French Revolution (part 2)

WEBVTT
Kind: captions
Language: en

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We left off the last video
at the end of 1789.

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The Bastille had been stormed in
July as Parisians wanted to

00:00:07.580 --> 00:00:10.340
get the weapons from the
Bastille and free a few

00:00:10.340 --> 00:00:13.150
political prisoners to, in
their minds, protect

00:00:13.150 --> 00:00:16.420
themselves from any tyranny
from Louis XVI.

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Louis XVI had reluctantly kind
of gone behind the scenes and

00:00:22.690 --> 00:00:25.100
said, OK, National Assembly, I'm
not going to get in your

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way anymore.

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Because he's seen the writing
on the walls that every time

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he's done something, it's only
led to even more extreme

00:00:32.580 --> 00:00:33.330
counteraction.

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So at the end of 1789, already
chaos has broken loose in a

00:00:38.790 --> 00:00:40.480
lot of France.

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The National Assembly, they're
in process of creating a

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constitution, which won't
fully happen until 1791.

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But they're starting to bring
things together in order to

00:00:51.240 --> 00:00:52.560
draft that constitution.

00:00:52.560 --> 00:00:56.380
But in August of 1789, they've
already done their version of

00:00:56.380 --> 00:00:57.670
the Declaration of
Independence.

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The Declaration of the Rights
of Man and the Citizen.

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So if everything was well, we
would just wait until a few

00:01:06.290 --> 00:01:08.055
years, we'd get a constitution,
and maybe we

00:01:08.055 --> 00:01:11.640
would have some type of a
constitutional monarchy.

00:01:11.640 --> 00:01:15.580
But unfortunate, especially
for Louis XVI,

00:01:15.580 --> 00:01:16.560
things weren't all well.

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As we mentioned, all of this
was propagated, all of this

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was started to begin with
because people were hungry.

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We have this fiscal crisis,
we have a famine.

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And so in October of 1789--
we're still in 1789-- October

00:01:32.686 --> 00:01:39.890
of 1789-- rumors started to
spread that Marie-Antoinette,

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the king's wife, that she was
hoarding grain at Versailles.

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So people started imagining
these big stacks of grain at

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Versailles, and this is in a
time where people couldn't get

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their bread.

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And bread was the main
staple of the diet.

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So there was actually a march
of peasant women onto

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Versailles.

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And they were armed.

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This is a depiction of the
peasant women marching on

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Versailles.

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And they went to Versailles, and
they actually were able to

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get into the building itself.

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And they demanded-- because they
were suspicious of what

00:02:10.560 --> 00:02:13.870
Louis XVI and Marie-Antoinette
were up to at Versailles--

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they demanded that they
move to Paris.

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So the women's march.

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And they were able to
get their demands.

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It resulted in Louis XVI and
wife, Marie-Antoinette, moving

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back to Paris, where
they couldn't do

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things like hoard grain.

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And they'll be surrounded
by all of the maybe

00:02:45.380 --> 00:02:47.800
not-so-friendly people
who could watch

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what they're doing.

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I think the main factor was that
people are hungry, rumors

00:02:54.740 --> 00:02:57.000
are spreading that the king
is hoarding grain.

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But there were also rumors that
the king was being very

00:02:59.780 --> 00:03:06.220
disrespectful to some of the
symbols of the new France, of

00:03:06.220 --> 00:03:07.330
the new National Assembly.

00:03:07.330 --> 00:03:09.390
So that also made
people angry.

00:03:09.390 --> 00:03:12.480
And across the board everyone
kind of knew, and including

00:03:12.480 --> 00:03:15.760
Louis XVI, that he wasn't really
into what was going on.

00:03:15.760 --> 00:03:18.970
He wasn't into this kind of
constitutional monarchy that

00:03:18.970 --> 00:03:21.740
was forming, this power
that was being lost to

00:03:21.740 --> 00:03:22.630
the National Assembly.

00:03:22.630 --> 00:03:25.080
So we have this very
uncomfortable situation

00:03:25.080 --> 00:03:31.420
entering into 1790, where the
king and queen are essentially

00:03:31.420 --> 00:03:36.680
in house arrest in a building
called the Tuileries in Paris.

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You have this National Assembly
drafting this

00:03:39.610 --> 00:03:41.320
constitution.

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They're charted to draft the
constitution up there.

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They all pledged at the
Tennis Court Oath.

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And at the same time, throughout
France, you have

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some counter insurgencies.

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This is France right here.

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Throughout France you have
counter insurgencies, people

00:03:54.030 --> 00:03:57.230
who don't like the Revolution
that's going on.

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And then those would
be subdued.

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And people are all plotting
one against each other.

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And then you have some
nobility, that says,

00:04:03.630 --> 00:04:04.330
gee, you know what?

00:04:04.330 --> 00:04:06.000
I don't like the way
that this is going.

00:04:06.000 --> 00:04:08.120
We've seen already a
lot of violence.

00:04:08.120 --> 00:04:09.260
People are angry.

00:04:09.260 --> 00:04:11.910
I'm just going to take my money
and whatever I can pack,

00:04:11.910 --> 00:04:14.060
and I'm just going to get
out of the country.

00:04:14.060 --> 00:04:16.450
I'm going to emigrate away
from the country.

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So you start having nobility
starting to leave France.

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They're called the Emigres.

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I know I'm not pronouncing
it correctly.

00:04:23.760 --> 00:04:26.120
But you see, you have this
notion of gee, I had it good

00:04:26.120 --> 00:04:28.620
in France, I'm not going to have
it good much longer, I'd

00:04:28.620 --> 00:04:34.770
better leave. And this same
idea, now that we get to 1791.

00:04:34.770 --> 00:04:37.100
So 1790 was just kind of
a bunch of unease.

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Now that we're at 1791, the
same idea of trying to get

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away from the danger got into
the heads of Louis XVI and

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Marie-Antoinette.

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But they couldn't leave
the country.

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They didn't trust
Great Britain.

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They didn't trust any of these
other countries to safely give

00:04:56.160 --> 00:04:57.340
them shelter.

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So one of their generals, who
was sympathetic to their

00:05:00.070 --> 00:05:02.690
cause, said, hey, at least
come here to the frontier

00:05:02.690 --> 00:05:04.770
areas and you could hide
from all of the unrest

00:05:04.770 --> 00:05:05.820
that's going on.

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So dressed as actual servants--
and it shows you

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what type of people they were--

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they dressed as servants.

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And they actually made their
servants dress as nobility to

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make them the targets in case
they were ambushed anyway on

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their way trying to
escape from Paris.

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Dressed as servants, the king
and queen-- the king tried to

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escape to this general's
estate.

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But when they were in Varennes,
right here, they

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were actually spotted.

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And then the people essentially
took them captive

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and brought them
back to Paris.

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So this is called, or you could
imagine this is the

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flight to Varennes, or the
flight away from Paris, or

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however you want to do it.

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So already, Louis XVI
started to see the

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writing on the wall.

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They tried to get away.

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But people brought them back.

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Now you can imagine, a lot
of people already did

00:06:04.560 --> 00:06:05.380
not like the king.

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They didn't like the notion
of even having a king.

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And the most revolutionary,
the most radical elements,

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were called the Jacobins.

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And after the king and queen
tried to escape and came back,

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they were like hey, gee,
what's the use of

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even having a king?

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You National Assembly, why are
you even trying to write some

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type of constitution that
gives any power

00:06:27.860 --> 00:06:29.520
whatsoever to a king?

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We should have a republic.

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Which is essentially-- there's
a lot of kind of nuanced

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definitions of what a republic
is, but the most simple one is

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it's a state without
a king, without an

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emperor, without a queen.

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So they're saying, we don't
need, you know-- you National

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Assembly, you think you're
being radical.

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But you're not being
radical enough.

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We want to eliminate the idea
of having a monarchy

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altogether.

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And the fact that Louis XVI
actually tried to run away, we

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view that as him abdicating
the throne.

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Abdication, or essentially
quitting.

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And they actually started
to organize in Paris.

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This right here is the
Champ-de-Mars.

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I know I'm saying it
completely wrong.

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This is a current
picture of it.

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And so they started taking
signatures in this kind of

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public park in Paris to
essentially say, we

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don't need a king.

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We want to essentially create
our own republic.

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That this National Assembly,
they're not radical enough.

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And so people started gathering
over here in the

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Champ-de-Mars and things
got a little ugly.

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So the actual troops were
sent in to kind of

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calm everyone down.

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And these were actually
troops controlled by

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the National Assembly.

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The people who are
mainly controlled

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by the Third Estate.

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But things got a little crazy.

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Rocks were thrown at
some of the troops.

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Some of the troops, at
first, they started

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firing in the air.

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But eventually when things
got really crazy, they

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fired into the crowd.

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And about 50 people died.

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And this was the massacre.

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Or the Champ-de-Mars Massacre.

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I know I'm saying it wrong.

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This isn't a video on French
pronunciation.

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But you could imagine, now
people are even angrier.

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People are still starving.

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That problem has
not gone away.

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The king and queen has been
kind of very reluctantly--

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everyone is suspicious of the
fact that they're probably

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going to try to come
back to power.

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They tried to run away.

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When the Jacobins, or
in general kind of

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revolutionaries, but they're led
by the Jacobins, when they

00:08:43.150 --> 00:08:46.340
start to suggest that, hey,
we should have a republic.

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We shouldn't even have a king.

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And they gather people here,
all of a sudden, the troops

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that are controlled by the
current National Assembly

00:08:52.510 --> 00:08:55.650
actually fire on the crowd, and
actually kill civilians

00:08:55.650 --> 00:08:56.510
for throwing rocks.

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And they might have
been big rocks.

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But you can imagine, this is
going to anger already hungry

00:09:01.400 --> 00:09:04.220
and already suppressed
people even more.

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And to make people even more
paranoid that the king and

00:09:07.940 --> 00:09:10.820
queen might eventually come
back to power, you had two

00:09:10.820 --> 00:09:14.110
major powers all of a sudden
trying to insert certain

00:09:14.110 --> 00:09:16.530
themselves into the
French Revolution.

00:09:16.530 --> 00:09:18.000
I'm going to do a little
bit of an aside here.

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Because this is something, at
least you when I first learned

00:09:20.810 --> 00:09:23.370
European history, I found
the most confusing.

00:09:23.370 --> 00:09:27.120
You have these states,
you can call them.

00:09:27.120 --> 00:09:31.820
You have Austria, which I've
highlighted in orange.

00:09:31.820 --> 00:09:34.700
The kind of map here
is a modern map.

00:09:34.700 --> 00:09:39.370
But in orange, I've kind of
shown what Austria was at that

00:09:39.370 --> 00:09:40.080
point in time.

00:09:40.080 --> 00:09:43.630
Around 1789, 1790, 1791.

00:09:43.630 --> 00:09:46.990
In this red color,
I have Prussia.

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I want to show you that these
are very different than our

00:09:48.860 --> 00:09:50.470
current notions of
one, Austria.

00:09:50.470 --> 00:09:53.090
Austria today is this modern
country right here.

00:09:53.090 --> 00:09:56.240
And Prussia doesn't even exist
as a modern country.

00:09:56.240 --> 00:09:59.120
And then you had this notion
of the Holy Roman Empire,

00:09:59.120 --> 00:10:04.020
which overlaps with these other
kingdoms, or empires, or

00:10:04.020 --> 00:10:05.110
whatever you want
to call them.

00:10:05.110 --> 00:10:07.730
And I want to do a little
bit of an aside here.

00:10:07.730 --> 00:10:11.240
The Holy Roman Empire, as
Voltaire famously said, is

00:10:11.240 --> 00:10:27.060
neither holy nor Roman
nor an empire.

00:10:27.060 --> 00:10:28.780
And he was right.

00:10:28.780 --> 00:10:32.710
It was really kind of a very
loose confederation of German

00:10:32.710 --> 00:10:35.090
kingdoms and states-- mainly
German kingdoms and states.

00:10:35.090 --> 00:10:38.300
As you can see, it kind of
coincides with modern Germany.

00:10:38.300 --> 00:10:41.770
And the two most influential
powers in the Holy Roman

00:10:41.770 --> 00:10:47.310
Empire, or actually the most
influential power in the Holy

00:10:47.310 --> 00:10:49.260
Roman Empire, was
the Austrians.

00:10:49.260 --> 00:10:52.080
And the ruler of the Austrians
had the title

00:10:52.080 --> 00:10:53.690
of Holy Roman Emperor.

00:10:53.690 --> 00:10:54.940
And that was Leopold II.

00:11:07.310 --> 00:11:11.430
But it's not like he was like
the Roman Emperors of old.

00:11:11.430 --> 00:11:13.310
The Roman Emperors of old
actually came out of Rome.

00:11:13.310 --> 00:11:17.880
Notice, nothing in the Holy
Roman Empire at that time, it

00:11:17.880 --> 00:11:18.890
had no control of Rome.

00:11:18.890 --> 00:11:20.870
So it was not Roman, we're
not talking about

00:11:20.870 --> 00:11:22.700
people who spoke Latin.

00:11:22.700 --> 00:11:25.960
We're talking about people who
spoke Germanic languages.

