Perpendicular radius bisects chord

WEBVTT
Kind: captions
Language: en

00:00:00.524 --> 00:00:01.940
In a previous
video, we've already

00:00:01.940 --> 00:00:04.930
shown that if we have some
circle centered at O right

00:00:04.930 --> 00:00:07.570
over here, and that
if OD is a radius,

00:00:07.570 --> 00:00:12.050
and it's a radius that
bisects chord AC-- so bisects

00:00:12.050 --> 00:00:14.370
means that it kind of
splits it in two, that AB

00:00:14.370 --> 00:00:17.450
is equal to BC-- we've
proven in a previous video

00:00:17.450 --> 00:00:21.480
that OD will be
perpendicular to AC.

00:00:21.480 --> 00:00:25.001
So we've proven that we can
assume that is perpendicular.

00:00:25.001 --> 00:00:26.750
And that video, if you
want to look it up,

00:00:26.750 --> 00:00:28.320
and you might want to
prove it yourself just

00:00:28.320 --> 00:00:29.944
out of interest, but
the video-- if you

00:00:29.944 --> 00:00:31.650
do a search on Khan
Academy for radius

00:00:31.650 --> 00:00:33.840
is perpendicular
to a chord, you'll

00:00:33.840 --> 00:00:35.520
hopefully find
the proof of that.

00:00:35.520 --> 00:00:37.750
What I want to do in this
video is go the other way.

00:00:37.750 --> 00:00:40.295
If we know that OD is
a radius and that it

00:00:40.295 --> 00:00:43.460
is perpendicular to chord AC,
what I want to do in this video

00:00:43.460 --> 00:00:45.487
is prove that it's bisecting it.

00:00:45.487 --> 00:00:47.320
So we're not assuming
that it's bisecting it

00:00:47.320 --> 00:00:48.350
in this video right over here.

00:00:48.350 --> 00:00:50.060
We're just assuming
that it's perpendicular.

00:00:50.060 --> 00:00:51.710
So we're essentially
going to go the other way.

00:00:51.710 --> 00:00:53.835
Here, we started with the
fact that it bisected it,

00:00:53.835 --> 00:00:55.940
and we established that
they were perpendicular.

00:00:55.940 --> 00:00:57.247
That was in the previous video.

00:00:57.247 --> 00:00:59.080
Now we're going to start
with the assumption

00:00:59.080 --> 00:01:02.807
that they're perpendicular and
then prove that they bisect.

00:01:02.807 --> 00:01:04.640
And just like we did
in that previous proof,

00:01:04.640 --> 00:01:06.610
we'll set up some
triangles here since we

00:01:06.610 --> 00:01:08.390
know a lot about triangles now.

00:01:08.390 --> 00:01:10.640
And we'll set up the
triangles by drawing two more

00:01:10.640 --> 00:01:16.450
radii, radius OC and radius OA.

00:01:16.450 --> 00:01:19.020
And that's useful
for us because we

00:01:19.020 --> 00:01:21.250
know that they're both
radii for the same circle.

00:01:21.250 --> 00:01:22.720
So they have to be
the same length.

00:01:22.720 --> 00:01:24.350
The radius doesn't
change on a circle.

00:01:24.350 --> 00:01:26.040
So those two things
are the same length.

00:01:26.040 --> 00:01:29.110
And you might already
see where this is going.

00:01:29.110 --> 00:01:30.926
Let me label this point here.

00:01:30.926 --> 00:01:32.550
Let me call this M
because we're hoping

00:01:32.550 --> 00:01:34.760
that ends up being
the midpoint of AC.

00:01:34.760 --> 00:01:37.850
Triangle AMO is
a right triangle.

00:01:37.850 --> 00:01:38.930
This is its hypotenuse.

00:01:38.930 --> 00:01:40.880
AO is its hypotenuse.

00:01:40.880 --> 00:01:43.020
Triangle OMC is
a right triangle,

00:01:43.020 --> 00:01:44.940
and this is its hypotenuse
right over there.

00:01:44.940 --> 00:01:46.930
And so already showed
that the hypotenuses

00:01:46.930 --> 00:01:52.830
have the same length, and both
of these right triangles share

00:01:52.830 --> 00:01:55.190
segment or side OM.

00:01:55.190 --> 00:01:58.580
So OM is clearly
equal to itself.

00:01:58.580 --> 00:02:00.990
And in a previous video,
not the same video

00:02:00.990 --> 00:02:02.822
where we explained this thing.

00:02:02.822 --> 00:02:05.030
In a previous video where
we talk about-- and I think

00:02:05.030 --> 00:02:05.844
you can look it up.

00:02:05.844 --> 00:02:07.510
I think the video is
called "More on why

00:02:07.510 --> 00:02:09.259
SSA is not a postulate."

00:02:09.259 --> 00:02:12.340
In that video, we say that
SSA is not a postulate.

00:02:12.340 --> 00:02:15.280
So SSA, not a
congruency postulate.

00:02:18.870 --> 00:02:21.220
But we did establish
in that video

00:02:21.220 --> 00:02:26.530
that RSH is a
congruency postulate.

00:02:26.530 --> 00:02:30.830
And RSH tells us that if we
have a right triangle-- that's

00:02:30.830 --> 00:02:33.620
where the R comes from-- if
we have a right triangle,

00:02:33.620 --> 00:02:35.760
and we have one of the
sides are congruent,

00:02:35.760 --> 00:02:38.170
and the hypotenuse
is congruent, then

00:02:38.170 --> 00:02:40.774
we have two congruent triangles.

00:02:40.774 --> 00:02:42.440
And if you look at
this right over here,

00:02:42.440 --> 00:02:43.640
we have two right triangles.

00:02:43.640 --> 00:02:45.210
AMO is a right triangle.

00:02:45.210 --> 00:02:46.940
CMO is a right triangle.

00:02:46.940 --> 00:02:49.904
They have one leg that's
congruent, right over here, MO,

00:02:49.904 --> 00:02:51.320
and then both of
their hypotenuses

00:02:51.320 --> 00:02:52.870
are congruent to each other.

00:02:52.870 --> 00:03:01.590
So, by RSH, we know
that triangle AMO

00:03:01.590 --> 00:03:11.832
is congruent to triangle
CMO by the RSH postulate.

00:03:11.832 --> 00:03:13.540
And so if we know that
they're congruent,

00:03:13.540 --> 00:03:16.530
then their corresponding
sides have to be congruent.

00:03:16.530 --> 00:03:19.660
So based on that,
we then know that AM

00:03:19.660 --> 00:03:20.980
is a corresponding side.

00:03:20.980 --> 00:03:23.130
Let me do that in
a different color.

00:03:23.130 --> 00:03:28.300
AM is a corresponding
side to MC.

00:03:28.300 --> 00:03:34.400
So we know that AM
must be equal to MC

00:03:34.400 --> 00:03:37.100
because they're
corresponding sides.

00:03:37.100 --> 00:03:38.350
These are corresponding sides.

00:03:38.350 --> 00:03:40.380
So congruency implies
that these are equal.

00:03:40.380 --> 00:03:45.190
And if those are equal, then we
know that OD is bisecting AC.

00:03:45.190 --> 00:03:47.190
So we've established
what we need to do.

00:03:47.190 --> 00:03:49.944
Another way that we could
have proven it without RSH,

00:03:49.944 --> 00:03:51.985
is just straight up with
the Pythagorean theorem.

00:03:55.990 --> 00:03:58.630
We already know, just by
setting up these two radii right

00:03:58.630 --> 00:04:02.959
over here, we know that OA--
so we draw a little line here.

00:04:02.959 --> 00:04:05.000
This is another way that
we could have proven it.

00:04:05.000 --> 00:04:09.270
We already know that
OA is equal to OC.

00:04:09.270 --> 00:04:11.820
And we also know that
OM is equal to itself.

00:04:11.820 --> 00:04:15.220
OM is clearly equal to OM.

00:04:15.220 --> 00:04:16.670
It's equal to itself.

00:04:16.670 --> 00:04:18.610
And we also know from
the Pythagorean theorem

00:04:18.610 --> 00:04:21.370
that AM-- do this
in a new color.

00:04:21.370 --> 00:04:35.570
We know that AM squared plus OM
squared is equal to OA squared.

00:04:35.570 --> 00:04:39.750
The length of the
two legs squared

00:04:39.750 --> 00:04:42.990
summed up is equal to the length
of the hypotenuse squared.

00:04:42.990 --> 00:04:45.830
So we know that for the left
triangle right over here,

00:04:45.830 --> 00:04:47.590
for AMO.

00:04:47.590 --> 00:04:50.580
And we can set up the
same relationship for CMO.

00:04:50.580 --> 00:04:53.110
We know in CMO
that-- and I'm going

00:04:53.110 --> 00:05:01.670
to try to do it corresponding--
that CM squared plus OM

00:05:01.670 --> 00:05:07.740
squared is equal to OC squared.

00:05:11.530 --> 00:05:13.000
Now we know a few things.

00:05:13.000 --> 00:05:16.040
We know that OA is equal to OC.

00:05:16.040 --> 00:05:18.780
So, for example, right over
here, where we have OA squared,

00:05:18.780 --> 00:05:20.195
we could replace this with OC.

00:05:25.070 --> 00:05:27.340
And then you can already
see where this is going.

00:05:27.340 --> 00:05:29.930
You can already see that CM and
AM are going to be the same.

00:05:29.930 --> 00:05:32.320
But if you want to do it a
little bit more formally,

00:05:32.320 --> 00:05:35.040
you can subtract OM squared from
both sides of this equation,

00:05:35.040 --> 00:05:41.350
and you'd get AM squared
is equal to OC squared--

00:05:41.350 --> 00:05:45.640
I've replaced this with OC
squared-- minus OM squared.

00:05:45.640 --> 00:05:48.090
So that's on the left-hand
side right over here.

00:05:48.090 --> 00:05:49.980
And then on the
right-hand side, if we

00:05:49.980 --> 00:05:53.450
subtract OM squared from both
sides, we have CM squared.

00:05:53.450 --> 00:06:00.232
CM squared is equal to OC
squared minus OM squared.

00:06:00.232 --> 00:06:01.940
And then we could take
the principal root

00:06:01.940 --> 00:06:03.610
of both sides of this
because we really

00:06:03.610 --> 00:06:05.410
care about the positive
root because we

00:06:05.410 --> 00:06:07.100
don't want to have
negative distances.

