:09.690 --> 00:01:11.590
So this length right over here,

00:01:11.600 --> 00:01:15.260
this length right over there is equal to that length over there

00:01:15.270 --> 00:01:18.760
And the point at which these medians intersect

00:01:18.770 --> 00:01:22.090
is called the centroid, the centroid

00:01:22.100 --> 00:01:23.360
And when you start studying physics,

00:01:23.370 --> 00:01:25.700
if you're actually and this was a uniform triangle

00:01:25.710 --> 00:01:26.790
and you were to throw it,

00:01:26.800 --> 00:01:30.010
it would rotate around the centroid right over here

00:01:30.020 --> 00:01:32.500
But we'll study it geometrically for now

00:01:32.510 --> 00:01:35.470
So let's call this ten centroid well, we've already gone up to F

00:01:35.480 --> 00:01:38.780
So let's call this centroid G right over there

00:01:38.790 --> 00:01:42.250
Now, what I wanna that by itself is neat that you have a centroid,

00:01:42.260 --> 00:01:44.330
that if you were to throw it, if it was a uniform mass,

00:01:44.340 --> 00:01:45.890
it would rotate around the centroid

00:01:45.900 --> 00:01:47.520
But what's even neater about this,

00:01:47.530 --> 00:01:49.980
is that we can see that we've divided this triangle

00:01:49.990 --> 00:01:51.820
into six smaller triangles

00:01:51.830 --> 00:01:53.010
What's really cool,

00:01:53.020 --> 00:01:56.280
even though these aren't congruent triangles necessarily,

00:01:56.290 --> 00:01:58.760
but they do have the same area

00:01:58.770 --> 00:02:01.170
And that's what we're going to prove in this video

00:02:01.180 --> 00:02:04.190
That these six triangles all have the same area

00:02:04.200 --> 00:02:07.500
Now, to start off, I'm just going to look at two

00:02:07.510 --> 00:02:09.650
I'm gonna look at different pairs of triangles

00:02:09.660 --> 00:02:12.280
So let's look at let's look at these two triangles right over here

00:02:12.290 --> 00:02:14.750
Let's look at those two triangles over here

00:02:14.760 --> 00:02:16.980
And to show that those two have the same area,

00:02:16.990 --> 00:02:20.200
we're just going to invoke a very simple principle

00:02:20.210 --> 00:02:23.880
So, imagine rotating, just those two triangles over

00:02:23.890 --> 00:02:26.610
So just rotating, just those two triangles over

00:02:26.620 --> 00:02:28.850
It would look something like this

00:02:28.860 --> 00:02:30.860
It would look something like this

00:02:30.870 --> 00:02:32.560
Let me try my best to draw it

00:02:32.570 --> 00:02:35.920
Where this would be point G I'll even try to color code it the same

00:02:35.930 --> 00:02:37.390
that is point G

00:02:37.400 --> 00:02:39.440
That is that side right over there

00:02:39.450 --> 00:02:41.230
This is point C

00:02:41.240 --> 00:02:44.640
This is point B, that is point B

00:02:44.650 --> 00:02:46.430
And then this right over here

00:02:46.440 --> 00:02:49.360
would be the second part of that median right over there

00:02:49.370 --> 00:02:51.360
That over there would be point D

00:02:51.370 --> 00:02:54.770
Now, we know and I didn't draw it I should, at nicely over here

00:02:54.780 --> 00:02:58.640
We know that this length is equal to this length right over here

00:02:58.650 --> 00:03:00.920
And these two triangles, if we're starting to think about the area,

00:03:00.930 --> 00:03:02.790
they have the same base

00:03:02.800 --> 00:03:04.250
And we know area

00:03:04.260 --> 00:03:07.430
Area is equal to ? base * height

00:03:07.440 --> 00:03:09.820
So, they definitely have the same base

00:03:09.830 --> 00:03:11.080
What about their heights?

00:03:11.090 --> 00:03:12.900
Well, they also have the same heights

00:03:12.910 --> 00:03:17.160
Both of their heights, both of their height is exactly that tall

00:03:17.170 --> 00:03:19.080
They both have the same height

00:03:19.090 --> 00:03:20.950
So both of them have the same base

00:03:20.960 --> 00:03:22.520
They both have the same height

00:03:22.530 --> 00:03:26.460
On an ob on an obtuse triangle right here, the altitude sits outside of it

00:03:26.470 --> 00:03:28.050
So that might be a little counterintuitive

00:03:28.060 --> 00:03:29.910
So if you have an obtuse triangle like that

00:03:29.920 --> 00:03:32.390
I say obtuse cause this is more than 90 degrees

00:03:32.400 --> 00:03:34.110
Your altitude, your height will actually

00:03:34.120 --> 00:03:36.310
sit outside of the triangle, but that's okay

00:03:36.320 --> 00:03:39.700
Both of these triangles have the same base, and the same height,

00:03:39.710 --> 00:03:41.310
so they must have the same area

00:03:41.320 --> 00:03:44.110
So if this one right here has area x,

00:03:44.120 --> 00:03:47.520
this one right here will have area x as well

00:03:47.530 --> 00:03:50.650
And you can use the exact same logic to say, well, look

00:03:50.660 --> 00:03:53.320
This guy and this guy, they have the same base,

00:03:53.330 --> 00:03:56.040
and they both have the same height

00:03:56.050 --> 00:03:59.620
So if this one right over here is of area y,

00:03:59.630 --> 00:04:02.490
then this one over here is also going to be of area y

00:04:02.500 --> 00:04:04.150
They will have the same area

00:04:04.160 --> 00:04:05.210
And then finally,

00:04:05.220 --> 00:04:07.810
we could do the same thing for these two characters up here