00:11:25.960 --> 00:11:27.340
And it wasn't an empire.

00:11:27.340 --> 00:11:32.040
That it wasn't a tightly knit
kind of governance structure.

00:11:32.040 --> 00:11:34.560
It was this loose confederation
of states.

00:11:34.560 --> 00:11:40.400
But what was the most
influential was the region

00:11:40.400 --> 00:11:43.640
that was under control of the
Habsburgs of Austria, or

00:11:43.640 --> 00:11:45.660
Leopold II.

00:11:45.660 --> 00:11:48.180
And not only was he in control,
or not only did he

00:11:48.180 --> 00:11:50.470
have the title of Holy Roman
Empire, and essentially had

00:11:50.470 --> 00:11:55.420
control of the Austrian, I guess
you could say Empire, at

00:11:55.420 --> 00:11:56.300
that point in time.

00:11:56.300 --> 00:11:58.390
He was also Marie-Antoinette's
brother.

00:12:01.570 --> 00:12:04.340
Leopold II, that's
her brother.

00:12:04.340 --> 00:12:10.780
So Leopold II and Frederick
William II of Prussia, which

00:12:10.780 --> 00:12:13.330
is another mainly
Germanic state.

00:12:13.330 --> 00:12:14.650
Let me do that in
a better color.

00:12:23.350 --> 00:12:26.950
They issued the Declaration
of Pillnitz.

00:12:26.950 --> 00:12:27.940
Let me write this down.

00:12:27.940 --> 00:12:31.240
So this is going to add even
more insult to injury to just

00:12:31.240 --> 00:12:33.670
the general population
of France.

00:12:33.670 --> 00:12:35.090
The Declaration of Pillnitz.

00:12:47.170 --> 00:12:50.890
And this was done in August. so
I just want to make it very

00:12:50.890 --> 00:12:51.910
clear what happened.

00:12:51.910 --> 00:12:55.220
In June of 1791, they tried
to escape, they

00:12:55.220 --> 00:12:56.700
were captured at Varennes.

00:12:56.700 --> 00:13:01.400
Then in July of 1791, you have
the Champ-de-Mars Massacre.

00:13:01.400 --> 00:13:04.270
So already, people are very
wary of the royals.

00:13:04.270 --> 00:13:06.810
The idea that we don't need
them is spreading.

00:13:06.810 --> 00:13:08.050
And people are getting
angrier.

00:13:08.050 --> 00:13:09.960
And then you have the
Declaration of Pillnitz by

00:13:09.960 --> 00:13:12.580
these foreign powers, one of
whom is essentially the

00:13:12.580 --> 00:13:15.290
brother of the current
French royalty.

00:13:15.290 --> 00:13:18.400
And that declaration is
essentially saying that they

00:13:18.400 --> 00:13:25.090
intend to bring the French
monarchy back to power.

00:13:25.090 --> 00:13:26.730
They don't say that they're
definitely going to do it in

00:13:26.730 --> 00:13:27.990
military terms or whatever.

00:13:27.990 --> 00:13:30.950
But it's a declaration that they
do not approve of what's

00:13:30.950 --> 00:13:31.880
going on in France.

00:13:31.880 --> 00:13:34.140
And even though they themselves
might have not

00:13:34.140 --> 00:13:36.920
taken it too seriously, the
people of France took it

00:13:36.920 --> 00:13:37.370
really seriously.

00:13:37.370 --> 00:13:42.140
You have these huge powers on
their border right here.

00:13:42.140 --> 00:13:48.670
You had the Austrians
and the Prussians.

00:13:48.670 --> 00:13:51.200
So this wasn't anything
that people

00:13:51.200 --> 00:13:52.920
could take very lightly.

00:13:52.920 --> 00:13:56.330
So it only increased the fear
that the royals were going to

00:13:56.330 --> 00:13:59.040
do something to come back
to full power and

00:13:59.040 --> 00:14:00.250
really suppress people.

00:14:00.250 --> 00:14:04.000
And it really gave even more
fuel for the Jacobins to kind

00:14:04.000 --> 00:14:07.530
of argue for some type
of a republic.

00:14:07.530 --> 00:14:09.080
So I'm going to leave you
there in this video.

00:14:09.080 --> 00:14:12.710
As you can see, we saw in the
first video, things got bad.

00:14:12.710 --> 00:14:14.960
Now they're getting really
worst. Chaos is

00:14:14.960 --> 00:14:15.740
breaking out in France.

00:14:15.740 --> 00:14:17.470
People are questioning
whether they even

00:14:17.470 --> 00:14:18.970
need a king or queen.

00:14:18.970 --> 00:14:21.520
Foreign powers are getting
involved, saying hey, they

00:14:21.520 --> 00:14:23.880
don't like what they're seeing
there, with kings and queens

00:14:23.880 --> 00:14:24.670
getting overthrown.

00:14:24.670 --> 00:14:26.690
Maybe that'll give ideas
to their people.

00:14:26.690 --> 00:14:28.630
And by the way, I'm your
brother, so I want to

00:14:28.630 --> 00:14:29.470
help you out too.

00:14:29.470 --> 00:14:31.080
That scares people even more.

00:14:31.080 --> 00:14:33.560
The current National Assembly,
which is kind of the beginning

00:14:33.560 --> 00:14:36.820
of the Revolution, they
themselves are on some level

00:14:36.820 --> 00:14:38.310
massacring people.

00:14:38.310 --> 00:14:42.850
So it's really leading to a
really tense and ugly time in

00:14:42.850 --> 00:14:43.480
French history.

00:14:43.480 --> 00:14:45.540
And you're going to see that
that's going to culminate with

00:14:45.540 --> 00:14:47.220
what's called the
Reign of Terror.

00:14:47.220 --> 00:14:50.110
And we're going to see that
in the next video.


French Revolution (part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:05.340
In this video we're going
to talk about the French

00:00:05.340 --> 00:00:06.750
Revolution.

00:00:06.750 --> 00:00:10.370
And what makes this especially
significant is that not only

00:00:10.370 --> 00:00:15.650
is this independence from a
monarchy-controlled empire,

00:00:15.650 --> 00:00:18.130
like in the American
independence, this is an

00:00:18.130 --> 00:00:21.230
actual overthrowing
of a monarchy.

00:00:21.230 --> 00:00:26.350
A monarchy that controls
a major world power.

00:00:26.350 --> 00:00:28.540
Depending on how you view it,
the American Revolution came

00:00:28.540 --> 00:00:32.770
first and kind of put out the
principles of self-governance

00:00:32.770 --> 00:00:34.730
and why do we need kings
and all of that.

00:00:34.730 --> 00:00:37.520
But the French Revolution was
the first time that those type

00:00:37.520 --> 00:00:41.160
of principles really took foot
in Europe and really overthrew

00:00:41.160 --> 00:00:42.050
a monarchy.

00:00:42.050 --> 00:00:45.600
So just to understand kind of
the environment in which this

00:00:45.600 --> 00:00:53.190
began, let's talk about what
France was like in 1789.

00:00:53.190 --> 00:00:56.260
Which most people kind of view
as the beginning of the

00:00:56.260 --> 00:00:57.190
Revolution.

00:00:57.190 --> 00:00:58.825
One, France was poor.

00:01:03.870 --> 00:01:06.210
Now, you wouldn't think that
France was poor, if you looked

00:01:06.210 --> 00:01:09.860
at Louis XVI, who was
king of France.

00:01:09.860 --> 00:01:12.400
If you looked at Louis XVI,
and the clothes he wore.

00:01:12.400 --> 00:01:19.400
If you looked at
Marie-Antoinette, his wife,

00:01:19.400 --> 00:01:20.590
they don't look poor.

00:01:20.590 --> 00:01:22.050
They lived in the palace of

00:01:22.050 --> 00:01:25.410
Versailles, which is ginormous.

00:01:25.410 --> 00:01:29.450
It's this massive palace, it
would compare to the greatest

00:01:29.450 --> 00:01:31.090
palaces in the world.

00:01:31.090 --> 00:01:32.430
They were living a
lavish lifestyle.

00:01:32.430 --> 00:01:34.360
Just in case you want to know
where this is, this is what's

00:01:34.360 --> 00:01:35.890
now almost a suburb of Paris.

00:01:35.890 --> 00:01:39.720
But at the time it was a village
20 or 30 kilometers

00:01:39.720 --> 00:01:40.790
away from Paris.

00:01:40.790 --> 00:01:42.630
So they don't seem to be poor.

00:01:42.630 --> 00:01:46.360
But the the actual government
of France is poor.

00:01:46.360 --> 00:01:48.190
And when I say poor,
they're in debt.

00:01:48.190 --> 00:01:51.420
They've just had two major
military adventures.

00:01:51.420 --> 00:01:52.775
One was the American
Revolution.

00:01:59.720 --> 00:02:01.880
They played a major part
in supporting the

00:02:01.880 --> 00:02:02.670
revolutionaries.

00:02:02.670 --> 00:02:05.070
Because they wanted to
stick it to their

00:02:05.070 --> 00:02:06.300
enemy, Great Britain.

00:02:06.300 --> 00:02:09.800
They wanted their empire
to shrink a little bit.

00:02:09.800 --> 00:02:14.250
So France sent significant
military help and resources.

00:02:14.250 --> 00:02:15.910
And you could imagine, that's
not a cheap thing when you're

00:02:15.910 --> 00:02:18.200
doing it across the
Atlantic Ocean.

00:02:18.200 --> 00:02:21.300
And even before the American
Revolution, the Seven Years'

00:02:21.300 --> 00:02:30.620
War that ended in 1763, this
really drained the amount of

00:02:30.620 --> 00:02:32.430
wealth that the French
government had.

00:02:32.430 --> 00:02:36.190
And for those of you who are
more American history focused,

00:02:36.190 --> 00:02:39.080
the Seven Years' War is really
the same thing as the French

00:02:39.080 --> 00:02:39.660
and Indian War.

00:02:39.660 --> 00:02:43.220
The French and Indian War was
the North American theater of

00:02:43.220 --> 00:02:44.140
the Seven Years' War.

00:02:44.140 --> 00:02:45.980
But the Seven Years' War is
the more general term.

00:02:45.980 --> 00:02:49.160
Because there was also a
conflict going on in Europe

00:02:49.160 --> 00:02:49.970
simultaneously.

00:02:49.970 --> 00:02:51.710
The French and Indian
War and it was just

00:02:51.710 --> 00:02:52.510
part of that conflict.

00:02:52.510 --> 00:02:56.090
And the Seven Years' actually
engulfed most of the powers of

00:02:56.090 --> 00:02:56.880
Europe at the time.

00:02:56.880 --> 00:03:00.440
So France had participated in
this, ended in 1763, you had

00:03:00.440 --> 00:03:01.770
the American Revolution.

00:03:01.770 --> 00:03:04.320
Both of these really just
drained the amount of funds

00:03:04.320 --> 00:03:05.730
that the government
itself had.

00:03:05.730 --> 00:03:07.890
At the same time, the French
people were starving.

00:03:12.950 --> 00:03:15.350
There was a generalized
famine at the time.

00:03:15.350 --> 00:03:17.650
They weren't producing enough
grain, people couldn't get

00:03:17.650 --> 00:03:18.690
their bread to eat.

00:03:18.690 --> 00:03:20.040
So you can imagine,
when people are

00:03:20.040 --> 00:03:21.650
starving they're not happy.

00:03:21.650 --> 00:03:24.970
And to kind of add insult to
injury, you would see your

00:03:24.970 --> 00:03:26.350
royals living like this.

00:03:26.350 --> 00:03:28.130
But even worse than the royals,
who you don't see

00:03:28.130 --> 00:03:29.610
every day, you saw
your nobility.

00:03:29.610 --> 00:03:33.990
Who is roughly a little over
1.5% of the population.

00:03:33.990 --> 00:03:36.500
But you saw the nobility really,
really, living it up.

00:03:45.720 --> 00:03:47.840
And the nobility, just so you
know, these are people with

00:03:47.840 --> 00:03:51.110
fancy titles who inherit land
and wealth from generation to

00:03:51.110 --> 00:03:52.120
generation.

00:03:52.120 --> 00:03:54.840
They don't dress too differently
from the king.

00:03:54.840 --> 00:03:57.950
And they essentially live in
smaller versions of the palace

00:03:57.950 --> 00:03:58.780
of Versailles.

00:03:58.780 --> 00:04:02.570
And if you're a peasant, you
work on their fields, do all

00:04:02.570 --> 00:04:04.540
the work, you send them
some of your crops

00:04:04.540 --> 00:04:06.590
and they pay no taxes.

00:04:06.590 --> 00:04:09.020
So from your point of view,
and it's not hard to

00:04:09.020 --> 00:04:11.040
understand why you would think
this, these are essentially

00:04:11.040 --> 00:04:14.660
kind of parasites who are
completely ignoring the fact

00:04:14.660 --> 00:04:16.670
that you are starving
and you're

00:04:16.670 --> 00:04:17.779
paying all of the taxes.

00:04:17.779 --> 00:04:20.550
You can imagine people weren't
too happy about that.

00:04:20.550 --> 00:04:22.680
And then to top it all off,
you had all of these

00:04:22.680 --> 00:04:25.450
philosophers hanging around
talking about the

00:04:25.450 --> 00:04:26.700
Enlightenment.