00:06:07.100 --> 00:06:09.100
So if you take the principal
root of both sides,

00:06:09.100 --> 00:06:12.430
this becomes AM is equal to
the principal root of that.

00:06:12.430 --> 00:06:16.070
And we also get that CM is equal
to the principal root of that.

00:06:16.070 --> 00:06:18.380
Well, these two
quantities are the same.

00:06:18.380 --> 00:06:20.310
So AM must be equal to CM.

00:06:23.810 --> 00:06:25.880
They both equal to this
quantity right here.

00:06:25.880 --> 00:06:28.680
So AM is equal to CM, and
it might be a bisector.

00:06:28.680 --> 00:06:30.370
And this is really common sense.

00:06:30.370 --> 00:06:35.274
If you know that two sides of
two different right triangles

00:06:35.274 --> 00:06:36.940
are going to be
congruent to each other,

00:06:36.940 --> 00:06:38.690
you can always use the
Pythagorean theorem

00:06:38.690 --> 00:06:39.720
to get the third side.

00:06:39.720 --> 00:06:42.019
And that third side is
uniquely constrained

00:06:42.019 --> 00:06:43.560
by the length of
the other two sides,

00:06:43.560 --> 00:06:45.049
because it's a right triangle.

00:06:45.049 --> 00:06:46.840
And so these are all
ways of really getting

00:06:46.840 --> 00:06:47.640
at the same thing.

00:06:47.640 --> 00:06:50.240
But now we can feel
pretty good about the fact

00:06:50.240 --> 00:06:53.480
that if OD is
perpendicular here,

00:06:53.480 --> 00:06:57.070
then it's definitely going
to bisect this chord.


Area of Koch snowflake (part 2) - advanced

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.345
In the last video we got as far as figuring out that the area of this Koch snowflake

00:00:04.345 --> 00:00:10.004
This thing that has an infinite perimeter, can be expressed as this infinite sum over here

00:00:10.004 --> 00:00:15.206
So our job in this video is to try to simplify this, and hopefully get a finite value

00:00:15.206 --> 00:00:19.721
Let's do our best to actually simplify this thing right over here

00:00:19.721 --> 00:00:24.355
So the easiest part of this thing to simplify is this right over here

00:00:24.355 --> 00:00:25.806
So let's just focus on that

00:00:25.806 --> 00:00:29.426
Then if we can get a value for this part that I am bracketing off

00:00:29.426 --> 00:00:32.731
Then we can just place that value here and simplify the rest of it

00:00:32.731 --> 00:00:42.112
So what I've just bracketed off can be re-written as three times four ninths

00:00:42.112 --> 00:00:45.974
plus four ninths squared

00:00:45.974 --> 00:00:51.774
plus four ninths to the third power

00:00:51.774 --> 00:00:53.791
And you can go on and on and on

00:00:53.791 --> 00:00:59.910
Plus four ninths to every other power, all the way through infinity!

00:00:59.910 --> 00:01:05.810
Lucky for us, there is a way to figure out this infinite (geometric) series

00:01:05.810 --> 00:01:11.395
There's a way to figure this out, and I've done several videos where we prove the general thing

00:01:11.395 --> 00:01:16.164
But I'll just do it by hand this time, just so that we don't have to resort to some magical formulas

00:01:16.164 --> 00:01:21.808
So let's say that we define some sum, this one over here (let's call it S)

00:01:21.808 --> 00:01:27.281
Let's say that S is equal to what we have in parentheses over here

00:01:27.281 --> 00:01:35.241
It's going to be equal to four ninths, plus four ninths squared, plus four ninths to the third

00:01:35.241 --> 00:01:38.748
all the way to infinity

00:01:38.748 --> 00:01:50.264
Now let's also say that we multiply S by four ninths

00:01:50.264 --> 00:01:56.299
What's four ninths S going to look like?

00:01:56.299 --> 00:02:00.051
So then, I'm just essentially multiplying every term here by four ninths

00:02:00.051 --> 00:02:03.404
So if I take this first term and multiply it by four ninths, what am I going to get?

00:02:03.404 --> 00:02:07.719
Well I'm going to get four ninths squared

00:02:07.719 --> 00:02:15.323
If I take the second term and multiply it by four ninths, I'm going to get four ninths to the third power

00:02:15.323 --> 00:02:17.283
And we are going to go all the way to infinity

00:02:17.283 --> 00:02:18.650
So this is interesting

00:02:18.650 --> 00:02:24.281
When I multiply four ninths times this I get all of the terms here except for this first four ninths

00:02:24.281 --> 00:02:30.633
Now, this is kind of the magic of how we can actually find the sum of an infinite geometric series

00:02:30.633 --> 00:02:37.173
We can subtract this term right over here (this pink line) from this green line

00:02:37.173 --> 00:02:41.805
If we do that, clearly this is equal to that and this is equal to that

00:02:41.805 --> 00:02:45.951
So if we subtract this from that its equivalent to subtracting the pink from the green

00:02:45.951 --> 00:02:58.715
So we get S minus four ninths S is equal to...

00:02:58.715 --> 00:03:04.880
Well, every other term, this guy minus this guy is going to cancel out

00:03:04.880 --> 00:03:09.066
And that's going to happen all the way to infinity

00:03:09.066 --> 00:03:17.299
and on the right hand side you're only going to be left with this four ninths over here

00:03:17.299 --> 00:03:25.787
Then this four ninths, we can (S is the same thing as nine over nine) write this as nine over nine S

00:03:25.787 --> 00:03:28.285
minus four ninths S is equal to four ninths

00:03:28.285 --> 00:03:32.714
So nine over nine minus four over nine of something gives us five over nine

00:03:32.714 --> 00:03:37.501
So this becomes five ninths S is equal to four ninths

00:03:37.501 --> 00:03:41.881
Then to solve for S (and this is kind of magical but it's actually quite logical)

00:03:41.881 --> 00:03:50.182
Multiply both sides times the inverse of this, so times nine fifths on both sides

00:03:50.182 --> 00:03:54.867
These guys cancel out, and we get S is equal to four fifths

00:03:54.867 --> 00:03:56.910
That's really neat!

00:03:56.910 --> 00:04:04.796
We've just shown that this whole thing over here is equal to four fifths

00:04:04.796 --> 00:04:09.320
So this entire bracket that we did over here is equal to three times four fifths

00:04:09.320 --> 00:04:19.918
This entire bracket is equal to twelve over five

00:04:19.918 --> 00:04:24.453
Now let's go to our original expression, so we don't lose track of what we are doing

00:04:24.453 --> 00:04:29.886
We have this square root of three times S squared over sixteen

00:04:29.886 --> 00:04:42.870
Then we have this four here plus the entire thing in brackets (which simplified to twelve fifths)

00:04:42.870 --> 00:04:52.913
Just to add these two together we can rewrite four as twenty fifths

00:04:52.913 --> 00:04:58.105
and then twenty over five plus twelve over five is thirty-two over five

00:04:58.105 --> 00:05:05.703
Let me write that down over here

00:05:05.703 --> 00:05:07.907
This is the home stretch now, this is very exciting!

00:05:07.907 --> 00:05:13.279
We are about to find the finite area of something that has an infinite perimeter!

00:05:13.279 --> 00:05:21.233
So it's going to be square root of three times S squared over sixteen

00:05:21.233 --> 00:05:27.036
Times thirty-two over five

00:05:27.036 --> 00:05:35.031
We can divide the thirty-two in the numerator by the sixteen there, which gives two

00:05:35.031 --> 00:05:42.562
We are left with the area of a Koch snowflake

00:05:42.562 --> 00:05:49.928
(where the initial equalateral triangle that we started with has each of its sides as length S)

00:05:49.928 --> 00:06:10.712
is, two times the square root of three times S squared all over five

00:06:10.712 --> 00:06:18.608
For example, if that first equalateral triangle had a side-length of one

00:06:18.608 --> 00:06:28.181
Then the area of this crazy thing (that has an infinite perimeter) would just be two square roots of three over five

00:06:28.181 --> 99:59:59.999
Anyway, I think that's kind of cool


Area of Koch snowflake (part 1) - advanced

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:03.310
We now know how to find the
area of an equilateral triangle.

00:00:03.310 --> 00:00:04.726
What I want to do
in this video is

00:00:04.726 --> 00:00:06.950
attempt to find the
area of a-- and I

00:00:06.950 --> 00:00:10.460
know I'm mispronouncing it-- a
"kawch," or "coach" snowflake.

00:00:10.460 --> 00:00:11.960
And the way you
construct one is you

00:00:11.960 --> 00:00:13.857
start with an
equilateral triangle.

00:00:13.857 --> 00:00:16.190
And then on each of the sides
you split them into thirds

00:00:16.190 --> 00:00:18.690
and in the middle third you
put another smaller equilateral

00:00:18.690 --> 00:00:19.827
triangle.

00:00:19.827 --> 00:00:20.910
And that's after one pass.

00:00:20.910 --> 00:00:23.409
And on the next pass, you do
that for all of the sides here.

00:00:23.409 --> 00:00:26.480
So a little one here, here,
here, here, here, here, here,

00:00:26.480 --> 00:00:27.505
here, here.

00:00:27.505 --> 00:00:28.880
I think you get
the general idea.

00:00:28.880 --> 00:00:29.790
That's the next pass.

00:00:29.790 --> 00:00:32.090
And on the next pass you do
it for all of these sides.

00:00:32.090 --> 00:00:33.230
And what's really
neat about this--

00:00:33.230 --> 00:00:34.840
and we showed this
in a previous video--

00:00:34.840 --> 00:00:37.330
is that you have a figure here
that has an infinite perimeter.

00:00:37.330 --> 00:00:39.329
But we're about to see
in this video it actually

00:00:39.329 --> 00:00:44.080
has a finite area, which is kind
of interesting to think about.

00:00:44.080 --> 00:00:47.680
So let's start with a
clean equilateral triangle

00:00:47.680 --> 00:00:49.510
right over here.

00:00:49.510 --> 00:00:53.140
We're going to assume that each
of the sides have length s.

00:00:53.140 --> 00:00:55.590
So it's going to be a
clean equilateral triangle.