00:04:07.820 --> 00:04:09.210
They both have the same base

00:04:09.220 --> 00:04:13.800
This was BF is equal to FA, and they both have the same height

00:04:13.810 --> 00:04:15.550
We drop an altitude like that

00:04:15.560 --> 00:04:19.590
And so if we call this area, if we call this area right over here Z,

00:04:19.600 --> 00:04:21.640
you could call that area Z, as well

00:04:21.650 --> 00:04:24.190
So, so far, we've shown that we can divide this

00:04:24.200 --> 00:04:27.630
into three pairs of triangles that have the same area

00:04:27.640 --> 00:04:29.830
But we wanna now show that they all have the same area

00:04:29.840 --> 00:04:32.730
And to do that, we can invoke this same principle,

00:04:32.740 --> 00:04:35.190
but we'll do it with different sets of triangles

00:04:35.200 --> 00:04:40.490
So now, let's look at triangle let's look at triangle BAE

00:04:40.500 --> 00:04:45.750
Look at triangle BAE, triangle BAE

00:04:45.760 --> 00:04:47.950
So the area of triangle BAE,

00:04:47.960 --> 00:04:54.120
area of BAE, is going to be equal to z + z + y

00:04:54.130 --> 00:04:56.770
z + z + y

00:04:56.780 --> 00:05:03.170
And let's look at the area of triangle BEC, BEC right over there

00:05:03.180 --> 00:05:04.750
That's going to be,

00:05:04.760 --> 00:05:08.060
this triangle right over here, is going to be x + x + y

00:05:08.070 --> 00:05:16.280
So the area of BEC is going to be x + x + y, but the same principle

00:05:16.290 --> 00:05:20.070
They both have the same base, and they both have the same height

00:05:20.080 --> 00:05:21.720
We could drop an altitude like this

00:05:21.730 --> 00:05:24.690
This one is obtuse, so that the altitude sits outside of it,

00:05:24.700 --> 00:05:26.420
but they have the exact same height

00:05:26.430 --> 00:05:29.040
So these two areas need to be equal to each other

00:05:29.050 --> 00:05:32.300
So you have Z, well, let me just add that up

00:05:32.310 --> 00:05:37.900
Now you have 2z + y is going to be equal to,

00:05:37.910 --> 00:05:42.510
is going to be equal to 2x + y

00:05:42.520 --> 00:05:43.930
2x + y

00:05:43.940 --> 00:05:45.790
Subtract y from both sides

00:05:45.800 --> 00:05:47.690
You get 2z is equal to 2x

00:05:47.700 --> 00:05:49.250
Divide both sides by 2

00:05:49.260 --> 00:05:52.330
You get z is equal to x

00:05:52.340 --> 00:05:56.600
And so we could say, we could write an x here, and an x there

00:05:56.610 --> 00:05:58.800
So we know that all of these will have the same area,

00:05:58.810 --> 00:06:00.740
but we still have to worry about these y's here

00:06:00.750 --> 00:06:01.480
And to do that,

00:06:01.490 --> 00:06:04.580
we just have to kind of rotate the way that we look at it

00:06:04.590 --> 00:06:08.790
And now look at triangle ADC

00:06:08.800 --> 00:06:10.480
Let me do that in a different color

00:06:10.490 --> 00:06:14.750
Triangle ADC, which I'm highlighting right over here

00:06:14.760 --> 00:06:22.590
Triangle ADC, whose area, so the area of ADC is going to be 2y + x,

00:06:22.600 --> 00:06:25.090
is equal to 2y + x

00:06:25.100 --> 00:06:29.470
And then look at triangle let's see, what color have I not used yet

00:06:29.480 --> 00:06:31.120
Let me use this green

00:06:31.130 --> 00:06:37.260
Triangle ADB, triangle ADB

00:06:37.270 --> 00:06:41.770
Triangle ADB is going to be equal to,

00:06:41.780 --> 00:06:44.190
well, you could say it's 2z + x,

00:06:44.200 --> 00:06:45.920
but we know that the z's are equal to x

00:06:45.930 --> 00:06:47.690
It's really just x + x + x

00:06:47.700 --> 00:06:50.910
It's really equal to ADB is equal to 3x

00:06:50.920 --> 00:06:52.560
And we have the same idea here

00:06:52.570 --> 00:06:57.610
ADB has this base, which is the same as ADC's base,

00:06:57.620 --> 00:06:59.420
and they both have the same height

00:06:59.430 --> 00:07:01.250
We can draw up an altitude like this,

00:07:01.260 --> 00:07:02.460
we can draw up an altitude like that

00:07:02.470 --> 00:07:03.330
They have the same height

00:07:03.340 --> 00:07:06.160
We're just invoking this principle over and over and over again

00:07:06.170 --> 00:07:08.630
So these things have to equal each other

00:07:08.640 --> 00:07:15.520
So we get 2y + x is equal to 3x, is equal to 3x

00:07:15.530 --> 00:07:17.410
Subtract x from both sides

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You get 2y is equal to 2x, or, or y is equal to x

00:07:25.150 --> 00:07:26.630
So that's a really neat result

00:07:26.640 --> 00:07:30.640
You go from each of the vertices of a triangle to the opposite side,

00:07:30.650 --> 00:07:33.390
and you bisect that opposite side, and you do that three times,

00:07:33.400 --> 00:07:34.730
you have three medians

00:07:34.740 --> 00:07:38.160
These lines are called medians, where they intersect as a centroid

00:07:38.170 --> 00:07:39.310
You know what's really cool is,

00:07:39.320 --> 00:07:44.920
it divides the triangle into six smaller triangles of equal area