00:04:30.520 --> 00:04:33.140
And this is kind of the whole
movement where people, and

00:04:33.140 --> 00:04:38.340
authors, and poets, and
philosophers, are starting to

00:04:38.340 --> 00:04:41.150
realize that, gee, maybe
we don't need kings.

00:04:41.150 --> 00:04:44.980
Maybe we don't need priests to
tell us what it means to be

00:04:44.980 --> 00:04:45.560
good or bad.

00:04:45.560 --> 00:04:49.420
Maybe people could
essentially rule

00:04:49.420 --> 00:04:50.830
themselves all of a sudden.

00:04:50.830 --> 00:04:54.620
And obviously, the biggest proof
of the Enlightenment was

00:04:54.620 --> 00:04:56.520
the American Revolution.

00:04:56.520 --> 00:04:59.980
That was kind of the first
example of people rising up

00:04:59.980 --> 00:05:01.960
and saying, we don't need
these kings anymore.

00:05:01.960 --> 00:05:03.460
We want to govern ourselves.

00:05:03.460 --> 00:05:05.080
For the people, by the people.

00:05:05.080 --> 00:05:06.750
So you also had kind of
this philosophical

00:05:06.750 --> 00:05:07.820
movement going around.

00:05:07.820 --> 00:05:10.480
Now if you ask me my opinion of
what the biggest thing was,

00:05:10.480 --> 00:05:12.810
I think the people starving,
you can never underestimate

00:05:12.810 --> 00:05:16.960
what people are willing to do
when they're actually hungry.

00:05:16.960 --> 00:05:18.440
And, this is kind
of more from the

00:05:18.440 --> 00:05:19.390
intellectual point of view.

00:05:19.390 --> 00:05:21.730
People said, oh there's
Enlightenment movement here.

00:05:21.730 --> 00:05:23.270
So this is the state
of France.

00:05:23.270 --> 00:05:24.980
They had a financial crisis.

00:05:24.980 --> 00:05:28.860
So a meeting was called, kind
of an emergency meeting, of

00:05:28.860 --> 00:05:33.560
the major groups of France to
try to resolve some of these

00:05:33.560 --> 00:05:37.340
problems. It's a fiscal crisis,
people are starving,

00:05:37.340 --> 00:05:38.240
what do you do?

00:05:38.240 --> 00:05:41.725
So they called the Convocation
of the Estates-General.

00:05:41.725 --> 00:05:43.060
Let me write that down.

00:05:54.310 --> 00:05:58.880
Which was a meeting of the
three estates of France.

00:05:58.880 --> 00:06:00.440
Now what are the three
estates of France?

00:06:00.440 --> 00:06:03.210
You can really just view them
as the three major social

00:06:03.210 --> 00:06:05.120
classes of France.

00:06:05.120 --> 00:06:09.690
The First Estate
was the clergy.

00:06:09.690 --> 00:06:16.790
The Second Estate
is the nobility.

00:06:16.790 --> 00:06:19.480
And then the Third Estate
is everyone else.

00:06:26.870 --> 00:06:28.910
And this gives you a sense
of how skewed the

00:06:28.910 --> 00:06:29.750
power structure was.

00:06:29.750 --> 00:06:32.550
Because people kind of grouped
the power as OK, these are the

00:06:32.550 --> 00:06:34.520
three groups and maybe they can
vote against each other.

00:06:34.520 --> 00:06:39.570
But this was only 0.5% of the
population, this is 1.5% of

00:06:39.570 --> 00:06:45.240
the population, this was
98% of the population.

00:06:45.240 --> 00:06:48.330
But these people had equal
weight with these guys.

00:06:48.330 --> 00:06:51.890
But these people had the burden
of most of the taxes.

00:06:51.890 --> 00:06:54.050
These are the people who are
doing all the work, producing

00:06:54.050 --> 00:06:56.650
all of France's wealth,
dying in the wars.

00:06:56.650 --> 00:07:00.570
But these guys, despite their
small population, have more

00:07:00.570 --> 00:07:03.350
weight than everybody else.

00:07:03.350 --> 00:07:05.760
So you had the Convocation of
the Estates-General, where

00:07:05.760 --> 00:07:10.170
representatives of these three
estates met at the Palace of

00:07:10.170 --> 00:07:13.940
Versailles to essentially figure
out what to do about

00:07:13.940 --> 00:07:14.870
this fiscal crisis.

00:07:14.870 --> 00:07:18.160
Now obviously, these people
right here, the Third Estate,

00:07:18.160 --> 00:07:18.970
they were angry.

00:07:18.970 --> 00:07:21.910
They were like look, we've taken
the burden on ourselves

00:07:21.910 --> 00:07:24.900
for much of the recent
history of France.

00:07:24.900 --> 00:07:27.530
We're tired of you guys getting
away with not paying

00:07:27.530 --> 00:07:30.230
taxes and just kind of
leeching off of us.

00:07:30.230 --> 00:07:33.230
They were afraid that even more
of the tax burden was

00:07:33.230 --> 00:07:34.050
going to be put on them.

00:07:34.050 --> 00:07:37.970
And the nobility, or the king,
or the clergy, that they

00:07:37.970 --> 00:07:39.240
wouldn't have to make
sacrifices.

00:07:39.240 --> 00:07:42.280
So they came in already angry.

00:07:42.280 --> 00:07:44.400
And so they really wanted to
meet in one big room together.

00:07:44.400 --> 00:07:47.460
Because they actually had
roughly 600 representatives.

00:07:47.460 --> 00:07:49.460
Which only the king at the
last minute agreed to.

00:07:49.460 --> 00:07:51.770
Before, it was only going to
be equal numbers of them.

00:07:51.770 --> 00:07:54.090
These guys had 300 roughly.

00:07:54.090 --> 00:07:55.550
These guys had 300 as well.

00:07:55.550 --> 00:07:57.580
These guys were able to say,
hey we're 98% of the

00:07:57.580 --> 00:07:59.740
population, maybe we should
have at least 600

00:07:59.740 --> 00:08:00.660
representative.

00:08:00.660 --> 00:08:03.050
But even there, they wanted
to meet in the same room.

00:08:03.050 --> 00:08:04.930
And essentially try to
make it so it's one

00:08:04.930 --> 00:08:06.410
representative, one vote.

00:08:06.410 --> 00:08:09.610
But obviously these other
estates, the clergy and the

00:08:09.610 --> 00:08:11.140
nobility, said no, let's
each vote as estates.

00:08:11.140 --> 00:08:14.200
And at the end of the day, these
guys lost. So they were

00:08:14.200 --> 00:08:19.440
essentially forced to kind of
organize independently as a

00:08:19.440 --> 00:08:19.850
Third Estate.

00:08:19.850 --> 00:08:21.020
So that made them
even angrier.

00:08:21.020 --> 00:08:23.960
So they met at an assembly hall
and said, if these guys

00:08:23.960 --> 00:08:27.210
are going to ignore us, not
only are we going to be in

00:08:27.210 --> 00:08:29.720
this room and start organizing
ourselves.

00:08:29.720 --> 00:08:31.990
But we're not going to call this
the Convocation of the

00:08:31.990 --> 00:08:33.340
Estates-General.

00:08:33.340 --> 00:08:35.289
We're going to declare
that we are the

00:08:35.289 --> 00:08:36.724
National Assembly of France.

00:08:45.390 --> 00:08:46.950
That we represent the people.

00:08:46.950 --> 00:08:48.600
We are essentially going
to become the

00:08:48.600 --> 00:08:50.680
parliamentary body of France.

00:08:50.680 --> 00:08:53.940
Instead of just being this
emergency Convocation of the

00:08:53.940 --> 00:08:54.500
Estates-General.

00:08:54.500 --> 00:08:56.750
And they actually got some
sympathy from some elements of

00:08:56.750 --> 00:08:58.970
the clergy and some elements
of the nobility.

00:08:58.970 --> 00:09:04.590
Now obviously, Louis XVI
was not amused by this

00:09:04.590 --> 00:09:05.620
whole turn of events.

00:09:05.620 --> 00:09:08.180
Here he was, he was an absolute
monarch, which means

00:09:08.180 --> 00:09:13.230
that he held pretty much all of
the power to do whatever he

00:09:13.230 --> 00:09:14.130
saw was fit.

00:09:14.130 --> 00:09:17.150
And all of a sudden you had this
group of upstarts taking

00:09:17.150 --> 00:09:19.910
advantage of this emergency
situation where he can't

00:09:19.910 --> 00:09:23.090
continue to buy as many silk
robes as he was before.

00:09:23.090 --> 00:09:24.840
They're taking advantage of
the situation to declare a

00:09:24.840 --> 00:09:26.600
National Assembly of France.

00:09:26.600 --> 00:09:29.880
To declare somehow that I'm
not an absolute monarch.

00:09:29.880 --> 00:09:32.580
That my power is going to be
taken by this assembly.

00:09:32.580 --> 00:09:33.310
So he wasn't happy.

00:09:33.310 --> 00:09:36.180
So when they took a break,
he locked the door of the

00:09:36.180 --> 00:09:36.820
assembly room.

00:09:36.820 --> 00:09:37.520
So they couldn't get in.

00:09:37.520 --> 00:09:39.690
And he said, oh I think
there needs to be some

00:09:39.690 --> 00:09:41.350
repairs in that room.

00:09:41.350 --> 00:09:42.660
Maybe you all can
assemble later.

00:09:42.660 --> 00:09:45.730
And that was kind of his
way of saying no.

00:09:45.730 --> 00:09:48.050
If you're declaring you're the
National Assembly of France,

00:09:48.050 --> 00:09:49.230
I'm not going to even
let you assemble.

00:09:49.230 --> 00:09:51.510
I'm not even going to let
you get in the room.

00:09:51.510 --> 00:09:56.170
So that clearly didn't do a lot
to make these guys, or in

00:09:56.170 --> 00:09:58.510
particular these guys,
any happier.

00:09:58.510 --> 00:09:59.640
People are hungry.

00:09:59.640 --> 00:10:01.950
These people are living
lavishly.

00:10:01.950 --> 00:10:03.900
They've already been
not allowed to

00:10:03.900 --> 00:10:06.310
vote in one room together.

00:10:06.310 --> 00:10:08.900
When they vote in their own
room, and declare themselves

00:10:08.900 --> 00:10:10.700
as representatives of the people
of France, which they

00:10:10.700 --> 00:10:14.350
really are, the king locks the
room, doesn't let them go in.

00:10:14.350 --> 00:10:18.030
So they go to an indoor tennis
court in Versailles.

00:10:18.030 --> 00:10:19.220
This is a picture of
it right here.

00:10:19.220 --> 00:10:20.680
This is an indoor
tennis court.

00:10:24.150 --> 00:10:27.310
And that gives you an idea of
how lavish Versailles was,

00:10:27.310 --> 00:10:32.050
that it had an indoor tennis
court in the late 1700s.

00:10:32.050 --> 00:10:34.770
And they proclaimed the
Tennis Court Oath.

00:10:41.790 --> 00:10:45.960
Where they proclaimed, not are
we only the National Assembly

00:10:45.960 --> 00:10:49.430
of France, but even more than
that, we all pledge to not

00:10:49.430 --> 00:10:52.830
stop until we create a
constitution of France.

00:10:56.650 --> 00:10:58.570
So they went from being
a National Assembly to

00:10:58.570 --> 00:11:00.820
essentially morphing into
a constituent assembly.

00:11:00.820 --> 00:11:02.070
We're going to create
a constitution.

00:11:15.590 --> 00:11:19.230
And they had sympathy from some
elements of the clergy

00:11:19.230 --> 00:11:20.340
and the nobility.

00:11:20.340 --> 00:11:22.510
So eventually Louis XVI,
he kind of saw the

00:11:22.510 --> 00:11:24.380
writing on the wall.

00:11:24.380 --> 00:11:25.690
The people are angry.

00:11:25.690 --> 00:11:27.550
And every time he tries
to mess with them,

00:11:27.550 --> 00:11:28.770
they only get angrier.

00:11:28.770 --> 00:11:30.920
And they only go to even
more extreme measures.

00:11:30.920 --> 00:11:34.355
So just to kind of make it seem
like he's going along, he

00:11:34.355 --> 00:11:35.900
says, OK that's cool, guys.

00:11:35.900 --> 00:11:36.620
Whatever you all want to do.

00:11:36.620 --> 00:11:38.850
Yeah, maybe I'm open to it,
we are in an emergency.

00:11:38.850 --> 00:11:40.620
And maybe it is unreasonable,
I have been a little bit

00:11:40.620 --> 00:11:41.090
unreasonable.

00:11:41.090 --> 00:11:43.760
So he lets them be, he lets
them assemble again.

00:11:43.760 --> 00:11:49.070
But while that's happening,
people start to notice that

00:11:49.070 --> 00:11:50.980
troops are converging
on Paris.

00:12:04.120 --> 00:12:07.260
And they're obviously being
sent there by the king.

00:12:07.260 --> 00:12:09.570
And not only are they just any
troops, a lot of the actual

00:12:09.570 --> 00:12:12.870
troops, even though they are
French troops, there under the

00:12:12.870 --> 00:12:15.330
authority of France's
military.