00:00:55.590 --> 00:00:58.380
Each of the sides-- let me
draw that a little bit neater.

00:00:58.380 --> 00:01:00.154
Each of the sides have length s.

00:01:00.154 --> 00:01:01.570
And so what I'm
going to do is I'm

00:01:01.570 --> 00:01:02.986
going to keep track
of two things.

00:01:02.986 --> 00:01:07.120
I'm going to keep track of the
sides on this triangle as it

00:01:07.120 --> 00:01:08.152
turns into a snowflake .

00:01:08.152 --> 00:01:10.110
I'm going to keep track
of the number of sides,

00:01:10.110 --> 00:01:11.734
and I'm going to keep
track of the area

00:01:11.734 --> 00:01:15.410
after each pass of adding
more smaller triangles.

00:01:15.410 --> 00:01:18.572
So this is going to be
our count of the area.

00:01:18.572 --> 00:01:21.030
Actually, let me give myself
a little bit more real estate,

00:01:21.030 --> 00:01:23.238
just because I have a feeling
I might need to use it.

00:01:23.238 --> 00:01:24.846
So this is the sides.

00:01:24.846 --> 00:01:25.970
I'm going to write up here.

00:01:25.970 --> 00:01:30.530
And then this is our running
count of the area down here.

00:01:30.530 --> 00:01:33.425
So right when we
start, we have 3 sides.

00:01:36.020 --> 00:01:38.622
And our area, we already
figured out in a previous video,

00:01:38.622 --> 00:01:40.580
is going to be-- if we
assume each of the sides

00:01:40.580 --> 00:01:47.970
are length s-- is going to
be square root of 3s squared

00:01:47.970 --> 00:01:48.970
over 4.

00:01:48.970 --> 00:01:49.670
Fair enough.

00:01:49.670 --> 00:01:52.140
That was just a simple
equilateral triangle.

00:01:52.140 --> 00:01:54.430
Now we're going to take
each of these sides,

00:01:54.430 --> 00:02:00.920
divide them into thirds, and
then in that middle third,

00:02:00.920 --> 00:02:03.285
we're going to add another
smaller equilateral triangle.

00:02:03.285 --> 00:02:06.199
So it'll look like that on
that side right over there.

00:02:06.199 --> 00:02:08.740
I want you to think about what
we're doing to each side right

00:02:08.740 --> 00:02:09.239
here.

00:02:09.239 --> 00:02:11.584
So before I did this,
this was just one side.

00:02:11.584 --> 00:02:12.750
Then I split it into thirds.

00:02:12.750 --> 00:02:15.320
And that middle
third, I essentially

00:02:15.320 --> 00:02:16.660
put two sides in there.

00:02:16.660 --> 00:02:18.290
I put an equilateral triangle.

00:02:18.290 --> 00:02:24.210
So one side is now turned into
one, two, three, four sides.

00:02:24.210 --> 00:02:28.400
So every time we do a pass
of making the snowflake more

00:02:28.400 --> 00:02:31.260
intricate, each side will
turn into four sides.

00:02:31.260 --> 00:02:35.670
So you could imagine if we
do this on all three sides,

00:02:35.670 --> 00:02:39.560
we have 4 times 3, which
is now twelve sides.

00:02:39.560 --> 00:02:43.100
So if you multiply
this times 4, this

00:02:43.100 --> 00:02:44.706
gets us to twelve sides now.

00:02:44.706 --> 00:02:46.330
We can count them
out just to make sure

00:02:46.330 --> 00:02:47.820
where our logic is correct.

00:02:47.820 --> 00:02:55.870
1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12 sides.

00:02:55.870 --> 00:02:57.740
And now what is the area, now?

00:02:57.740 --> 00:03:00.430
Well it's going to be the
area of our original yellow

00:03:00.430 --> 00:03:04.030
equilateral triangle, plus the
area of each of these smaller

00:03:04.030 --> 00:03:04.530
ones.

00:03:04.530 --> 00:03:06.720
And what's the area of
each of these smaller ones?

00:03:06.720 --> 00:03:08.670
Well first of all, we
have three of them.

00:03:08.670 --> 00:03:11.830
There are three of each
of these smaller ones

00:03:11.830 --> 00:03:13.750
and then we use the
formula for the area

00:03:13.750 --> 00:03:15.660
of an equilateral
triangle again.

00:03:15.660 --> 00:03:19.950
So it's going to be square
root of 3 times s squared.

00:03:19.950 --> 00:03:22.380
But now the length of
each of the sides of each

00:03:22.380 --> 00:03:24.450
of these smaller
equilateral triangles--

00:03:24.450 --> 00:03:26.620
they aren't s anymore,
they are s over 3.

00:03:26.620 --> 00:03:30.380
Remember, this length right
over here is s over 3.

00:03:30.380 --> 00:03:32.840
So this is going to
be s over 3, as well.

00:03:32.840 --> 00:03:35.420
Every pass, the sides of
the equilateral triangle

00:03:35.420 --> 00:03:38.060
become 1/3 of the previous pass.

00:03:38.060 --> 00:03:39.970
So it's not going to
be s squared anymore.

00:03:39.970 --> 00:03:43.420
It's going to be
s over 3 squared.

00:03:43.420 --> 00:03:47.570
And then all of that over 4.

00:03:47.570 --> 00:03:49.330
And then let's do another pass.

00:03:49.330 --> 00:03:52.620
So we're going to add these
triangles right over here.

00:03:52.620 --> 00:03:56.946
Going to add these
right over there.

00:03:56.946 --> 00:03:58.820
And this is the last
pass where I'll actually

00:03:58.820 --> 00:04:01.980
attempt to draw all of
the triangles over there.

00:04:01.980 --> 00:04:04.460
So how many sides
am I going to have,

00:04:04.460 --> 00:04:06.570
first of all, after
I do another pass?

00:04:06.570 --> 00:04:08.690
Well, the previous
pass, I had 12 sides.

00:04:08.690 --> 00:04:11.560
Each of those 12 sides are now
going to turn to 4 new sides

00:04:11.560 --> 00:04:13.640
when I add these little
orange bumps there.

00:04:13.640 --> 00:04:15.925
So I'm going to multiply
it times 4 again.

00:04:15.925 --> 00:04:17.300
I'm going to
multiply it times 4.

00:04:17.300 --> 00:04:22.170
So now I'm going
to have 48 sides.

00:04:22.170 --> 00:04:23.380
And how many new triangles?

00:04:23.380 --> 00:04:23.980
So what's the area?

00:04:23.980 --> 00:04:25.570
Well it's going to
be the yellow area

00:04:25.570 --> 00:04:28.230
plus the blue area
plus the orange area.

00:04:28.230 --> 00:04:30.820
So how many new orange
triangles do I have.?

00:04:30.820 --> 00:04:32.500
Well I'm adding a
new orange triangle

00:04:32.500 --> 00:04:34.910
to each of the sides
for the previous pass.

00:04:34.910 --> 00:04:37.260
In the previous
pass I had 12 sides.

00:04:37.260 --> 00:04:40.075
So now I'm going to add
12 orange triangles.

00:04:42.869 --> 00:04:44.160
And actually let me write that.

00:04:44.160 --> 00:04:46.420
I'll just write 12
orange triangles.

00:04:46.420 --> 00:04:49.420
But it's really I just
multiplied it times 4.

00:04:49.420 --> 00:04:52.980
And then I'm going to have
times the square root of 3.

00:04:52.980 --> 00:04:56.020
And now this isn't going
to be s over 3 any more.

00:04:56.020 --> 00:04:57.485
These are now going
to be s over 9.

00:04:57.485 --> 00:05:01.120
These have 1/3 of the dimensions
of these blue triangle.

00:05:01.120 --> 00:05:04.130
So this is going to be s over 9.

00:05:04.130 --> 00:05:08.979
s over 9 squared over 4.

00:05:08.979 --> 00:05:11.020
And so I think you might
start to see the pattern

00:05:11.020 --> 00:05:13.535
building if we do another
pass after this one.

00:05:13.535 --> 00:05:15.100
Move to the right a little bit.

00:05:15.100 --> 00:05:16.450
What will that look like?

00:05:16.450 --> 00:05:19.355
Let me do this in a different
color that I haven't used yet.

00:05:19.355 --> 00:05:21.540
Let me see I haven't
used this pink, yet.

00:05:21.540 --> 00:05:25.270
So now I'm going to have the
previous number of sides,

00:05:25.270 --> 00:05:28.000
that's my number of
new triangles, 48

00:05:28.000 --> 00:05:33.650
times the square root
of 3 times s over-- now

00:05:33.650 --> 00:05:38.640
these are going to be even
1/3 of this-- s over 27

00:05:38.640 --> 00:05:39.970
to the second power.

00:05:39.970 --> 00:05:41.720
All of that over 4.

00:05:41.720 --> 00:05:44.170
And I'm going to keep adding
an infinite number of terms

00:05:44.170 --> 00:05:47.400
of this to get the area
of a true koch snowflake.

00:05:47.400 --> 00:05:50.510
So I'm just going to keep doing
this over, and over again.

00:05:50.510 --> 00:05:54.180
So the trick really is
finding this infinite sum

00:05:54.180 --> 00:05:57.920
and seeing if we get a
finite number over here.

00:05:57.920 --> 00:06:00.460
So the first thing I want to
do, just to simplify, well,

00:06:00.460 --> 00:06:04.960
let me just rewrite it a little
bit differently over here.

00:06:04.960 --> 00:06:07.686
So the first thing
that might be obvious

00:06:07.686 --> 00:06:09.310
is that we can factor
out a square root

00:06:09.310 --> 00:06:11.310
of 3s squared over 4.

00:06:11.310 --> 00:06:13.400
So let me factor that out.

00:06:13.400 --> 00:06:16.750
So if we factor out
a square root of 3s

00:06:16.750 --> 00:06:19.680
squared over 4 from
all of the terms.

00:06:19.680 --> 00:06:23.120
Then this term right over
here will become a 1.

00:06:23.120 --> 00:06:26.324
This term right over here
is going to become a 3.

00:06:26.324 --> 00:06:28.240
Let's see, we factored
out a square root of 3.

00:06:28.240 --> 00:06:29.790
We factor out a four.

00:06:29.790 --> 00:06:32.050
And we factored
out the s squared.

00:06:32.050 --> 00:06:34.150
We factor out only
the s squared.