00:12:15.330 --> 00:12:18.260
They're actually
foreign troops.

00:12:18.260 --> 00:12:21.050
So, if you think about it, these
would be the ideal types

00:12:21.050 --> 00:12:24.590
of troops to put down any
type of insurgency,

00:12:24.590 --> 00:12:25.720
or any type of rebellion.

00:12:25.720 --> 00:12:28.430
Or even better, to go
in and dissolve

00:12:28.430 --> 00:12:29.230
the National Assembly.

00:12:29.230 --> 00:12:30.760
So people start getting
a little bit

00:12:30.760 --> 00:12:32.290
paranoid, you can imagine.

00:12:32.290 --> 00:12:38.050
Now on top of that, Louis XVI's
main financial adviser,

00:12:38.050 --> 00:12:43.340
Necker, Jacques Necker.

00:12:43.340 --> 00:12:45.680
He was sympathetic to the Third
Estate, to the plight of

00:12:45.680 --> 00:12:48.275
the Third Estate.

00:12:48.275 --> 00:12:55.770
And he said hey, Mr. King, I
think it's reasonable for you

00:12:55.770 --> 00:12:58.170
to essentially budget your
expenses a little bit better.

00:12:58.170 --> 00:13:00.630
And maybe a little bit less
of a lavish lifestyle.

00:13:00.630 --> 00:13:05.070
Considering the state of the
government's budget.

00:13:05.070 --> 00:13:09.330
And the state of the people of
France, they're starving.

00:13:09.330 --> 00:13:10.510
Why don't you do that
a little bit?

00:13:10.510 --> 00:13:12.370
But Louis XVI, instead
of taking his

00:13:12.370 --> 00:13:14.240
advice, he fired him.

00:13:14.240 --> 00:13:15.610
He fired the financial
adviser.

00:13:22.200 --> 00:13:26.100
So taken together, troops are
converging on Paris, you have

00:13:26.100 --> 00:13:31.020
this Tennis Court Oath, Louis
XVI has fired his adviser,

00:13:31.020 --> 00:13:32.760
people are going hungry.

00:13:32.760 --> 00:13:34.070
They're genuinely
going hungry.

00:13:34.070 --> 00:13:36.820
People in Paris said, the king
is going to try to suppress us

00:13:36.820 --> 00:13:39.300
again, this is no good.

00:13:39.300 --> 00:13:40.540
And especially if he does
it with troops,

00:13:40.540 --> 00:13:42.520
we have to arm ourselves.

00:13:42.520 --> 00:13:45.070
So they stormed the Bastille.

00:13:45.070 --> 00:13:47.890
This right here is a picture
of the Bastille.

00:13:52.020 --> 00:13:56.640
And this is most famous, when
you when you first learn about

00:13:56.640 --> 00:13:59.630
it, or maybe this is the first
time you're learning about it.

00:13:59.630 --> 00:14:01.910
They put political prisoners
there and they freed the

00:14:01.910 --> 00:14:02.700
political prisoners.

00:14:02.700 --> 00:14:04.280
But in reality, there
were only seven

00:14:04.280 --> 00:14:06.400
prisoners in the Bastille.

00:14:06.400 --> 00:14:08.670
So it's not like thousands and
thousands of political

00:14:08.670 --> 00:14:10.580
prisoners were being held there
and there were freed.

00:14:10.580 --> 00:14:14.150
The real value of the Bastille
to the revolutionaries, we

00:14:14.150 --> 00:14:17.150
could say, is that there
were weapons there.

00:14:17.150 --> 00:14:20.440
There was a major arms
cache there.

00:14:20.440 --> 00:14:22.750
And so by storming the Bastille
and getting the

00:14:22.750 --> 00:14:26.420
weapons, they all of a sudden
could essentially fend off any

00:14:26.420 --> 00:14:28.390
type of threat that the
troops would have.

00:14:28.390 --> 00:14:30.970
But this is also kind of the
very beginning of the real

00:14:30.970 --> 00:14:32.205
chaos of the French
Revolution.

00:14:32.205 --> 00:14:34.490
And as we're going to see over
the next several years, the

00:14:34.490 --> 00:14:36.750
chaos only gets worse
and worse.

00:14:36.750 --> 00:14:39.500
It's almost on a lot of levels
a lot worse than the American

00:14:39.500 --> 00:14:40.050
Revolution.

00:14:40.050 --> 00:14:43.070
Because what actually happened
in the cities and what fellow

00:14:43.070 --> 00:14:45.630
Frenchman started doing to do
each other was really on many

00:14:45.630 --> 00:14:46.900
levels barbaric.

00:14:46.900 --> 00:14:48.730
And you actually saw it here for
the first time, where the

00:14:48.730 --> 00:14:51.190
governor of the Bastille, the
guy who was in charge of it,

00:14:51.190 --> 00:14:53.670
he had the standoff between
the troops.

00:14:53.670 --> 00:14:55.350
And he eventually called
for a ceasefire.

00:14:55.350 --> 00:14:57.050
Because he's like, oh there's
too much bloodshed.

00:14:57.050 --> 00:15:00.790
But once the revolutionaries got
to him, they stabbed them,

00:15:00.790 --> 00:15:03.110
they cut his head off, and
they put it on a pike.

00:15:03.110 --> 00:15:05.440
Then they went back to the mayor
of Paris, they shot him.

00:15:05.440 --> 00:15:08.560
So clearly, things were really
getting out of hand.

00:15:08.560 --> 00:15:12.130
But most people associate the
storming of the Bastille as

00:15:12.130 --> 00:15:16.640
kind of the landmark event
of the French Revolution.

00:15:16.640 --> 00:15:19.650
Even today, people celebrate
Bastille Day.

00:15:19.650 --> 00:15:26.670
And that is July 14, 1789.

00:15:26.670 --> 00:15:28.890
So just to give you a sense
of how quickly all of this

00:15:28.890 --> 00:15:31.150
happened, the Convocation
of the

00:15:31.150 --> 00:15:33.570
Estates-General, that was in May.

00:15:33.570 --> 00:15:36.580
The Tennis Court Oath
was in June.

00:15:36.580 --> 00:15:40.800
And then in July, you have the
storming of the Bastille.

00:15:40.800 --> 00:15:45.000
And then in August, just to kind
of complete the idea that

00:15:45.000 --> 00:15:47.640
we are definitely in a
revolutionary period.

00:15:50.150 --> 00:15:53.180
The National Assembly, that
started off at the tennis

00:15:53.180 --> 00:15:57.160
courts with the Third Estate,
they declared their equivalent

00:15:57.160 --> 00:15:58.780
of the Declaration
of Independence.

00:15:58.780 --> 00:16:12.530
They declared their Declaration
of the Rights of

00:16:12.530 --> 00:16:18.600
Man and of the Citizen.

00:16:22.720 --> 00:16:25.080
Which was essentially their
version of the Declaration of

00:16:25.080 --> 00:16:25.840
Independence.

00:16:25.840 --> 00:16:28.750
And it essentially put
everything into question of

00:16:28.750 --> 00:16:31.980
what is life, liberty, and
the pursuit of happiness?

00:16:31.980 --> 00:16:33.820
I'm using words from the
American Revolution.

00:16:33.820 --> 00:16:35.380
But this was their Declaration
of Independence.

00:16:35.380 --> 00:16:38.660
It wasn't a constitution, it
was just a statement of the

00:16:38.660 --> 00:16:42.010
things that they think need
to govern any type of

00:16:42.010 --> 00:16:43.190
constitution or country.

00:16:43.190 --> 00:16:46.400
Or the ideas that any country
should be based on.

00:16:46.400 --> 00:16:48.590
So I'm going to leave
you there.

00:16:48.590 --> 00:16:51.390
We've really now started
the French Revolution.

00:16:51.390 --> 00:16:53.320
And now, you're going to see
that over the next several

00:16:53.320 --> 00:16:56.550
years, it's only going to get
bloodier and bloodier and even

00:16:56.550 --> 00:16:57.520
more complex.

00:16:57.520 --> 00:16:59.750
And when everything is said and
done, it's actually not

00:16:59.750 --> 00:17:01.670
going to end that
well in terms of

00:17:01.670 --> 00:17:03.590
giving people liberty.


Introduction to Compound Interest

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.250
[MUSIC PLAYING]

00:00:08.870 --> 00:00:10.830
What I want to do in this video
is talk a little bit

00:00:10.830 --> 00:00:16.490
about compounding interest. And
then, have a little bit of

00:00:16.490 --> 00:00:20.050
a discussion of a way to
quickly, kind of an

00:00:20.050 --> 00:00:21.530
approximate way to
figure out how

00:00:21.530 --> 00:00:22.890
quickly something compounds.

00:00:22.890 --> 00:00:26.340
And then we'll actually see how
good of an approximation

00:00:26.340 --> 00:00:27.490
this really is.

00:00:27.490 --> 00:00:29.940
So just as a review, let's say
I'm running some type of a

00:00:29.940 --> 00:00:34.730
bank and I tell you that I am
offering 10% interest that

00:00:34.730 --> 00:00:35.980
compounds annually.

00:00:41.860 --> 00:00:43.830
That's usually not the
case in a real bank.

00:00:43.830 --> 00:00:46.080
You would probably compound
continuously.

00:00:46.080 --> 00:00:47.620
But I'm just going to
keep it a simple

00:00:47.620 --> 00:00:49.040
example, compounding annually.

00:00:49.040 --> 00:00:51.430
There are other videos on
compounding continuously.

00:00:51.430 --> 00:00:53.110
This makes the math
a little simpler.

00:00:53.110 --> 00:00:58.780
And all that means is that,
let's say today you deposit

00:00:58.780 --> 00:01:01.540
$100 in that bank account.

00:01:01.540 --> 00:01:05.129
If we wait one year and you
just keep that in the bank

00:01:05.129 --> 00:01:11.280
account, then you'll have
your $100 plus 10%

00:01:11.280 --> 00:01:13.190
on your $100 deposit.

00:01:13.190 --> 00:01:17.330
10% of 100 is going
to be another $10.

00:01:17.330 --> 00:01:23.200
So after a year, you're
going to have $110.

00:01:23.200 --> 00:01:25.550
You can just say I added
10% to the 100.

00:01:25.550 --> 00:01:30.930
And after two years, or a year
after that first year, after

00:01:30.930 --> 00:01:35.900
two years, you're going to get
10% not just on the $100.

00:01:35.900 --> 00:01:38.500
You're going to get
10% on the $110.

00:01:38.500 --> 00:01:40.890
So you get 10% on $110.

00:01:40.890 --> 00:01:43.040
You're going to get
another $11.

00:01:43.040 --> 00:01:44.870
So 10% of $110 is $11.

00:01:44.870 --> 00:01:49.490
So you're going to get $110--
that was, you can imagine,

00:01:49.490 --> 00:01:51.960
your deposit entering
your second year.

00:01:51.960 --> 00:01:53.950
And then you get plus
10% on that.

00:01:53.950 --> 00:01:55.980
Not 10% on your initial
deposit.

00:01:55.980 --> 00:01:57.880
That's why we say
it compounds.

00:01:57.880 --> 00:02:01.970
You get interest on the interest
from previous years.

00:02:01.970 --> 00:02:05.290
So $110 plus now $11.

00:02:05.290 --> 00:02:07.380
So every year, the amount of
interest we're getting-- if we

00:02:07.380 --> 00:02:08.639
don't withdraw anything--
goes up.

00:02:08.639 --> 00:02:12.950
So now we have $121.

00:02:12.950 --> 00:02:14.570
And I could just keep
doing that.

00:02:14.570 --> 00:02:18.400
And the general way to figure
out how much you have after

00:02:18.400 --> 00:02:21.170
let's say, n years, is
you multiply it.

00:02:23.960 --> 00:02:25.860
I'll use a little bit
of algebra here.

00:02:25.860 --> 00:02:27.790
So let's say this
is my original

00:02:27.790 --> 00:02:30.240
deposit, or my principal.

00:02:30.240 --> 00:02:31.840
However you want to view it.

00:02:31.840 --> 00:02:35.090
After x years, so after 1 year,
you would just multiply

00:02:35.090 --> 00:02:37.570
it to get-- to this number
right here, you

00:02:37.570 --> 00:02:40.260
multiply it by 1.1.

00:02:40.260 --> 00:02:41.090
Actually, let me
do it this way.

00:02:41.090 --> 00:02:43.040
I don't want to be
too abstract.

00:02:43.040 --> 00:02:46.570
So just to get the math here, so
to get to this number right

00:02:46.570 --> 00:02:48.340
here, we just multiplied.

00:02:48.340 --> 00:02:53.510
That number right there is
100 times 1 plus 10%.

00:02:53.510 --> 00:02:56.780
Or you could say 1.1.

00:02:56.780 --> 00:02:58.780
Now this number right here
is going to be this

00:02:58.780 --> 00:03:01.130
110 times 1.1 again.

00:03:01.130 --> 00:03:02.370
So it's this.

00:03:02.370 --> 00:03:08.170
It's the 100 times 1.1, which
was this number right there.