00:06:34.150 --> 00:06:42.102
So now it's going to have
plus 3 times 1/3 squared.

00:06:42.102 --> 00:06:43.310
That's all we have left here.

00:06:43.310 --> 00:06:44.280
We have the 1/3 squared.

00:06:44.280 --> 00:06:45.279
And then we have this 3.

00:06:45.279 --> 00:06:47.150
And I'm not simplifying
this on purpose,

00:06:47.150 --> 00:06:49.110
so that we see a pattern emerge.

00:06:49.110 --> 00:06:53.490
And then this next term, right
over here, plus-- so this 12

00:06:53.490 --> 00:06:55.000
is still going to be there.

00:06:55.000 --> 00:06:56.970
But I'm going to write
that as 3 times 4.

00:07:00.124 --> 00:07:02.290
Let's see, we're factoring
out the square root of 3,

00:07:02.290 --> 00:07:06.100
we're factoring out the 4, we're
factoring out the s squared.

00:07:06.100 --> 00:07:08.290
And so we're going to
be left with 3 squared.

00:07:08.290 --> 00:07:10.840
That's what this is
down here, squared.

00:07:10.840 --> 00:07:16.230
So this is 1/3 squared.

00:07:16.230 --> 00:07:18.360
And then that, squared.

00:07:18.360 --> 00:07:20.870
So that's what we're left
with, with that orange term.

00:07:20.870 --> 00:07:22.570
And then we go to the pink term.

00:07:22.570 --> 00:07:24.250
This pink term.

00:07:24.250 --> 00:07:26.620
48 is just 3 times 4 times 4.

00:07:26.620 --> 00:07:28.540
Three times 4.

00:07:28.540 --> 00:07:30.952
I'll write 4 squared here,
because each time we're

00:07:30.952 --> 00:07:32.410
going to multiply
it times 4 again.

00:07:32.410 --> 00:07:34.285
So the next one's going
to be 4 to the third,

00:07:34.285 --> 00:07:37.870
because we're really-- each
side turns into four sides.

00:07:37.870 --> 00:07:39.460
That's where that came from.

00:07:39.460 --> 00:07:42.120
4 squared, we're factoring
out the square root of 3,

00:07:42.120 --> 00:07:44.130
we're factoring
out the four, we're

00:07:44.130 --> 00:07:45.750
factoring out the s squared.

00:07:45.750 --> 00:07:49.380
And all we're left is 1 over
3 to the third power, squared.

00:07:49.380 --> 00:07:54.200
So times 1 over 3 to the
third power, squared.

00:07:54.200 --> 00:07:56.560
And we're just going to keep
going like that forever.

00:08:00.080 --> 00:08:03.820
So on each step, we're
multiplying by 4,

00:08:03.820 --> 00:08:06.620
and we're also multiplying--
I guess we say,

00:08:06.620 --> 00:08:08.420
the power of this
4 is incrementing.

00:08:08.420 --> 00:08:11.310
So there's really 4 to
the 0-th power here.

00:08:11.310 --> 00:08:13.510
We have a 1 here you
can imagine, implicitly.

00:08:13.510 --> 00:08:15.600
The 4 to the first
power, 4 squared,

00:08:15.600 --> 00:08:17.770
then it'll go 4 to the third.

00:08:17.770 --> 00:08:19.240
This power is also
incrementing-- 3

00:08:19.240 --> 00:08:21.220
to the first, 3 to the
second, 3 to the third.

00:08:21.220 --> 00:08:24.900
But we see that this power
is always one more than that.

00:08:24.900 --> 00:08:28.270
And it'll be much easier to
calculate this infinite--

00:08:28.270 --> 00:08:30.840
what's going to turn into an
infinite-- geometric series,

00:08:30.840 --> 00:08:33.200
if those are actually
the same power.

00:08:33.200 --> 00:08:35.575
So what I want to do is I want
to increase the power of 4

00:08:35.575 --> 00:08:36.490
in all of those.

00:08:36.490 --> 00:08:39.230
But I can't just willy-nilly
multiply everything by 4.

00:08:39.230 --> 00:08:41.270
If I'm going to multiply
everything by 4,

00:08:41.270 --> 00:08:43.864
I also need to divide
everything by 4.

00:08:43.864 --> 00:08:46.030
So what I'm going to do in
this step right over here

00:08:46.030 --> 00:08:50.300
is I'm going to multiply
and divide everything by 4.

00:08:50.300 --> 00:08:52.980
So if we divide by 4, I
can do that on the outside.

00:08:52.980 --> 00:08:56.290
So let me multiply 1/4
times this right over here.

00:08:56.290 --> 00:08:57.882
And so I'm dividing
by 4 out here.

00:08:57.882 --> 00:08:59.590
And then I'm going to
multiply this by 4.

00:08:59.590 --> 00:09:01.048
And so I'm not
going to be changing

00:09:01.048 --> 00:09:04.490
the value of the actual thing.

00:09:04.490 --> 00:09:14.620
This is going to be 4 plus
3 times 4 plus 3 times 4

00:09:14.620 --> 00:09:15.520
to the third.

00:09:15.520 --> 00:09:18.830
And so what was cool about
this is now that the power of 4

00:09:18.830 --> 00:09:20.265
and the power of
this 3 down here

00:09:20.265 --> 00:09:21.635
are going to be the same power.

00:09:21.635 --> 00:09:23.010
But it still seems
a little weird

00:09:23.010 --> 00:09:24.670
because we're taking
this 1 over 3 squared

00:09:24.670 --> 00:09:25.795
and then we're squaring it.

00:09:28.490 --> 00:09:29.917
And here we just
have to realize--

00:09:29.917 --> 00:09:31.500
so this is always
going to be squared.

00:09:31.500 --> 00:09:33.350
And this is the thing
that's incrementing.

00:09:33.350 --> 00:09:38.760
But in general, if I
have 1 over 3 to the n,

00:09:38.760 --> 00:09:42.400
and I'm squaring it,
this is equal to 1 over 3

00:09:42.400 --> 00:09:45.340
to the 2n power.

00:09:45.340 --> 00:09:47.192
So I'm just multiplying it by 2.

00:09:47.192 --> 00:09:48.900
If I'm raising something
to the exponent,

00:09:48.900 --> 00:09:50.300
then raising that
to an exponent,

00:09:50.300 --> 00:09:54.400
that's just raising it to
the n times 2 exponent.

00:09:54.400 --> 00:09:58.830
And this is the exact same thing
as 1 over 3 squared, raised

00:09:58.830 --> 00:10:00.390
to the nth power.

00:10:00.390 --> 00:10:02.550
So we can actually switch
these two exponents

00:10:02.550 --> 00:10:04.172
in a very legitimate way.

00:10:04.172 --> 00:10:05.630
And then let me
rewrite everything.

00:10:05.630 --> 00:10:07.130
Because I don't want to
do too much on this one

00:10:07.130 --> 00:10:08.100
step right over here.

00:10:08.100 --> 00:10:10.010
So this part right
over here gives us

00:10:10.010 --> 00:10:14.260
square root of 3s
squared over 16.

00:10:14.260 --> 00:10:15.940
And then that's
going to be times--

00:10:15.940 --> 00:10:18.000
I'll open and close
the parentheses.

00:10:18.000 --> 00:10:21.070
So then we have 4 plus.

00:10:21.070 --> 00:10:25.890
Then in blue, I'll write 3
times 4 to the first power.

00:10:25.890 --> 00:10:30.720
And then I can rewrite
this as times 1/3.

00:10:30.720 --> 00:10:34.470
We could view this
is 1/3 squared.

00:10:34.470 --> 00:10:37.660
We could view this as 1 over
3 to the first power, squared,

00:10:37.660 --> 00:10:40.300
or we could view this as 1 over
3 squared to the first power.

00:10:40.300 --> 00:10:41.758
And I'm going to
write it that way.

00:10:41.758 --> 00:10:45.550
So times 1/9 to the first power.

00:10:45.550 --> 00:10:50.550
And then plus 3 times 4 squared.

00:10:50.550 --> 00:10:56.850
And then this we can write as,
times 1/9 to the second power.

00:10:56.850 --> 00:10:58.620
And then this one
we could write,

00:10:58.620 --> 00:11:05.380
plus 3 times 4 to
the third times--

00:11:05.380 --> 00:11:07.920
we could write this is 1
over 27 to the second power.

00:11:07.920 --> 00:11:10.780
But we could also write this
based on what we saw over here.

00:11:10.780 --> 00:11:13.330
We could write this as 1 over
3 squared to the third power,

00:11:13.330 --> 00:11:14.579
based on this right over here.

00:11:14.579 --> 00:11:15.630
Let me make this clear.

00:11:15.630 --> 00:11:18.640
1 over 3 to the third,
to the second power.

00:11:18.640 --> 00:11:23.180
This is the same thing as 1 over
3 squared to the third power.

00:11:23.180 --> 00:11:25.050
That's what we showed
right over here.

00:11:25.050 --> 00:11:30.630
So this is the equivalent
to 1/9 to the third power.

00:11:30.630 --> 00:11:33.500
And now we start to
see the pattern is

00:11:33.500 --> 00:11:35.090
starting to clean
up a little bit.

00:11:35.090 --> 00:11:36.550
And let me just
do one more step.

00:11:36.550 --> 00:11:39.250
And then we'll finish
this in the next video.

00:11:39.250 --> 00:11:44.610
So this is equal to square root
of 3s squared over 16 times

00:11:44.610 --> 00:11:53.910
4 plus 3 times, this is
4/9, plus the next term is

00:11:53.910 --> 00:11:59.010
three times 4/9 squared.

00:11:59.010 --> 00:12:06.759
And then we have plus 3 times
4 over 9 to the third power.

00:12:06.759 --> 00:12:08.550
And then we're just
going to keep going on,

00:12:08.550 --> 00:12:10.460
and on, and on, and
on taking the 3 times

00:12:10.460 --> 00:12:14.080
4/9 to the successively
larger, and larger powers.

00:12:14.080 --> 00:12:17.255
So this is what we have to find
the sum of to find our area.

00:12:17.255 --> 00:12:19.130
And we're going to do
that in the next video.

00:12:19.130 --> 00:12:21.046
And we're going to use
some of the tools we've

00:12:21.046 --> 00:12:23.180
used to find the sums of
infinite geometric series.