00:03:08.170 --> 00:03:11.400
And now we're going to multiply
that times 1.1 again.

00:03:11.400 --> 00:03:14.830
And remember, where does
the 1.1 come from?

00:03:14.830 --> 00:03:21.340
1.1 is the same thing as
100% plus another 10%.

00:03:21.340 --> 00:03:22.040
Right?

00:03:22.040 --> 00:03:22.780
That's what we're getting.

00:03:22.780 --> 00:03:27.800
We have 100% of our original
deposit plus another 10%.

00:03:27.800 --> 00:03:30.170
So we're multiplying by 1.1.

00:03:30.170 --> 00:03:31.170
Here we're doing that twice.

00:03:31.170 --> 00:03:33.620
We multiply by 1.1 twice.

00:03:33.620 --> 00:03:37.410
So after three years, how
much money do we have?

00:03:37.410 --> 00:03:44.030
So after three years, we're
going to have 100 times 1.1 to

00:03:44.030 --> 00:03:47.390
the third power.

00:03:47.390 --> 00:03:50.320
After n years-- now we're
getting a little abstract

00:03:50.320 --> 00:03:55.430
here-- we're going to have 100
times 1.1 to the nth power.

00:03:55.430 --> 00:03:59.100
And now you could imagine, this
is not easy to calculate.

00:03:59.100 --> 00:04:02.790
This was all the situation where
we're dealing with 10%.

00:04:02.790 --> 00:04:06.280
If we're dealing in a world
where let's say it's 7%.

00:04:06.280 --> 00:04:08.700
So let's say this is a different
reality here, where

00:04:08.700 --> 00:04:16.019
we have 7% compounding annual
interest. Then after 1 year,

00:04:16.019 --> 00:04:20.589
we would have 100 times--
instead of 1.1, it'd be 100%

00:04:20.589 --> 00:04:25.940
plus 7% or 1.07.

00:04:25.940 --> 00:04:27.380
Let's go to three years.

00:04:27.380 --> 00:04:29.390
After three years, I could
do two in between.

00:04:29.390 --> 00:04:35.450
It'd be 100 times 1.07
to the third power.

00:04:35.450 --> 00:04:37.230
Or 1.07 times itself
three times.

00:04:37.230 --> 00:04:40.290
After n years, it'd be 1.07
to the nth power.

00:04:40.290 --> 00:04:42.860
So I think you get the sense
here that, although the idea

00:04:42.860 --> 00:04:46.020
is reasonably simple, to
actually calculate compounding

00:04:46.020 --> 00:04:47.950
interest is actually
pretty difficult.

00:04:47.950 --> 00:04:51.240
And even more, let's say I were
to ask you, how long does

00:04:51.240 --> 00:05:04.790
it take to double your money?

00:05:04.790 --> 00:05:08.110
So if you were to just use this
math right here, you'd

00:05:08.110 --> 00:05:11.660
have to say gee, to double my
money, I would have to start

00:05:11.660 --> 00:05:15.300
with $100 and I'm going to
multiply that times-- let's

00:05:15.300 --> 00:05:15.790
say, whatever.

00:05:15.790 --> 00:05:19.880
Let's say it's a 10% interest.
1.1 or 1.10-- depending on how

00:05:19.880 --> 00:05:24.540
you want to view it-- to the x
is equal to-- well I'm going

00:05:24.540 --> 00:05:25.700
to double my money
so it's going to

00:05:25.700 --> 00:05:27.900
have to equal to $200.

00:05:27.900 --> 00:05:30.390
And now I'm going to have to
solve for x and I'm going to

00:05:30.390 --> 00:05:32.060
have to do some logarithms
here.

00:05:32.060 --> 00:05:33.770
You can divide both
sides by 100.

00:05:33.770 --> 00:05:37.490
You get 1.1 to the
x is equal to 2.

00:05:37.490 --> 00:05:40.010
I just divided both
sides by 100.

00:05:40.010 --> 00:05:42.890
And then you could take the
logarithm of both sides base

00:05:42.890 --> 00:05:44.980
1.1, and you get x.

00:05:44.980 --> 00:05:48.060
And I'm showing you that this
is complicated on purpose.

00:05:48.060 --> 00:05:49.880
And if any of this is confusing,
there's multiple

00:05:49.880 --> 00:05:51.840
videos on how to solve these.

00:05:51.840 --> 00:05:55.780
You get x is equal to
log base 1.1 of 2.

00:05:55.780 --> 00:05:58.120
And most of us cannot do
this in our heads.

00:05:58.120 --> 00:06:00.360
So although the idea is simple,
how long will it take

00:06:00.360 --> 00:06:02.295
for me to double my money?

00:06:02.295 --> 00:06:06.280
To actually solve it, to get
the exact answer, is not an

00:06:06.280 --> 00:06:08.950
easy thing to do.

00:06:08.950 --> 00:06:10.730
If you have a simple calculator,
you can kind of

00:06:10.730 --> 00:06:13.370
keep incrementing the number
of years until you get a

00:06:13.370 --> 00:06:14.950
number that's close.

00:06:14.950 --> 00:06:16.280
But no straightforward
way to do it.

00:06:16.280 --> 00:06:17.500
And this is with 10%.

00:06:17.500 --> 00:06:20.050
If we're doing it with
9.3% it just

00:06:20.050 --> 00:06:23.090
becomes even more difficult.

00:06:23.090 --> 00:06:25.100
So what I'm going to do in the
next video is I'm going to

00:06:25.100 --> 00:06:28.840
explain something called the
rule of 72, which is an

00:06:28.840 --> 00:06:31.670
approximate way to figure
out how long

00:06:31.670 --> 00:06:32.700
to answer this question.

00:06:32.700 --> 00:06:35.650
How long does it take to double
your-- I forgot the

00:06:35.650 --> 00:06:37.340
word, the most important word.

00:06:37.340 --> 00:06:40.730
How long does it take to
double your money?

00:06:40.730 --> 00:06:43.540
And we'll see how good of an
approximation it is in that

00:06:43.540 --> 00:06:44.346
next video.

00:06:44.346 --> 00:06:45.596
[MUSIC PLAYING]


Proof: Right triangles inscribed in circles

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.640
Let's say we have a circle,
and then we have a

00:00:03.640 --> 00:00:05.280
diameter of the circle.

00:00:05.280 --> 00:00:09.080
Let me draw my best diameter.

00:00:09.080 --> 00:00:09.760
That's pretty good.

00:00:09.760 --> 00:00:12.580
This right here is the diameter
of the circle or it's a

00:00:12.580 --> 00:00:14.700
diameter of the circle.

00:00:14.700 --> 00:00:16.110
That's a diameter.

00:00:16.110 --> 00:00:19.220
Let's say I have a triangle
where the diameter is one side

00:00:19.220 --> 00:00:26.040
of the triangle, and the angle
opposite that side, it's

00:00:26.040 --> 00:00:28.960
vertex, sits some place
on the circumference.

00:00:28.960 --> 00:00:34.200
So, let's say, the angle or the
angle opposite of this diameter

00:00:34.200 --> 00:00:35.260
sits on that circumference.

00:00:35.260 --> 00:00:38.020
So the triangle
looks like this.

00:00:38.020 --> 00:00:44.160
The triangle looks like that.

00:00:44.160 --> 00:00:47.170
What I'm going to show you
in this video is that

00:00:47.170 --> 00:00:50.700
this triangle is going
to be a right triangle.

00:00:54.290 --> 00:00:57.040
The 90 degree side is going
to be the side that is

00:00:57.040 --> 00:00:58.550
opposite this diameter.

00:00:58.550 --> 00:01:00.340
I don't want to label it
just yet because that would

00:01:00.340 --> 00:01:02.140
ruin the fun of the proof.

00:01:02.140 --> 00:01:05.070
Now let's see what we
can do to show this.

00:01:05.070 --> 00:01:08.910
Well, we have in our tool kit
the notion of an inscribed

00:01:08.910 --> 00:01:12.970
angle, it's relation to
a central angle that

00:01:12.970 --> 00:01:14.830
subtends the same arc.

00:01:14.830 --> 00:01:15.720
So let's look at that.

00:01:15.720 --> 00:01:18.950
So let's say that this is an
inscribed angle right here.

00:01:18.950 --> 00:01:22.760
Let's call this theta.

00:01:22.760 --> 00:01:25.070
Now let's say that
that's the center of

00:01:25.070 --> 00:01:27.370
my circle right there.

00:01:27.370 --> 00:01:30.190
Then this angle right here
would be a central angle.

00:01:30.190 --> 00:01:32.620
Let me draw another triangle
right here, another

00:01:32.620 --> 00:01:33.460
line right there.

00:01:33.460 --> 00:01:35.130
This is a central
angle right here.

00:01:35.130 --> 00:01:38.190
This is a radius.

00:01:38.190 --> 00:01:40.070
This is the same radius
-- actually this

00:01:40.070 --> 00:01:41.230
distance is the same.

00:01:41.230 --> 00:01:44.480
But we've learned several
videos ago that look, this

00:01:44.480 --> 00:01:48.710
angle, this inscribed angle,
it subtends this arc up here.

00:01:52.420 --> 00:01:55.850
The central angle that subtends
that same arc is going

00:01:55.850 --> 00:01:57.400
to be twice this angle.

00:01:57.400 --> 00:01:59.040
We proved that
several videos ago.

00:01:59.040 --> 00:02:02.150
So this is going to be 2theta.

00:02:02.150 --> 00:02:05.260
It's the central angle
subtending the same arc.

00:02:05.260 --> 00:02:10.120
Now, this triangle right here,
this one right here, this

00:02:10.120 --> 00:02:11.620
is an isosceles triangle.

00:02:11.620 --> 00:02:13.800
I could rotate it and
draw it like this.

00:02:16.480 --> 00:02:22.163
If I flipped it over it would
look like that, that, and then

00:02:22.163 --> 00:02:25.000
the green side would
be down like that.

00:02:25.000 --> 00:02:28.980
And both of these sides
are of length r.

00:02:28.980 --> 00:02:31.160
This top angle is 2theta.

00:02:31.160 --> 00:02:33.530
So all I did is I took it
and I rotated it around to

00:02:33.530 --> 00:02:35.060
draw it for you this way.

00:02:35.060 --> 00:02:37.050
This side is that
side right there.

00:02:37.050 --> 00:02:41.660
Since its two sides are equal,
this is isosceles, so these to

00:02:41.660 --> 00:02:43.980
base angles must be the same.

00:02:47.580 --> 00:02:49.820
That and that must be the same,
or if I were to draw it up

00:02:49.820 --> 00:02:55.150
here, that and that must be
the exact same base angle.

00:02:55.150 --> 00:02:58.150
Now let me see, I already
used theta, maybe I'll

00:02:58.150 --> 00:02:59.800
use x for these angles.

00:02:59.800 --> 00:03:05.230
So this has to be x,
and that has to be x.

00:03:05.230 --> 00:03:08.000
So what is x going
to be equal to?

00:03:08.000 --> 00:03:12.120
Well, x plus x plus 2theta
have to equal 180 degrees.

00:03:12.120 --> 00:03:13.970
They're all in the
same triangle.

00:03:13.970 --> 00:03:15.770
So let me write that down.

00:03:15.770 --> 00:03:23.010
We get x plus x plus 2theta,
all have to be equal to 180

00:03:23.010 --> 00:03:30.880
degrees, or we get 2x plus
2theta is equal to 180 degrees,

00:03:30.880 --> 00:03:35.970
or we get 2x is equal
to 180 minus 2theta.

00:03:35.970 --> 00:03:42.980
Divide both sides by 2, you get
x is equal to 90 minus theta.

00:03:42.980 --> 00:03:50.590
So x is equal to
90 minus theta.

00:03:50.590 --> 00:03:52.890
Now let's see what else
we could do with this.

00:03:52.890 --> 00:03:55.130
Well we could look at this
triangle right here.

00:03:55.130 --> 00:03:59.160
This triangle, this side over
here also has this distance

00:03:59.160 --> 00:04:01.930
right here is also a
radius of the circle.

00:04:01.930 --> 00:04:04.080
This distance over here we've
already labeled it, is

00:04:04.080 --> 00:04:05.060
a radius of a circle.

00:04:05.060 --> 00:04:08.870
So once again, this is also
an isosceles triangle.

00:04:08.870 --> 00:04:12.770
These two sides are equal,
so these two base angles

00:04:12.770 --> 00:04:13.500
have to be equal.

00:04:13.500 --> 00:04:17.160
So if this is theta,
this is also going to

00:04:17.160 --> 00:04:17.895
be equal to theta.

00:04:17.895 --> 00:04:20.770
And actually, we use that
information, we use to actually

00:04:20.770 --> 00:04:25.100
show that first result about
inscribed angles and the

00:04:25.100 --> 00:04:27.340
relation between them and
central angles subtending

00:04:27.340 --> 00:04:27.980
the same arc.

00:04:27.980 --> 00:04:29.670
So if this is theta, that's
theta because this is

00:04:29.670 --> 00:04:31.120
an isosceles triangle.

00:04:31.120 --> 00:04:36.150
So what is this whole
angle over here?

00:04:36.150 --> 00:04:39.850
Well it's going to be theta
plus 90 minus theta.