00:12:23.180 --> 00:12:25.236
But we're going to
re-do it in this video

00:12:25.236 --> 00:12:26.860
just so that you
don't have to remember

00:12:26.860 --> 00:12:29.990
that formula or that proof.


Area of an equilateral triangle

WEBVTT
Kind: captions
Language: en

00:00:00.597 --> 00:00:02.430
Let's say that this
triangle right over here

00:00:02.430 --> 00:00:04.850
is equilateral, which
means all of its sides

00:00:04.850 --> 00:00:05.730
have the same length.

00:00:05.730 --> 00:00:08.040
And let's say that
that length is s.

00:00:08.040 --> 00:00:09.790
What I want to do in
this video is come up

00:00:09.790 --> 00:00:12.270
with a way of
figuring out the area

00:00:12.270 --> 00:00:16.896
of this equilateral
triangle, as a function of s.

00:00:16.896 --> 00:00:18.520
And to do that, I'm
just going to split

00:00:18.520 --> 00:00:19.980
this equilateral in two.

00:00:19.980 --> 00:00:24.340
I'm just going to drop an
altitude from this top vertex

00:00:24.340 --> 00:00:25.590
right over here.

00:00:25.590 --> 00:00:28.490
This is going to be
perpendicular to the base.

00:00:28.490 --> 00:00:31.160
And it's also going to
bisect this top angle.

00:00:31.160 --> 00:00:33.497
So this angle is going to
be equal to that angle.

00:00:33.497 --> 00:00:35.080
And we showed all
of this in the video

00:00:35.080 --> 00:00:37.190
where we proved
the relationships

00:00:37.190 --> 00:00:40.730
between the sides of
a 30-60-90 triangle.

00:00:40.730 --> 00:00:43.410
Well, in a regular equilateral
triangle, all of the angles

00:00:43.410 --> 00:00:44.640
are 60 degrees.

00:00:44.640 --> 00:00:46.960
So this one right over here
is going to be 60 degrees,

00:00:46.960 --> 00:00:49.160
let me do that in
a different color.

00:00:49.160 --> 00:00:52.120
This one down here is
going to be 60 degrees.

00:00:52.120 --> 00:00:54.520
This one down here is
going to be 60 degrees.

00:00:54.520 --> 00:00:56.390
And then this one up
here is 60 degrees,

00:00:56.390 --> 00:00:57.610
but we just split it in two.

00:00:57.610 --> 00:01:01.270
So this angle is going
to be 30 degrees.

00:01:01.270 --> 00:01:03.489
And then this angle is
going to be 30 degrees.

00:01:03.489 --> 00:01:05.030
And then the other
thing that we know

00:01:05.030 --> 00:01:06.820
is that this altitude
right over here

00:01:06.820 --> 00:01:10.240
also will bisect
this side down here.

00:01:10.240 --> 00:01:13.340
So that this length is
equal to that length.

00:01:13.340 --> 00:01:15.590
And we showed all of this a
little bit more rigorously

00:01:15.590 --> 00:01:18.290
on that 30-60-90 triangle video.

00:01:18.290 --> 00:01:21.480
But what this tells us is well,
if this entire length was s,

00:01:21.480 --> 00:01:23.230
because all three sides
are going to be s,

00:01:23.230 --> 00:01:24.940
it's an equilateral
triangle, then

00:01:24.940 --> 00:01:27.320
each of these, so this
part right over here,

00:01:27.320 --> 00:01:29.610
is going to be s/2.

00:01:29.610 --> 00:01:31.970
And if this length
is s/2, we can

00:01:31.970 --> 00:01:34.750
use what we know about
30-60-90 triangles

00:01:34.750 --> 00:01:37.840
to figure out this
side right over here.

00:01:37.840 --> 00:01:40.807
So to figure out what
the actual altitude is.

00:01:40.807 --> 00:01:42.640
And the reason why I
care about the altitude

00:01:42.640 --> 00:01:44.870
is because the
area of a triangle

00:01:44.870 --> 00:01:50.280
is 1/2 times the base times the
height, or times the altitude.

00:01:50.280 --> 00:01:52.400
So this is s/2,
the shortest side.

00:01:52.400 --> 00:01:56.990
The side opposite the
30 degree angle is s/2.

00:01:56.990 --> 00:01:59.180
Then the side opposite
the 60 degree angle

00:01:59.180 --> 00:02:02.010
is going to be square
root of 3 times that.

00:02:02.010 --> 00:02:06.660
So it's going to be
square root of 3 s over 2.

00:02:06.660 --> 00:02:08.270
And we know that
because the ratio

00:02:08.270 --> 00:02:15.450
of the sides of a 30-60-90
triangle, if the side

00:02:15.450 --> 00:02:17.250
opposite the 30
degree side is 1,

00:02:17.250 --> 00:02:18.970
then the side opposite
the 60 degree side

00:02:18.970 --> 00:02:21.060
is going to be square
root of 3 times that.

00:02:21.060 --> 00:02:24.380
And the side opposite the 90
degree side, or the hypotenuse,

00:02:24.380 --> 00:02:26.350
is going to be 2 times that.

00:02:26.350 --> 00:02:28.390
So it's 1 to square
root of 3 to 2.

00:02:28.390 --> 00:02:30.610
So this is the shortest
side right over here.

00:02:30.610 --> 00:02:33.570
That's the side opposite
the 30 degree side.

00:02:33.570 --> 00:02:35.240
The side opposite
the 60 degree side

00:02:35.240 --> 00:02:37.310
is going to be square
root of 3 times this.

00:02:37.310 --> 00:02:39.780
So square root of 3 s over 2.

00:02:39.780 --> 00:02:42.710
So now we just
need to figure out

00:02:42.710 --> 00:02:49.290
what the area of this triangle
is, using area of our triangle

00:02:49.290 --> 00:02:52.190
is equal to 1/2 times the
base, times the height

00:02:52.190 --> 00:02:53.390
of the triangle.

00:02:53.390 --> 00:02:55.180
Well, what is the
base of the triangle?

00:02:55.180 --> 00:02:58.600
Well, the entire base of the
triangle right over here is s.

00:02:58.600 --> 00:03:00.130
So that is going to be s.

00:03:00.130 --> 00:03:02.059
And what is the height
of the triangle?

00:03:02.059 --> 00:03:03.350
Well, we just figured that out.

00:03:03.350 --> 00:03:08.200
It is the square root
of 3 times s over 2.

00:03:08.200 --> 00:03:10.971
And we just multiply
that out, and we get,

00:03:10.971 --> 00:03:13.470
let's see, in the numerator you
get a square root of 3 times

00:03:13.470 --> 00:03:14.830
1 times s times s.

00:03:14.830 --> 00:03:18.580
That is the square root
of 3 times s squared.

00:03:18.580 --> 00:03:20.800
And the denominator,
we have a 2 times a 2.

00:03:20.800 --> 00:03:22.570
All of that over 4.

00:03:22.570 --> 00:03:25.090
So for example, if you have
an equilateral triangle where

00:03:25.090 --> 00:03:27.546
each of the sides
was 1, then its area

00:03:27.546 --> 00:03:28.920
would be square
root of 3 over 4.

00:03:28.920 --> 00:03:30.250
If you had an equilateral
triangle where

00:03:30.250 --> 00:03:32.030
each of the sides
were 2, then this

00:03:32.030 --> 00:03:34.939
would be 2 squared over
4, which is just 1.

00:03:34.939 --> 00:03:36.480
So it would just be
square root of 3.

00:03:36.480 --> 00:03:38.790
So we just found out
a generalizable way

00:03:38.790 --> 00:03:42.190
to figure out the area of
an equilateral triangle.


30-60-90 triangle example problem

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:02.610
So we have this rectangle
right over here,

00:00:02.610 --> 00:00:04.902
and we're told that the
length of AB is equal to 1.

00:00:04.902 --> 00:00:06.360
So that's labeled
right over there.

00:00:06.360 --> 00:00:07.820
AB is equal to 1.

00:00:07.820 --> 00:00:13.440
And then they tell us that
BE and BD trisect angle ABC.

00:00:13.440 --> 00:00:20.330
So BE and BD trisect angle ABC.

00:00:20.330 --> 00:00:26.030
So trisect means dividing
it into 3 equal angles.

00:00:26.030 --> 00:00:28.920
So that means that this
angle is equal to this angle

00:00:28.920 --> 00:00:30.242
is equal to that angle.

00:00:30.242 --> 00:00:31.700
And what they want
us to figure out

00:00:31.700 --> 00:00:34.580
is, what is the perimeter
of triangle BED?

00:00:37.680 --> 00:00:39.400
So it's kind of
this middle triangle

00:00:39.400 --> 00:00:41.480
in the rectangle
right over here.

00:00:41.480 --> 00:00:43.857
So at first this seems
like a pretty hard problem,

00:00:43.857 --> 00:00:46.440
because you're like well, what
is the width of this rectangle.

00:00:46.440 --> 00:00:47.648
How can I even start on this?

00:00:47.648 --> 00:00:49.176
They've only given
us one side here.

00:00:49.176 --> 00:00:51.300
But they've actually given
us a lot of information,

00:00:51.300 --> 00:00:53.390
given that we do know
this is a rectangle.

00:00:53.390 --> 00:00:56.430
We have four sides, and
that we have four angles.

00:00:56.430 --> 00:00:58.640
The sides are all
parallel to each other

00:00:58.640 --> 00:01:00.559
and that the angles
are all 90 degrees.

00:01:00.559 --> 00:01:02.100
Which is more than
enough information

00:01:02.100 --> 00:01:04.207
to know that this is
definitely a rectangle.

00:01:04.207 --> 00:01:06.290
And so one thing we do
know is that opposite sides

00:01:06.290 --> 00:01:08.020
of a rectangle are
the same length.

00:01:08.020 --> 00:01:11.410
So if this side is 1, then
this side right over there

00:01:11.410 --> 00:01:12.670
is also 1.

00:01:12.670 --> 00:01:15.714
The other thing we know is
that this angle is trisected.

00:01:15.714 --> 00:01:17.630
Now we know what the
measure of this angle is.

00:01:17.630 --> 00:01:20.130
It was a right angle, it
was a 90 degree angle.

00:01:20.130 --> 00:01:22.720
So if it's divided into three
equal parts, that tells us

00:01:22.720 --> 00:01:25.320
that this angle right
over here is 30 degrees,

00:01:25.320 --> 00:01:27.490
this angle right over
here is 30 degrees,

00:01:27.490 --> 00:01:31.230
and then this angle right
over here is 30 degrees.

00:01:31.230 --> 00:01:32.730
And then we see
that we're dealing

00:01:32.730 --> 00:01:34.700
with a couple of
30-60-90 triangles.

00:01:34.700 --> 00:01:38.480
This one is 30, 90, so this
other side right over here

00:01:38.480 --> 00:01:39.720
needs to be 60 degrees.

00:01:43.310 --> 00:01:46.900
This triangle right over here,
you have 30, you have 90,

00:01:46.900 --> 00:01:49.030
so this one has
to be 60 degrees.

00:01:49.030 --> 00:01:53.134
They have to add up to
180, 30-60-90 triangle.

00:01:53.134 --> 00:01:55.550
And you can also figure out
the measures of this triangle,

00:01:55.550 --> 00:01:58.180
although it's not going
to be a right triangle.

00:01:58.180 --> 00:02:00.810
But knowing what we know
about 30-60-90 triangles,

00:02:00.810 --> 00:02:02.440
if we just have
one side of them,

00:02:02.440 --> 00:02:04.650
we can actually figure
out the other sides.

00:02:04.650 --> 00:02:08.060
So for example, here we
have the shortest side.

00:02:08.060 --> 00:02:11.800
We have the side opposite
of the 30 degree side.

00:02:11.800 --> 00:02:15.680
Now, if the 30 degree side
is 1, then the 60 degree side

00:02:15.680 --> 00:02:17.940
is going to be square
root of 3 times that.

00:02:17.940 --> 00:02:19.480
So this length
right over here is

00:02:19.480 --> 00:02:20.870
going to be square root of 3.

00:02:20.870 --> 00:02:23.290
And that's pretty useful
because we now just figured out

00:02:23.290 --> 00:02:26.600
the length of the entire
base of this rectangle

00:02:26.600 --> 00:02:27.480
right over there.

00:02:27.480 --> 00:02:30.380
And we just used our knowledge
of 30-60-90 triangles.

00:02:30.380 --> 00:02:32.040
If that was a little
bit mysterious,

00:02:32.040 --> 00:02:33.540
how I came up with
that, I encourage

00:02:33.540 --> 00:02:34.620
you to watch that video.

00:02:34.620 --> 00:02:40.900
We know that 30-60-90
triangles, their sides

00:02:40.900 --> 00:02:45.530
are in the ratio of 1 to
square root of 3 to 2.

00:02:45.530 --> 00:02:48.090
So this is 1, this
is a 30 degree side,

00:02:48.090 --> 00:02:50.150
this is going to be square
root of 3 times that.

00:02:50.150 --> 00:02:51.570
And the hypotenuse
right over here

00:02:51.570 --> 00:02:53.200
is going to be 2 times that.

00:02:53.200 --> 00:02:54.620
So this length
right over here is

00:02:54.620 --> 00:02:58.180
going to be 2 times this
side right over here.

00:02:58.180 --> 00:03:00.292
So 2 times 1 is just 2.

00:03:00.292 --> 00:03:01.500
So that's pretty interesting.

00:03:01.500 --> 00:03:03.166
Let's see if we can
do something similar

00:03:03.166 --> 00:03:04.520
with this side right over here.

00:03:04.520 --> 00:03:07.770
Here the 1 is not the side
opposite the 30 degree side.

00:03:07.770 --> 00:03:10.650
Here the 1 is the side
opposite the 60 degree side.

00:03:14.850 --> 00:03:17.780
So once again, if we
multiply this side

00:03:17.780 --> 00:03:21.410
times square root of 3, we
should get this side right over

00:03:21.410 --> 00:03:21.930
here.

00:03:21.930 --> 00:03:27.650
This is the 60, remember this
1, this is the 60 degree side.

00:03:27.650 --> 00:03:32.290
So this has to be 1 square
root of 3 of this side.

00:03:32.290 --> 00:03:35.300
Let me write this down, 1
over the square root of 3.

00:03:35.300 --> 00:03:38.280
And the whole reason, the
way I was able to get this

00:03:38.280 --> 00:03:41.140
is, well, whatever this
side, if I multiply it

00:03:41.140 --> 00:03:43.680
by the square root of 3, I
should get this side right over

00:03:43.680 --> 00:03:44.040
here.

00:03:44.040 --> 00:03:45.748
I should get the 60
degree side, the side

00:03:45.748 --> 00:03:47.350
opposite the 60 degree angle.

00:03:47.350 --> 00:03:50.360
Or if I take the 60 degree
side, if I divide it

00:03:50.360 --> 00:03:53.420
by the square root of 3 I should
get the shortest side, the 30

00:03:53.420 --> 00:03:54.330
degree side.

00:03:54.330 --> 00:03:56.630
So if I start with the
60 degree side, divide

00:03:56.630 --> 00:03:59.580
by the square root of 3, I
get that right over there.

00:03:59.580 --> 00:04:01.280
And then the
hypotenuse is always

00:04:01.280 --> 00:04:06.949
going to be twice the side
opposite the 30 degree angle.

00:04:06.949 --> 00:04:08.990
So this is the side opposite
the 30 degree angle.

00:04:08.990 --> 00:04:11.170
The hypotenuse is
always twice that.

00:04:11.170 --> 00:04:13.240
So this is the side opposite
the 30 degree angle.

00:04:13.240 --> 00:04:15.110
The hypotenuse is
going to be twice that.

00:04:15.110 --> 00:04:18.816
It is going to be 2 over
the square root of 3.

00:04:18.816 --> 00:04:19.940
So we're doing pretty good.

00:04:19.940 --> 00:04:21.140
We have to figure
out the perimeter

00:04:21.140 --> 00:04:22.800
of this inner triangle
right over here.

00:04:22.800 --> 00:04:24.780
We already figured
out one length is 2.

00:04:24.780 --> 00:04:27.290
We figured out another length
is 2 square roots of 3.

00:04:27.290 --> 00:04:30.614
And then all we have to really
figure out is, what ED is.

00:04:30.614 --> 00:04:32.030
And we can do that
because we know

00:04:32.030 --> 00:04:35.390
that AD is going to be
the same thing as BC.

00:04:35.390 --> 00:04:37.560
We know that this entire
length, because we're

00:04:37.560 --> 00:04:41.670
dealing with a rectangle,
is the square root of 3.

00:04:41.670 --> 00:04:43.420
If that entire length
is square root of 3,

00:04:43.420 --> 00:04:47.420
if this AE is 1 over
the square root of 3,

00:04:47.420 --> 00:04:48.840
then this length
right over here,

00:04:48.840 --> 00:04:52.310
ED is going to be
square root of 3 minus 1

00:04:52.310 --> 00:04:53.740
over the square root of 3.

00:04:53.740 --> 00:04:56.425
That length minus that
length right over there.

00:04:56.425 --> 00:04:58.800
And how to find the perimeter
is pretty straight forward.

00:04:58.800 --> 00:05:01.220
We just have to add these
things up and simplify it.

00:05:01.220 --> 00:05:04.440
So it's going to be,
just let me write

00:05:04.440 --> 00:05:09.890
this, perimeter
of triangle BED is

00:05:09.890 --> 00:05:11.730
equal to-- This is
short for perimeter.

00:05:11.730 --> 00:05:13.990
I just didn't feel like
writing the whole word.--

00:05:13.990 --> 00:05:19.920
is equal to 2 over the square
root of 3 plus square root of 3

00:05:19.920 --> 00:05:26.990
minus 1 over the square
root of 3 plus 2.

00:05:26.990 --> 00:05:29.400
And now this just boils down
to simplifying radicals.

00:05:29.400 --> 00:05:31.024
You could take a
calculator out and get

00:05:31.024 --> 00:05:32.891
some type of decimal
approximation for it.

00:05:32.891 --> 00:05:35.390
Let's see, if we have 2 square
root of 3 minus 1 square root

00:05:35.390 --> 00:05:39.800
of 3, that will leave us with
1 over the square root of 3.

00:05:42.467 --> 00:05:44.800
2 over the square of 3 minus
1 over the square root of 3

00:05:44.800 --> 00:05:46.330
is 1 over the square root of 3.

00:05:46.330 --> 00:05:50.730
And then you have the
square root of 3 plus 2.

00:05:50.730 --> 00:05:52.782
And let's see, I can
rationalize this.

00:05:52.782 --> 00:05:54.740
If I multiply the numerator
and the denominator

00:05:54.740 --> 00:05:57.640
by the square root of 3,
this gives me the square root

00:05:57.640 --> 00:06:01.400
of 3 over 3 plus the
square root of 3,

00:06:01.400 --> 00:06:04.640
which I could rewrite
that as plus 3

00:06:04.640 --> 00:06:06.490
square roots of 3 over 3.

00:06:06.490 --> 00:06:06.990
Right?

00:06:06.990 --> 00:06:12.350
I just multiplied this
times 3 over 3 plus 2.

00:06:12.350 --> 00:06:16.410
And so this gives us-- this
is the drum roll part now-- so

00:06:16.410 --> 00:06:19.480
one square root of 3
plus 3 square roots of 3,

00:06:19.480 --> 00:06:24.890
and all of that over 3, gives
us 4 square roots of 3 over 3

00:06:24.890 --> 00:06:26.012
plus 2.

00:06:26.012 --> 00:06:27.220
Or you could put the 2 first.

00:06:27.220 --> 00:06:30.080
Some people like to write
the non-irrational part

00:06:30.080 --> 00:06:31.535
before the irrational part.

00:06:31.535 --> 00:06:32.160
But we're done.

00:06:32.160 --> 00:06:33.800
We figured out the perimeter.

00:06:33.800 --> 00:06:38.400
We figured out the perimeter
of this inner triangle BED,

00:06:38.400 --> 00:06:40.220
right there.


The golden ratio

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:02.960
What I want to explore
in this video is,

00:00:02.960 --> 00:00:05.540
given some length
of string or a line

00:00:05.540 --> 00:00:07.540
or some line segment
right here, b,

00:00:07.540 --> 00:00:13.600
can I set up an a, so
that the ratio of a to b

00:00:13.600 --> 00:00:17.530
is equal to the ratio
of the sum of these two

00:00:17.530 --> 00:00:19.230
to the longer side?