00:04:39.850 --> 00:04:41.650
That angle right there's
going to be theta

00:04:41.650 --> 00:04:44.690
plus 90 minus theta.

00:04:44.690 --> 00:04:46.270
Well, the thetas cancel out.

00:04:46.270 --> 00:04:49.690
So no matter what, as long as
one side of my triangle is the

00:04:49.690 --> 00:04:53.070
diameter, and then the angle or
the vertex of the angle

00:04:53.070 --> 00:04:56.620
opposite sits opposite of
that side, sits on the

00:04:56.620 --> 00:05:01.780
circumference, then this angle
right here is going to be a

00:05:01.780 --> 00:05:08.750
right angle, and this is going
to be a right triangle.

00:05:08.750 --> 00:05:11.640
So if I just were to draw
something random like this --

00:05:11.640 --> 00:05:16.010
if I were to just take a point
right there, like that, and

00:05:16.010 --> 00:05:19.750
draw it just like that,
this is a right angle.

00:05:19.750 --> 00:05:23.220
If I were to draw something
like that and go out like

00:05:23.220 --> 00:05:25.240
that, this is a right angle.

00:05:25.240 --> 00:05:27.860
For any of these I could
do this exact same proof.

00:05:27.860 --> 00:05:30.090
And in fact, the way I drew it
right here, I kept it very

00:05:30.090 --> 00:05:33.810
general so it would apply
to any of these triangles.


Area of inscribed equilateral triangle (some basic trig used)

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:03.140
What I want to do in this video
is use some of the results from

00:00:03.140 --> 00:00:05.960
the last several videos to
do some pretty neat things.

00:00:05.960 --> 00:00:10.000
So let's say this is a circle,
and I have an inscribed

00:00:10.000 --> 00:00:12.130
equilateral triangle
in this circle.

00:00:12.130 --> 00:00:17.340
So all the vertices of
this triangle sit on the

00:00:17.340 --> 00:00:18.825
circumference of the circle.

00:00:18.825 --> 00:00:24.170
So I'm going to try my best to
draw an equilateral triangle.

00:00:24.170 --> 00:00:26.960
I think that's about as good as
I'm going to be able to do.

00:00:26.960 --> 00:00:28.620
And when I say equilateral
that means all of these

00:00:28.620 --> 00:00:29.910
sides are the same length.

00:00:29.910 --> 00:00:33.060
So if this is side length a,
then this is side length a,

00:00:33.060 --> 00:00:36.610
and that is also a
side of length a.

00:00:36.610 --> 00:00:44.010
And let's say we know that the
radius of this circle is 2.

00:00:44.010 --> 00:00:45.925
I'm just picking a number,
just to do this problem.

00:00:45.925 --> 00:00:49.600
So let's say the radius
of this circle is 2.

00:00:49.600 --> 00:00:51.700
So from the center to the
circumference at any

00:00:51.700 --> 00:00:55.910
point, this distance, the
radius, is equal to 2.

00:00:55.910 --> 00:01:01.780
Now, what I'm going to ask you
is using some of the results of

00:01:01.780 --> 00:01:04.020
the last few videos and a
little bit of basic

00:01:04.020 --> 00:01:06.940
trigonometry-- and if the word
"trigonometry" scares you,

00:01:06.940 --> 00:01:09.570
you'll just need to know maybe
the first two or three videos

00:01:09.570 --> 00:01:11.710
in the trigonometry playlist to
be able to understand

00:01:11.710 --> 00:01:12.840
what I do here.

00:01:12.840 --> 00:01:18.830
What I want to do is figure out
the area of the region inside

00:01:18.830 --> 00:01:21.080
the circle and outside
of the triangle.

00:01:21.080 --> 00:01:25.690
So I want to figure out the
area of that little space, that

00:01:25.690 --> 00:01:30.940
space, and this space combined.

00:01:30.940 --> 00:01:33.490
So the obvious way to do
this is to say, well I can

00:01:33.490 --> 00:01:36.670
figure out the area of
the circle pretty easily.

00:01:36.670 --> 00:01:40.215
Area of the circle.

00:01:40.215 --> 00:01:43.740
And that's going to be
equal to pi r squared.

00:01:43.740 --> 00:01:48.840
Or pi times 2 squared,
which is equal to 4 pi.

00:01:48.840 --> 00:01:53.040
And I could subtract from 4
pi the area of the triangle.

00:01:53.040 --> 00:01:55.450
So we need to figure out
the area of the triangle.

00:01:55.450 --> 00:02:00.760
What is the area
of the triangle?

00:02:03.930 --> 00:02:07.260
Well, from several videos ago I
showed you Heron's formula,

00:02:07.260 --> 00:02:10.720
where if you know the lengths
of the sides of a triangle

00:02:10.720 --> 00:02:12.070
you can figure out the area.

00:02:12.070 --> 00:02:14.180
But we don't know the lengths
of the sides just yet.

00:02:14.180 --> 00:02:16.560
Once we do maybe we can
figure out the area.

00:02:16.560 --> 00:02:18.740
Let me apply Heron's
formula not knowing it.

00:02:18.740 --> 00:02:21.950
So let me just say that the
lengths of this equilateral--

00:02:21.950 --> 00:02:23.760
the lengths of the
sides-- are a.

00:02:23.760 --> 00:02:31.450
Applying Heron's formula, we
first define our variable

00:02:31.450 --> 00:02:38.220
s as being equal to a
plus a plus a, over 2.

00:02:38.220 --> 00:02:42.070
Or that's the same
thing as 3a over 2.

00:02:42.070 --> 00:02:46.380
And then the area of this
triangle, in terms of a.

00:02:46.380 --> 00:02:52.910
So the area is going to be
equal to the square root of

00:02:52.910 --> 00:02:59.310
s, which is 3a over
2, times s minus a.

00:02:59.310 --> 00:03:03.820
So that's 3a over 2 minus a.

00:03:03.820 --> 00:03:07.060
Or I could just
write, 2a over 2.

00:03:07.060 --> 00:03:08.970
Right? a is the same
thing as 2a over 2.

00:03:08.970 --> 00:03:10.740
You could cancel
those out and get a.

00:03:10.740 --> 00:03:13.170
And then I'm going to
do that three times.

00:03:13.170 --> 00:03:16.000
So instead of just multiplying
that out three times for each

00:03:16.000 --> 00:03:18.640
of the sides, by Heron's
formula I could just say

00:03:18.640 --> 00:03:20.700
to the third power.

00:03:20.700 --> 00:03:22.000
So what's this going
to be equal to?

00:03:22.000 --> 00:03:31.050
This is going to be equal to
the square root of 3a over 2.

00:03:31.050 --> 00:03:34.070
And then this right here
is going to be equal

00:03:34.070 --> 00:03:36.810
to 3a minus 2a, is a.

00:03:36.810 --> 00:03:42.010
So a/2 to the third power.

00:03:42.010 --> 00:03:44.860
And so this is going to be
equal to-- I'll arbitrarily

00:03:44.860 --> 00:03:46.490
switch colors.

00:03:46.490 --> 00:03:53.560
We have 3a times a to the
third, which is 3a to the

00:03:53.560 --> 00:03:58.170
fourth, over 2 times
2 to the third.

00:03:58.170 --> 00:04:03.400
Well that's 2 to the
fourth power, or 16.

00:04:03.400 --> 00:04:03.680
Right?

00:04:03.680 --> 00:04:07.100
2 times 2 to the third
is 2 to the fourth.

00:04:07.100 --> 00:04:07.890
That's 16.

00:04:07.890 --> 00:04:10.660
And then if we take the square
root of the numerator and the

00:04:10.660 --> 00:04:14.150
denominator, this is going to
be equal to the square root of

00:04:14.150 --> 00:04:16.690
a to the fourth is a squared.

00:04:16.690 --> 00:04:21.390
a squared times, well I'll just
write the square root of 3,

00:04:21.390 --> 00:04:24.860
over the square root of the
denominator, which is just 4.

00:04:24.860 --> 00:04:30.130
So if we know a, using Heron's
formula we know what the area

00:04:30.130 --> 00:04:32.720
of this equilateral
triangle is.

00:04:32.720 --> 00:04:35.080
So how can we figure out a?

00:04:35.080 --> 00:04:37.770
So what else do we know about
equilateral triangles?

00:04:37.770 --> 00:04:42.690
Well we know that all of
these angles are equal.

00:04:42.690 --> 00:04:45.720
And since they must add
up to 180 degrees, they

00:04:45.720 --> 00:04:48.210
all must be 60 degrees.

00:04:48.210 --> 00:04:51.890
That's 60 degrees,
that's 60 degrees, and

00:04:51.890 --> 00:04:54.090
that is 60 degrees.

00:04:54.090 --> 00:04:56.980
Now let's see if we can use the
last video, where I talked

00:04:56.980 --> 00:05:01.720
about the relationship
between an inscribed angle

00:05:01.720 --> 00:05:02.800
and a central angle.

00:05:02.800 --> 00:05:04.560
So this is an inscribed
angle right here.

00:05:04.560 --> 00:05:09.620
It's vertex is sitting
on the circumference.

00:05:09.620 --> 00:05:16.630
And so it is subtending
this arc right here.

00:05:20.500 --> 00:05:25.000
And the central angle that
is subtending that same arc

00:05:25.000 --> 00:05:26.330
is this one right here.

00:05:29.880 --> 00:05:33.740
The central angles subtending
that same arc is that

00:05:33.740 --> 00:05:34.960
one right there.

00:05:34.960 --> 00:05:39.170
So based on what we saw in the
last video, the central angle

00:05:39.170 --> 00:05:41.980
that subtends the same arc is
going to be double of

00:05:41.980 --> 00:05:43.040
the inscribed angle.

00:05:43.040 --> 00:05:47.230
So this angle right here is
going to be 120 degrees.

00:05:47.230 --> 00:05:48.860
Let me just put an arrow there.

00:05:48.860 --> 00:05:50.860
120 degrees.

00:05:50.860 --> 00:05:52.440
It's double of that one.

00:05:52.440 --> 00:05:56.110
Now, if I were to exactly
bisect this angle right here.

00:05:56.110 --> 00:05:58.140
So I go halfway through the
angle, and I want to just go

00:05:58.140 --> 00:06:01.260
straight down like that.

00:06:01.260 --> 00:06:03.310
What are these two
angles going to be?

00:06:03.310 --> 00:06:04.440
Well, they're going
to be 60 degrees.

00:06:04.440 --> 00:06:05.760
I'm bisecting that angle.

00:06:05.760 --> 00:06:10.480
That is 60 degrees, and that
is 60 degrees right there.

00:06:10.480 --> 00:06:14.450
And we know that I'm
splitting this side in two.

00:06:14.450 --> 00:06:17.080
This is an isosceles triangle.

00:06:17.080 --> 00:06:19.040
This is a radius right here.

00:06:19.040 --> 00:06:21.030
Radius r is equal to 2.

00:06:21.030 --> 00:06:24.530
This is a radius right
here of r is equal to 2.

00:06:24.530 --> 00:06:26.090
So this whole triangle
is symmetric.

00:06:26.090 --> 00:06:28.540
If I go straight down the
middle, this length right

00:06:28.540 --> 00:06:33.100
here is going to be
that side divided by 2.

00:06:33.100 --> 00:06:36.280
That side right there is going
to be that side divided by 2.

00:06:36.280 --> 00:06:37.240
Let me draw that over here.

00:06:37.240 --> 00:06:39.890
If I just take an isosceles
triangle, any isosceles

00:06:39.890 --> 00:06:44.850
triangle, where this side is
equivalent to that side.

00:06:44.850 --> 00:06:47.300
Those are our radiuses
in this example.

00:06:47.300 --> 00:06:49.530
And this angle is going to
be equal to that angle.

00:06:49.530 --> 00:06:51.790
If I were to just go straight
down this angle right

00:06:51.790 --> 00:06:55.260
here, I would split that
opposite side in two.

00:06:55.260 --> 00:06:56.880
So these two lengths
are going to be equal.

00:06:56.880 --> 00:06:59.120
In this case if the whole
thing is a, each of these

00:06:59.120 --> 00:07:01.140
are going to be a/2.

00:07:01.140 --> 00:07:04.420
Now, let's see if we can use
this and a little bit of

00:07:04.420 --> 00:07:08.620
trigonometry to find the
relationship between a and r.

00:07:08.620 --> 00:07:12.050
Because if we're able to solve
for a using r, then we can then

00:07:12.050 --> 00:07:14.640
put that value of a in here and
we'll get the area

00:07:14.640 --> 00:07:15.690
of our triangle.

00:07:15.690 --> 00:07:17.600
And then we could subtract
that from the area of the

00:07:17.600 --> 00:07:20.070
circle, and we're done.

00:07:20.070 --> 00:07:22.050
We will have solved
the problem.

00:07:22.050 --> 00:07:24.610
So let's see if we can do that.

00:07:24.610 --> 00:07:29.340
So we have an angle
here of 60 degrees.

00:07:29.340 --> 00:07:32.050
Half of this whole central
angle right there.

00:07:32.050 --> 00:07:35.890
If this angle is 60 degrees,
we have a/2 that's

00:07:35.890 --> 00:07:37.380
opposite to this angle.