00:00:19.230 --> 00:00:22.840
So it's equal to the
ratio of a plus b to a.

00:00:22.840 --> 00:00:25.040
So I want to sit and think
about this a little bit.

00:00:25.040 --> 00:00:27.780
I want to see is can I
construct some a that's

00:00:27.780 --> 00:00:29.880
on this ratio,
this perfect ratio

00:00:29.880 --> 00:00:32.810
that I'm somehow
referring to right here,

00:00:32.810 --> 00:00:35.470
so that the ratio of the
longer side to the shorter side

00:00:35.470 --> 00:00:40.086
is equal to the ratio of the
whole thing to the longer side.

00:00:40.086 --> 00:00:42.460
And let's just assume that we
can find a ratio like that.

00:00:42.460 --> 00:00:44.430
And we'll call it phi.

00:00:44.430 --> 00:00:47.550
We'll use the Greek letter
phi for that ratio over there.

00:00:47.550 --> 00:00:51.980
So let's see what we can learn
about this special ratio phi.

00:00:51.980 --> 00:00:55.560
Well if phi is equal to a over
b, which is equal to a plus b

00:00:55.560 --> 00:00:58.310
over a, we know that
a plus b over a is

00:00:58.310 --> 00:01:06.040
the same thing as a
over a plus b over a.

00:01:06.040 --> 00:01:07.120
a over a is just 1.

00:01:07.120 --> 00:01:09.830
And b over a is just the
inverse of this statement

00:01:09.830 --> 00:01:10.540
right over here.

00:01:10.540 --> 00:01:12.940
So b over a-- this thing
right here over here

00:01:12.940 --> 00:01:17.130
is phi-- so b over a is
going to be 1 over phi.

00:01:17.130 --> 00:01:19.250
This is going to be 1 over phi.

00:01:19.250 --> 00:01:20.580
So this is interesting.

00:01:20.580 --> 00:01:22.540
We've now set up a
number, which we're

00:01:22.540 --> 00:01:30.350
going to call this special
ratio, phi is equal to 1 plus 1

00:01:30.350 --> 00:01:31.775
over phi.

00:01:31.775 --> 00:01:35.260
Well just that is kind of a
neat statement right over there.

00:01:35.260 --> 00:01:37.130
First of all, you
could, if you subtract 1

00:01:37.130 --> 00:01:40.230
from both sides of this,
you get phi minus 1

00:01:40.230 --> 00:01:42.220
is equal to its inverse.

00:01:42.220 --> 00:01:44.840
That seems to be a pretty
neat property of any number

00:01:44.840 --> 00:01:46.960
that if I just
subtract 1 from it,

00:01:46.960 --> 00:01:48.850
I get its
multiplicative inverse.

00:01:48.850 --> 00:01:51.550
And so that, already that
seems kind of intriguing.

00:01:51.550 --> 00:01:53.150
But then even this
statement over here

00:01:53.150 --> 00:01:56.460
is kind of interesting because
we've defined phi in terms of 1

00:01:56.460 --> 00:01:58.140
plus 1 over phi.

00:01:58.140 --> 00:02:00.170
So we can actually
think of it this way.

00:02:00.170 --> 00:02:07.322
We could say that phi is
equal to 1 plus 1 over phi.

00:02:07.322 --> 00:02:09.030
But instead of writing
phi, we're saying,

00:02:09.030 --> 00:02:14.020
wait, phi is just 1
plus 1 over-- instead

00:02:14.020 --> 00:02:18.540
of saying phi-- I could say,
well, that's just 1 plus 1 over

00:02:18.540 --> 00:02:22.160
and I could just write phi again
or I could just keep on going.

00:02:22.160 --> 00:02:24.110
I could just keep on
going like this forever.

00:02:24.110 --> 00:02:27.110
I could say that's
1 over 1 plus 1 over

00:02:27.110 --> 00:02:30.330
and just keep on going on
and on and on, forever.

00:02:30.330 --> 00:02:32.550
And this is a recursive
definition of a function,

00:02:32.550 --> 00:02:35.270
or a recursive
definition of a variable,

00:02:35.270 --> 00:02:37.240
where it's defined
in terms of itself.

00:02:37.240 --> 00:02:39.410
But even this seems like
a pretty neat property.

00:02:39.410 --> 00:02:41.410
But we want to get a
little bit further into it.

00:02:41.410 --> 00:02:44.120
We actually eventually want
to figure out what phi is.

00:02:44.120 --> 00:02:47.840
What is the value of phi, this
weird number, this weird ratio

00:02:47.840 --> 00:02:49.990
that we're beginning to explore?

00:02:49.990 --> 00:02:53.100
So let's see if we can turn
it into a quadratic equation

00:02:53.100 --> 00:02:56.300
that we can solve using
fairly traditional methods.

00:02:56.300 --> 00:02:57.980
And the easiest
way to do that is

00:02:57.980 --> 00:03:00.820
to multiply both sides
of this equation by phi.

00:03:00.820 --> 00:03:03.542
And then you get
phi squared-- let

00:03:03.542 --> 00:03:08.800
me write a little bit
different-- phi squared is

00:03:08.800 --> 00:03:12.990
equal to phi plus 1.

00:03:12.990 --> 00:03:16.760
phi squared is
equal to phi plus 1.

00:03:16.760 --> 00:03:19.260
And then, actually, I'm going
to take a little bit of a side

00:03:19.260 --> 00:03:19.500
here.

00:03:19.500 --> 00:03:21.060
But even this is
interesting, because then

00:03:21.060 --> 00:03:23.280
if we take the square root
of both sides of this,

00:03:23.280 --> 00:03:25.710
you get-- let me scroll
down a little bit--

00:03:25.710 --> 00:03:29.760
you get phi is equal to the
square root of-- and I'll just

00:03:29.760 --> 00:03:34.880
switch the order here-- the
square root of 1 plus phi.

00:03:34.880 --> 00:03:37.340
So once again, we can set up
another recursive definition.

00:03:37.340 --> 00:03:42.752
phi is equal to the
square root of 1 plus phi.

00:03:42.752 --> 00:03:44.460
And I could write phi
there, but hey, phi

00:03:44.460 --> 00:03:48.970
is equal to the
square root of 1 plus

00:03:48.970 --> 00:03:50.760
and I could write phi
there, but hey, phi

00:03:50.760 --> 00:03:53.831
is just equal to the square root
of 1 plus, the square root of 1

00:03:53.831 --> 00:03:54.330
plus.

00:03:54.330 --> 00:03:56.960
And we could just keep going
on and on like this forever.

00:03:56.960 --> 00:03:57.900
So even this is neat.

00:03:57.900 --> 00:03:59.858
The same number that can
be expressed this way,

00:03:59.858 --> 00:04:02.020
the same number where if
I just subtract 1 from it,

00:04:02.020 --> 00:04:03.150
you get its inverse.

00:04:03.150 --> 00:04:05.100
It can also be
expressed in these kind

00:04:05.100 --> 00:04:08.870
of recursive square roots
underneath each other.

00:04:08.870 --> 00:04:11.760
So this is already starting
to get very, very, very

00:04:11.760 --> 00:04:13.710
intriguing, but let's
get back to business.

00:04:13.710 --> 00:04:18.630
Let's actually solve for this
magic number, this magic ratio

00:04:18.630 --> 00:04:20.060
that we started thinking about.

00:04:20.060 --> 00:04:22.130
And really from a
very simple idea,

00:04:22.130 --> 00:04:24.990
that the ratio of the longer
side to the shorter side

00:04:24.990 --> 00:04:29.750
is equal to the ratio of the sum
of the two to the longer side.

00:04:29.750 --> 00:04:31.930
So let's just solve this
as a traditional quadratic.

00:04:31.930 --> 00:04:33.721
Let's get everything
on the left-hand side.

00:04:33.721 --> 00:04:36.390
So we're going to subtract
phi plus 1 from both sides.

00:04:36.390 --> 00:04:42.850
And we get phi squared minus
phi minus 1 is equal to 0.

00:04:42.850 --> 00:04:45.270
And we can solve
for phi now using

00:04:45.270 --> 00:04:48.550
the quadratic formula, which
we've proven in other videos.

00:04:48.550 --> 00:04:50.680
You can prove using
completing the square.

00:04:50.680 --> 00:04:54.540
But the quadratic formula
you say, negative b.

00:04:54.540 --> 00:04:56.950
Negative b is the coefficient
on this term right here.

00:04:56.950 --> 00:04:58.570
So let me just
write it down, a is

00:04:58.570 --> 00:05:00.990
equal to 1, that's the
coefficient on this term.

00:05:00.990 --> 00:05:04.320
b is equal to negative 1, that's
the coefficient on this term.

00:05:04.320 --> 00:05:06.615
c is equal to negative 1,
that's the coefficient,

00:05:06.615 --> 00:05:09.180
or it's really the constant
term right over there.

00:05:09.180 --> 00:05:11.449
So the solutions to
this, phi-- and we're

00:05:11.449 --> 00:05:13.740
actually only going to care
about the positive solution

00:05:13.740 --> 00:05:15.844
because we're thinking
about a positive--

00:05:15.844 --> 00:05:17.510
when we go to our
original problem here,

00:05:17.510 --> 00:05:19.759
we're assuming that these
are both positive distances,

00:05:19.759 --> 00:05:22.310
so we care about a positive
value right over here.

00:05:22.310 --> 00:05:27.240
We get phi is equal to-- do
it in orange-- negative b.

00:05:27.240 --> 00:05:30.740
Well negative negative
1 is 1 plus or minus

00:05:30.740 --> 00:05:33.680
the square root of b squared.

00:05:33.680 --> 00:05:38.200
b squared is going
to be 1 minus 4ac.

00:05:38.200 --> 00:05:40.480
a is 1, c is negative 1.

00:05:40.480 --> 00:05:42.570
So negative 4 times
negative 1 is positive 4.

00:05:42.570 --> 00:05:47.270
So 1 plus 4, all
of that over 2a.

00:05:47.270 --> 00:05:49.390
So a is 1, so all
of that over 2.

00:05:49.390 --> 00:05:50.967
So phi is equal to 1.