00:07:37.380 --> 00:07:43.480
So we have an opposite
is equal to a/2.

00:07:43.480 --> 00:07:44.860
And we also have
the hypotenuse.

00:07:44.860 --> 00:07:45.040
Right?

00:07:45.040 --> 00:07:46.870
This is a right
triangle right here.

00:07:46.870 --> 00:07:49.830
You're just going straight
down, and you're bisecting

00:07:49.830 --> 00:07:50.840
that opposite side.

00:07:50.840 --> 00:07:52.640
This is a right triangle.

00:07:52.640 --> 00:07:54.000
So we can do a little
trigonometry.

00:07:54.000 --> 00:08:02.550
Our opposite is a/2, the
hypotenuse is equal to r.

00:08:02.550 --> 00:08:05.020
This is the hypotenuse, right
here, of our right triangle.

00:08:05.020 --> 00:08:06.360
So that is equal to 2.

00:08:06.360 --> 00:08:12.440
So what trig ratio is the
ratio of an angle's opposite

00:08:12.440 --> 00:08:14.920
side to hypotenuse?

00:08:14.920 --> 00:08:18.910
So some of you all might get
tired of me doing this all

00:08:18.910 --> 00:08:22.030
the time, but SOH CAH TOA.

00:08:22.030 --> 00:08:27.070
SOH-- sin of an angle is
equal to the opposite

00:08:27.070 --> 00:08:28.620
over the hypotenuse.

00:08:28.620 --> 00:08:29.580
So let me scroll
down a little bit.

00:08:29.580 --> 00:08:31.270
I'm running out of space.

00:08:31.270 --> 00:08:38.700
So the sin of this angle right
here, the sin of 60 degrees, is

00:08:38.700 --> 00:08:42.070
going to be equal to the
opposite side, is going to be

00:08:42.070 --> 00:08:45.800
equal to a/2, over the
hypotenuse, which is

00:08:45.800 --> 00:08:48.140
our radius-- over 2.

00:08:48.140 --> 00:08:54.510
Which is equal to a/2
divided by 2 is a/4.

00:08:54.510 --> 00:08:56.880
And what is sin of 60 degrees?

00:08:56.880 --> 00:08:59.720
And if the word "sin" looks
completely foreign to you,

00:08:59.720 --> 00:09:04.150
watch the first several videos
on the trigonometry playlist.

00:09:04.150 --> 00:09:06.240
It shouldn't be too daunting.

00:09:06.240 --> 00:09:08.310
sin of 60 degrees you
might remember from your

00:09:08.310 --> 00:09:10.680
30-60-90 triangles.

00:09:10.680 --> 00:09:13.210
So let me draw one right there.

00:09:13.210 --> 00:09:15.705
So that is a 30-60-90 triangle.

00:09:15.705 --> 00:09:21.540
If this is 60 degrees, that
is 30 degrees, that is 90.

00:09:21.540 --> 00:09:26.660
You might remember that this is
of length 1, this is going to

00:09:26.660 --> 00:09:29.520
be of length 1/2, and this
is going to be of length

00:09:29.520 --> 00:09:31.370
square root of 3 over 2.

00:09:31.370 --> 00:09:35.300
So the sin of 60 degrees is
opposite over hypotenuse.

00:09:35.300 --> 00:09:37.770
Square root of 3 over 2 over 1.

00:09:37.770 --> 00:09:40.940
sin of 60 degrees.

00:09:40.940 --> 00:09:42.840
If you don't have a calculator,
you could just use this--

00:09:42.840 --> 00:09:44.930
is square root of 3 over 2.

00:09:44.930 --> 00:09:48.890
So this right here is
square root of 3 over 2.

00:09:48.890 --> 00:09:51.280
Now we can solve for a.

00:09:51.280 --> 00:09:56.920
Square root of 3 over
2 is equal to a/4.

00:09:56.920 --> 00:09:59.610
Let's multiply both sides by 4.

00:09:59.610 --> 00:10:01.640
So you get this 4 cancels out.

00:10:01.640 --> 00:10:03.445
You multiply 4 here.

00:10:03.445 --> 00:10:04.480
This becomes a 2.

00:10:04.480 --> 00:10:05.660
This becomes a 1.

00:10:05.660 --> 00:10:09.250
You get a is equal to
2 square roots of 3.

00:10:09.250 --> 00:10:11.000
We're in the home stretch.

00:10:11.000 --> 00:10:15.420
We just figured out the length
of each of these sides.

00:10:15.420 --> 00:10:17.460
We used Heron's formula to
figure out the area of the

00:10:17.460 --> 00:10:19.080
triangle in terms
of those lengths.

00:10:19.080 --> 00:10:22.110
So we just substitute this
value of a into there

00:10:22.110 --> 00:10:24.670
to get our actual area.

00:10:24.670 --> 00:10:30.380
So our triangle's area
is equal to a squared.

00:10:30.380 --> 00:10:31.690
What's a squared?

00:10:31.690 --> 00:10:37.780
That is 2 square roots
of 3 squared, times the

00:10:37.780 --> 00:10:42.710
square root of 3 over 4.

00:10:42.710 --> 00:10:45.470
We just did a squared times
the square root of 3 over 4.

00:10:45.470 --> 00:10:51.950
This is going to be equal
to 4 times 3 times the

00:10:51.950 --> 00:10:53.930
square of 3 over 4.

00:10:53.930 --> 00:10:55.350
These 4's cancel.

00:10:55.350 --> 00:10:58.360
So the area of our triangle
we got is 3 times the

00:10:58.360 --> 00:11:00.770
square root of 3.

00:11:00.770 --> 00:11:03.160
So the area here is 3
square roots of 3.

00:11:03.160 --> 00:11:06.480
That's the area of
this entire triangle.

00:11:06.480 --> 00:11:08.810
Now, to go back to what this
question was all about.

00:11:08.810 --> 00:11:12.970
The area of this orange area
outside of the triangle

00:11:12.970 --> 00:11:14.530
and inside of the circle.

00:11:14.530 --> 00:11:18.460
Well, the area of
our circle is 4 pi.

00:11:18.460 --> 00:11:23.340
And from that we subtract
the area of the triangle,

00:11:23.340 --> 00:11:25.170
3 square roots of 3.

00:11:25.170 --> 00:11:27.390
And we are done.

00:11:27.390 --> 00:11:28.670
This is our answer.

00:11:28.670 --> 00:11:35.270
This is the area of this
orange region right there.

00:11:35.270 --> 00:11:37.800
Anyway, hopefully
you found that fun.


Inscribed angle theorem proof

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:03.450
What I want to do in this video
is to prove one of the more

00:00:03.450 --> 00:00:08.980
useful results in geometry, and
that's that an inscribed angle

00:00:08.980 --> 00:00:14.950
is just an angle who's vertex
sits on the circumference

00:00:14.950 --> 00:00:17.080
of the circle.

00:00:17.080 --> 00:00:19.800
So that is our inscribed angle.

00:00:19.800 --> 00:00:24.950
I'll denote it by si -- I'll
use the si for inscribed angle

00:00:24.950 --> 00:00:27.170
and angles in this video.

00:00:27.170 --> 00:00:33.530
That si, the inscribed angle,
is going to be exactly 1/2 of

00:00:33.530 --> 00:00:37.880
the central angle that
subtends the same arc.

00:00:37.880 --> 00:00:40.730
So I just used a lot a fancy
words, but I think you'll

00:00:40.730 --> 00:00:41.650
get what I'm saying.

00:00:41.650 --> 00:00:42.820
So this is si.

00:00:42.820 --> 00:00:44.470
It is an inscribed angle.

00:00:44.470 --> 00:00:48.710
It sits, it's vertex sits
on the circumference.

00:00:48.710 --> 00:00:52.570
And if you draw out the to rays
that come out from this angle

00:00:52.570 --> 00:00:56.040
or the two cords that define
this angle, it intersects the

00:00:56.040 --> 00:00:57.340
circle at the other end.

00:00:57.340 --> 00:01:00.390
And if you look at the part of
the circumference of the circle

00:01:00.390 --> 00:01:03.730
that's inside of it, that
is the arc that is

00:01:03.730 --> 00:01:06.160
subtended by si.

00:01:06.160 --> 00:01:09.010
It's all very fancy words,
but I think the idea is

00:01:09.010 --> 00:01:09.920
pretty straightforward.

00:01:09.920 --> 00:01:28.485
This right here is the arc
subtended by si, where si is

00:01:28.485 --> 00:01:31.560
that inscribed angle right over
there, the vertex sitting

00:01:31.560 --> 00:01:32.400
on the circumference.

00:01:32.400 --> 00:01:37.920
Now, a central angle is an
angle where the vertex is

00:01:37.920 --> 00:01:39.460
sitting at the center
of the circle.

00:01:39.460 --> 00:01:41.880
So let's say that this right
here -- I'll try to eyeball

00:01:41.880 --> 00:01:45.510
it -- that right there is
the center of the circle.

00:01:45.510 --> 00:01:51.360
So let me draw a central angle
that subtends this same arc.

00:01:51.360 --> 00:01:58.470
So that looks like a central
angle subtending that same arc.

00:01:58.470 --> 00:01:59.390
Just like that.

00:01:59.390 --> 00:02:01.440
Let's call this theta.

00:02:01.440 --> 00:02:06.030
So this angle is si, this
angle right here is theta.

00:02:06.030 --> 00:02:10.120
When I'm going to prove in this
video is that si is always

00:02:10.120 --> 00:02:14.050
going to be equal
to 1/2 of theta.

00:02:14.050 --> 00:02:18.220
So if I were to tell you that
si is equal to, I don't know,

00:02:18.220 --> 00:02:21.330
25 degrees, then you would
immediately know that theta

00:02:21.330 --> 00:02:23.090
must be equal to 50 degrees.

00:02:23.090 --> 00:02:26.080
Or if I told you that theta was
80 degrees, then you would

00:02:26.080 --> 00:02:29.300
immediately know that
si was 40 degrees.

00:02:29.300 --> 00:02:31.500
So let's actually proved this.

00:02:31.500 --> 00:02:34.520
So let me clear this.

00:02:34.520 --> 00:02:37.730
So a good place to start,
or the place I'm going to

00:02:37.730 --> 00:02:40.460
start, is a special case.

00:02:40.460 --> 00:02:45.250
I'm going to draw an inscribed
angle, but one of the cords

00:02:45.250 --> 00:02:47.910
that define it is going to be
the diameter of the circle.

00:02:47.910 --> 00:02:50.526
So this isn't going to be the
general case, this is going

00:02:50.526 --> 00:02:51.320
to be a special case.

00:02:51.320 --> 00:02:55.325
So let me see, this is the
center right here of my circle.

00:02:55.325 --> 00:02:59.030
I'm trying to eyeball it.

00:02:59.030 --> 00:03:00.770
Center looks like that.

00:03:00.770 --> 00:03:04.210
So let me draw a diameter.

00:03:04.210 --> 00:03:06.440
So the diameter
looks like that.

00:03:06.440 --> 00:03:09.410
Then let me define
my inscribed angle.

00:03:09.410 --> 00:03:11.860
This diameter is
one side of it.

00:03:11.860 --> 00:03:15.910
And then the other side
maybe is just like that.

00:03:15.910 --> 00:03:20.520
So let me call this
right here si.

00:03:20.520 --> 00:03:27.120
If that's si, this length right
here is a radius -- that's

00:03:27.120 --> 00:03:29.330
our radius of our circle.

00:03:29.330 --> 00:03:33.080
Then this length right here is
also going to be the radius of

00:03:33.080 --> 00:03:35.760
our circle going from the
center to the circumference.

00:03:35.760 --> 00:03:38.130
Your circumference is defined
by all of the points that are

00:03:38.130 --> 00:03:40.340
exactly a radius away
from the center.

00:03:40.340 --> 00:03:43.610
So that's also a radius.

00:03:43.610 --> 00:03:47.920
Now, this triangle right here
is an isosceles triangle.

00:03:47.920 --> 00:03:49.890
It has two sides
that are equal.

00:03:49.890 --> 00:03:51.880
Two sides that are
definitely equal.

00:03:51.880 --> 00:03:54.630
We know that when we have two
sides being equal, their

00:03:54.630 --> 00:03:57.290
base angles are also equal.

00:03:57.290 --> 00:04:00.640
So this will also
be equal to si.

00:04:00.640 --> 00:04:02.130
You might not recognize
it because it's

00:04:02.130 --> 00:04:03.180
tilted up like that.

00:04:03.180 --> 00:04:05.720
But I think many of us when we
see a triangle that looks like

00:04:05.720 --> 00:04:10.940
this, if I told you this is r
and that is r, that these two

00:04:10.940 --> 00:04:17.860
sides are equal, and if this is
si, then you would also

00:04:17.860 --> 00:04:20.830
know that this angle is
also going to be si.

00:04:20.830 --> 00:04:23.930
Base angles are equivalent
on an isosceles triangle.

00:04:23.930 --> 00:04:26.720
So this is si, that is also si.

00:04:26.720 --> 00:04:29.770
Now, let me look at
the central angle.

00:04:29.770 --> 00:04:32.710
This is the central angle
subtending the same arc.

00:04:32.710 --> 00:04:35.920
Let's highlight the arc that
they're both subtending.

00:04:35.920 --> 00:04:40.300
This right here is the arc that
they're both going to subtend.

00:04:40.300 --> 00:04:44.350
So this is my central
angle right there, theta.

00:04:44.350 --> 00:04:49.000
Now if this angle is theta,
what's this angle going to be?

00:04:49.000 --> 00:04:50.620
This angle right here.

00:04:50.620 --> 00:04:53.010
Well, this angle is
supplementary to theta,

00:04:53.010 --> 00:04:56.640
so it's 180 minus theta.

00:04:56.640 --> 00:04:59.560
When you add these two angles
together you go 180 degrees

00:04:59.560 --> 00:05:01.750
around or the kind
of formal line.

00:05:01.750 --> 00:05:03.790
They're supplementary
to each other.

00:05:03.790 --> 00:05:06.740
Now we also know that these
three angles are sitting

00:05:06.740 --> 00:05:08.260
inside of the same triangle.

00:05:08.260 --> 00:05:12.030
So they must add up
to 180 degrees.

00:05:12.030 --> 00:05:19.300
So we get si -- this si plus
that si plus si plus this

00:05:19.300 --> 00:05:25.420
angle, which is 180 minus
theta plus 180 minus theta.

00:05:25.420 --> 00:05:29.130
These three angles must
add up to 180 degrees.

00:05:29.130 --> 00:05:31.740
They're the three
angles of a triangle.

00:05:31.740 --> 00:05:34.605
Now we could subtract
180 from both sides.

00:05:37.140 --> 00:05:43.260
Si plus si is 2 si minus
theta is equal to 0.

00:05:43.260 --> 00:05:44.840
Add theta to both sides.

00:05:44.840 --> 00:05:48.770
You get 2 si is equal to theta.

00:05:48.770 --> 00:05:52.850
Multiply both sides by 1/2
or divide both sides by 2.

00:05:52.850 --> 00:05:56.680
You get si is equal
to 1/2 of theta.

00:05:56.680 --> 00:06:00.070
So we just proved what we set
out to prove for the special

00:06:00.070 --> 00:06:07.120
case where our inscribed angle
is defined, where one on the

00:06:07.120 --> 00:06:11.200
rays, if you want to view these
lines as rays, where one of the

00:06:11.200 --> 00:06:15.220
rays that defines this
inscribed angle is

00:06:15.220 --> 00:06:17.180
along the diameter.

00:06:17.180 --> 00:06:19.200
The diameter forms
part of that ray.

00:06:19.200 --> 00:06:21.720
So this is a special
case where one edge is

00:06:21.720 --> 00:06:23.760
sitting on the diameter.

00:06:23.760 --> 00:06:27.660
So already we could
generalize this.

00:06:27.660 --> 00:06:30.580
So now that we know that if
this is 50 that this is

00:06:30.580 --> 00:06:32.820
going to be 100 degrees
and likewise, right?

00:06:32.820 --> 00:06:37.460
Whatever si is or whatever
theta is, si's going to be 1/2

00:06:37.460 --> 00:06:40.450
of that, or whatever si is,
theta is going to

00:06:40.450 --> 00:06:41.830
be 2 times that.

00:06:41.830 --> 00:06:44.110
And now this will
apply for any time.

00:06:44.110 --> 00:06:55.440
We could use this notion any
time that -- so just using that

00:06:55.440 --> 00:06:59.460
result we just got, we can now
generalize it a little bit,

00:06:59.460 --> 00:07:02.890
although this won't apply
to all inscribed angles.

00:07:02.890 --> 00:07:05.090
Let's have an inscribed
angle that looks like this.

00:07:10.680 --> 00:07:12.980
So this situation, the center,
you can kind of view it as

00:07:12.980 --> 00:07:15.470
it's inside of the angle.

00:07:15.470 --> 00:07:17.150
That's my inscribed angle.

00:07:17.150 --> 00:07:18.890
And I want to find a
relationship between this

00:07:18.890 --> 00:07:22.450
inscribed angle and the central
angle that's subtending

00:07:22.450 --> 00:07:24.360
to same arc.

00:07:24.360 --> 00:07:29.880
So that's my central angle
subtending the same arc.

00:07:29.880 --> 00:07:33.550
Well, you might say, hey, gee,
none of these ends or these

00:07:33.550 --> 00:07:37.310
cords that define this angle,
neither of these are diameters,

00:07:37.310 --> 00:07:40.400
but what we can do is
we can draw a diameter.

00:07:40.400 --> 00:07:43.300
If the center is within
these two cords we

00:07:43.300 --> 00:07:46.100
can draw a diameter.

00:07:46.100 --> 00:07:48.920
We can draw a diameter
just like that.

00:07:48.920 --> 00:07:51.680
If we draw a diameter just like
that, if we define this angle

00:07:51.680 --> 00:07:55.430
as si 1, that angle as si 2.

00:07:55.430 --> 00:07:58.320
Clearly si is the sum
of those two angles.

00:07:58.320 --> 00:08:04.350
And we call this angle theta
1, and this angle theta 2.

00:08:04.350 --> 00:08:07.240
We immediately you know that,
just using the result I just

00:08:07.240 --> 00:08:12.540
got, since we have one side of
our angles in both cases being

00:08:12.540 --> 00:08:18.260
a diameter now, we know
that si 1 is going to be

00:08:18.260 --> 00:08:22.010
equal to 1/2 theta 1.

00:08:22.010 --> 00:08:24.870
And we know that si 2 is
going to be 1/2 theta 2.

00:08:24.870 --> 00:08:30.140
Si 2 is going to
be 1/2 theta 2.

00:08:30.140 --> 00:08:39.850
So si, which is si 1 plus si 2,
so si 1 plus si 2 is going to

00:08:39.850 --> 00:08:41.120
be equal to these two things.

00:08:41.120 --> 00:08:47.580
1/2 theta 1 plus 1/2 theta 2.

00:08:47.580 --> 00:08:51.180
Si 1 plus si 2, this is equal
to the first inscribed

00:08:51.180 --> 00:08:53.850
angle that we want to deal
with, just regular si.

00:08:53.850 --> 00:08:54.980
That's si.

00:08:54.980 --> 00:08:58.350
And this right here, this
is equal to 1/2 times

00:08:58.350 --> 00:09:00.960
theta 1 plus theta 2.

00:09:00.960 --> 00:09:03.960
What's theta 1 plus theta 2?

00:09:03.960 --> 00:09:06.470
Well that's just our
original theta that

00:09:06.470 --> 00:09:08.490
we were dealing with.

00:09:08.490 --> 00:09:12.080
So now we see that si
is equal to 1/2 theta.

00:09:12.080 --> 00:09:14.710
So now we've proved it for a
slightly more general case

00:09:14.710 --> 00:09:20.020
where our center is inside
of the two rays that

00:09:20.020 --> 00:09:21.640
define that angle.

00:09:21.640 --> 00:09:27.100
Now, we still haven't addressed
a slightly harder situation or

00:09:27.100 --> 00:09:33.660
a more general situation where
if this is the center of our

00:09:33.660 --> 00:09:39.420
circle and I have an inscribed
angle where the center isn't

00:09:39.420 --> 00:09:40.990
sitting inside of
the two cords.

00:09:40.990 --> 00:09:41.820
Let me draw that.

00:09:41.820 --> 00:09:48.800
So that's going to be my
vertex, and I'll switch colors,

00:09:48.800 --> 00:09:51.540
so let's say that is one of the
cords that defines the

00:09:51.540 --> 00:09:53.320
angle, just like that.

00:09:53.320 --> 00:09:57.860
And let's say that is the
other cord that defines

00:09:57.860 --> 00:09:59.170
the angle just like that.

00:09:59.170 --> 00:10:02.500
So how do we find the
relationship between, let's

00:10:02.500 --> 00:10:07.910
call, this angle right
here, let's call it si 1.

00:10:07.910 --> 00:10:13.050
How do we find the relationship
between si 1 and the central

00:10:13.050 --> 00:10:16.160
angle that subtends
this same arc?

00:10:16.160 --> 00:10:19.530
So when I talk about the same
arc, that's that right there.

00:10:19.530 --> 00:10:22.720
So the central angle that
subtends the same arc

00:10:22.720 --> 00:10:23.660
will look like this.

00:10:28.150 --> 00:10:32.910
Let's call that theta 1.

00:10:32.910 --> 00:10:36.770
What we can do is use what we
just learned when one side of

00:10:36.770 --> 00:10:39.350
our inscribed angle
is a diameter.

00:10:39.350 --> 00:10:41.135
So let's construct that.

00:10:41.135 --> 00:10:44.260
So let me draw a diameter here.

00:10:44.260 --> 00:10:47.010
The result we want still is
that this should be 1/2 of

00:10:47.010 --> 00:10:48.180
this, but let's prove it.

00:10:48.180 --> 00:10:57.560
Let's draw a diameter
just like that.

00:10:57.560 --> 00:11:09.490
Let me call this angle right
here, let me call that si 2.

00:11:09.490 --> 00:11:14.770
And it is subtending this arc
right there -- let me do

00:11:14.770 --> 00:11:16.140
that in a darker color.

00:11:16.140 --> 00:11:19.770
It is subtending this
arc right there.

00:11:19.770 --> 00:11:22.360
So the central angle that
subtends that same arc,

00:11:22.360 --> 00:11:25.300
let me call that theta 2.

00:11:25.300 --> 00:11:30.890
Now, we know from the earlier
part of this video that si

00:11:30.890 --> 00:11:37.600
2 is going to be equal
to 1/2 theta 2, right?

00:11:37.600 --> 00:11:40.760
They share -- the
diameter is right there.

00:11:40.760 --> 00:11:44.300
The diameter is one of the
cords that forms the angle.

00:11:44.300 --> 00:11:47.500
So si 2 is going to be
equal to 1/2 theta 2.

00:11:50.140 --> 00:11:52.810
This is exactly what we've been
doing in the last video, right?

00:11:52.810 --> 00:11:55.430
This is an inscribed angle.

00:11:55.430 --> 00:11:59.550
One of the cords that define
is sitting on the diameter.

00:11:59.550 --> 00:12:02.740
So this is going to be 1/2 of
this angle, of the central

00:12:02.740 --> 00:12:05.980
angle that subtends
the same arc.

00:12:05.980 --> 00:12:09.000
Now, let's look at
this larger angle.

00:12:09.000 --> 00:12:11.680
This larger angle right here.

00:12:11.680 --> 00:12:14.240
Si 1 plus si 2.

00:12:14.240 --> 00:12:22.720
Right, that larger angle
is si 1 plus si 2.

00:12:22.720 --> 00:12:28.680
Once again, this subtends this
entire arc right here, and it

00:12:28.680 --> 00:12:32.100
has a diameter as one of the
cords that defines

00:12:32.100 --> 00:12:34.310
this huge angle.

00:12:34.310 --> 00:12:37.380
So this is going to be 1/2
of the central angle that

00:12:37.380 --> 00:12:38.580
subtends the same arc.

00:12:38.580 --> 00:12:42.270
We're just using what we've
already shown in this video.

00:12:42.270 --> 00:12:47.390
So this is going to be equal to
1/2 of this huge central angle

00:12:47.390 --> 00:12:51.370
of theta 1 plus theta 2.

00:12:54.310 --> 00:12:56.530
So far we've just used
everything that we've learned

00:12:56.530 --> 00:12:58.160
earlier in this video.

00:12:58.160 --> 00:13:03.160
Now, we already know that si
2 is equal to 1/2 theta 2.

00:13:03.160 --> 00:13:05.630
So let me make that
substitution.

00:13:05.630 --> 00:13:07.030
This is equal to that.

00:13:07.030 --> 00:13:15.330
So we can say that si 1 plus
-- instead of si 2 I'll write

00:13:15.330 --> 00:13:26.630
1/2 theta 2 is equal to 1/2
theta 1 plus 1/2 theta 2.

00:13:30.340 --> 00:13:34.020
We can subtract 1/2 theta
2 from both sides, and

00:13:34.020 --> 00:13:35.740
we get our result.

00:13:35.740 --> 00:13:40.900
Si 1 is equal to 1/2 theta one.

00:13:40.900 --> 00:13:41.970
And now we're done.

00:13:41.970 --> 00:13:44.990
We have proven the situation
that the inscribed angle is

00:13:44.990 --> 00:13:50.680
always 1/2 of the central angle
that subtends the same arc,

00:13:50.680 --> 00:13:53.980
regardless of whether the
center of the circle is inside

00:13:53.980 --> 00:13:58.990
of the angle, outside of the
angle, whether we have a

00:13:58.990 --> 00:14:00.950
diameter on one side.

00:14:00.950 --> 00:14:05.860
So any other angle can be
constructed as a sum of

00:14:05.860 --> 00:14:08.300
any or all of these that
we've already done.

00:14:08.300 --> 00:14:10.190
So hopefully you found this
useful and now we can actually

00:14:10.190 --> 00:14:14.630
build on this result to do some
more interesting

00:14:14.630 --> 00:14:16.460
geometry proofs.