00:05:50.967 --> 00:05:53.300
And once again, we only care
about the positive solution

00:05:53.300 --> 00:05:53.870
here.

00:05:53.870 --> 00:05:55.800
This is going to be
the square root of 5.

00:05:55.800 --> 00:05:57.180
If you have 1 minus
the square root of 5,

00:05:57.180 --> 00:05:58.710
you're going to get a
negative in the numerator.

00:05:58.710 --> 00:06:00.690
So we only care about
the positive solution.

00:06:00.690 --> 00:06:04.510
1 plus the square
root of 5 over 2.

00:06:04.510 --> 00:06:08.870
So this seems like a
pretty interesting number.

00:06:08.870 --> 00:06:10.370
Let's actually take
a calculator out

00:06:10.370 --> 00:06:15.370
and see if we can get the first
few places of this magic number

00:06:15.370 --> 00:06:16.320
phi.

00:06:16.320 --> 00:06:18.640
So let me get my calculator out.

00:06:18.640 --> 00:06:20.500
Let's just actually evaluate it.

00:06:20.500 --> 00:06:22.480
And you might recognize
that square root of 5

00:06:22.480 --> 00:06:23.641
is an irrational number.

00:06:23.641 --> 00:06:26.140
And so this whole thing is going
to be an irrational number,

00:06:26.140 --> 00:06:28.140
but I'll prove that in
another video, which

00:06:28.140 --> 00:06:29.710
means it never repeats.

00:06:29.710 --> 00:06:31.129
It goes on and on
and on forever.

00:06:31.129 --> 00:06:32.420
But let's actually evaluate it.

00:06:32.420 --> 00:06:41.210
So it's 1 plus the square
root of 5 divided by 2.

00:06:41.210 --> 00:06:45.120
So it says 1.6180339.

00:06:45.120 --> 00:06:46.280
So let me put that aside.

00:06:48.980 --> 00:06:50.154
Let me write it down.

00:06:50.154 --> 00:06:51.570
And this is where
it starts to get

00:06:51.570 --> 00:06:53.430
really interesting
and mysterious.

00:06:53.430 --> 00:07:01.542
So this number right
over here is 1.618033988

00:07:01.542 --> 00:07:04.460
and it just keeps on going
on and on and on, keeps

00:07:04.460 --> 00:07:07.020
on never terminating,
never repeating.

00:07:07.020 --> 00:07:09.120
So that by itself,
it's this cool number.

00:07:09.120 --> 00:07:11.670
It's this ratio that has all
of these neat properties, which

00:07:11.670 --> 00:07:14.600
are pretty crazy anyway
that you express it.

00:07:14.600 --> 00:07:17.240
But what's really neat is if
we revisit this thing right

00:07:17.240 --> 00:07:18.250
over here.

00:07:18.250 --> 00:07:20.700
Because what is 1
over phi going to be?

00:07:20.700 --> 00:07:27.610
So 1 over phi, which we
sometimes denote with a capital

00:07:27.610 --> 00:07:30.210
phi.

00:07:30.210 --> 00:07:32.860
We already know 1 over
phi is just phi minus 1.

00:07:32.860 --> 00:07:34.710
So we actually can
do this in our heads.

00:07:34.710 --> 00:07:41.019
1 over this is just
going to be 0.618033988.

00:07:41.019 --> 00:07:41.560
I don't know.

00:07:41.560 --> 00:07:43.720
There's just something
wacky about that,

00:07:43.720 --> 00:07:45.630
that the inverse of
the number is really

00:07:45.630 --> 00:07:48.750
just the decimals left over
after you get rid of the 1.

00:07:48.750 --> 00:07:52.040
That, by itself, is
kind of a crazy idea.

00:07:52.040 --> 00:07:54.060
But it gets even crazier
because this number

00:07:54.060 --> 00:07:55.430
is showing up everywhere.

00:07:55.430 --> 00:07:58.360
And as you might imagine
from the title of this video,

00:07:58.360 --> 00:08:02.850
this phi right over here, this
is called the golden ratio.

00:08:02.850 --> 00:08:05.260
This is the golden ratio
and it shows up everywhere.

00:08:08.350 --> 00:08:09.330
It shows up in art.

00:08:09.330 --> 00:08:10.600
It shows up in music.

00:08:10.600 --> 00:08:12.640
It shows up in nature.

00:08:12.640 --> 00:08:15.060
And just to get an idea of
where it shows up in nature,

00:08:15.060 --> 00:08:17.250
it shows up in very pure ideas.

00:08:17.250 --> 00:08:20.700
So if I were to just
draw a perfect star,

00:08:20.700 --> 00:08:24.030
if I were to just draw a
regular star like this.

00:08:24.030 --> 00:08:25.480
Let me just draw it like this.

00:08:25.480 --> 00:08:27.800
I'll draw it right over here.

00:08:27.800 --> 00:08:29.390
So this is just a regular star.

00:08:29.390 --> 00:08:31.310
All the lengths are equal.

00:08:31.310 --> 00:08:34.429
So I want to draw it a
little bit better than that.

00:08:34.429 --> 00:08:37.600
So if I just draw a star
like this right over here

00:08:37.600 --> 00:08:40.299
or sometimes this would
be called a pentagram,

00:08:40.299 --> 00:08:43.120
some amazing things
start to happen here.

00:08:43.120 --> 00:08:49.110
The ratio of this pink
side to this blue length

00:08:49.110 --> 00:08:51.880
right over here, that's
the golden ratio.

00:08:51.880 --> 00:08:56.140
The ratio of this
magenta to this pink

00:08:56.140 --> 00:08:59.580
is the golden ratio, as
it should, by definition.

00:08:59.580 --> 00:09:04.710
Now the ratio of the
magenta to this orange

00:09:04.710 --> 00:09:06.350
is also the golden ratio.

00:09:06.350 --> 00:09:08.950
It just keeps on showing up
in a ton of different ways

00:09:08.950 --> 00:09:11.000
when you look at a
pentagram like this.

00:09:11.000 --> 00:09:13.370
If you look at something
like a pentagon,

00:09:13.370 --> 00:09:15.480
a regular pentagon
where all the angles

00:09:15.480 --> 00:09:17.760
are the same and all
the sides are the same,

00:09:17.760 --> 00:09:20.460
a regular pentagon.

00:09:20.460 --> 00:09:24.970
If you take any of the
diagonals of a regular pentagon,

00:09:24.970 --> 00:09:28.130
so right over here, if you take
this diagonal right over here,

00:09:28.130 --> 00:09:31.117
the ratio of this green side
to--and when I'm talking about

00:09:31.117 --> 00:09:32.950
the diagonals, ones
that actually aren't one

00:09:32.950 --> 00:09:36.450
of the edges-- the ratio of
any of the diagonals to any

00:09:36.450 --> 00:09:40.290
of the sides is once
again, this golden ratio.

00:09:40.290 --> 00:09:42.600
So it keeps showing
up on and on and on.

00:09:42.600 --> 00:09:45.370
And we can do interesting
things with the golden ratio.

00:09:45.370 --> 00:09:47.230
Let's say that we had
a rectangle, where

00:09:47.230 --> 00:09:50.100
the ratio of the width to the
height is the golden ratio.

00:09:50.100 --> 00:09:52.930
So let's try that out.

00:09:52.930 --> 00:09:55.760
So let's say that
this is its height.

00:09:55.760 --> 00:09:57.400
This is its width.

00:09:57.400 --> 00:10:00.480
And that the ratio--
So let's call this a.

00:10:00.480 --> 00:10:01.510
Let's call this b.

00:10:01.510 --> 00:10:05.390
And the ratio of a
to b is equal to phi.

00:10:05.390 --> 00:10:08.330
That 1.61 so on and so forth.

00:10:08.330 --> 00:10:10.040
Let me scroll down a little bit.

00:10:10.040 --> 00:10:12.060
So that is going
to be equal to phi.

00:10:12.060 --> 00:10:14.860
So that's something
interesting to do.

00:10:14.860 --> 00:10:17.400
Maybe that's a nice looking
rectangle of some sort.

00:10:17.400 --> 00:10:18.910
But let me put
out a square here.

00:10:18.910 --> 00:10:22.790
So let me separate this
into a b by b square.

00:10:22.790 --> 00:10:25.590
So this is a b by b
square right over here.

00:10:25.590 --> 00:10:29.320
And then-- actually let me do
it a little bit, let me draw it

00:10:29.320 --> 00:10:31.600
a little bit differently,
this rectangle actually

00:10:31.600 --> 00:10:35.200
isn't exactly the way I
would want to draw it--

00:10:35.200 --> 00:10:38.670
so the ratio might look
a little bit like this.

00:10:38.670 --> 00:10:41.690
So the ratio of the
width to the length,

00:10:41.690 --> 00:10:44.190
or the width to the height, is
going to be the golden ratio.

00:10:44.190 --> 00:10:49.190
So a over b is going to
be that golden ratio.

00:10:49.190 --> 00:10:52.350
And let me separate out a
little b by b square over here.

00:10:55.810 --> 00:10:58.080
So this has width b as well.

00:10:58.080 --> 00:11:02.650
And so this distance right over
here is going to be a minus b.

00:11:02.650 --> 00:11:08.400
So now it is a b by
a minus b square.

00:11:08.400 --> 00:11:10.820
Actually, I should say,
we have a b by b square,

00:11:10.820 --> 00:11:11.770
right over here.

00:11:11.770 --> 00:11:13.040
This is b by b.

00:11:13.040 --> 00:11:16.240
And then we're left with a
b by a minus b rectangle.

00:11:16.240 --> 00:11:21.550
Now wouldn't it be cool if
this was also the golden ratio?

00:11:21.550 --> 00:11:22.810
And so let's try it out.

00:11:22.810 --> 00:11:26.430
Let's find the ratio
of b to a minus b.

00:11:26.430 --> 00:11:30.000
So the ratio of b to a minus b.

00:11:30.000 --> 00:11:33.920
Well, that's going to be equal
to 1 over the ratio of a minus

00:11:33.920 --> 00:11:35.570
b to b.

00:11:35.570 --> 00:11:38.070
I just took the reciprocal
of this right over here.

00:11:38.070 --> 00:11:42.260
And this is just going to
be equal to 1 over a over b.

00:11:42.260 --> 00:11:45.570
Let me write this,
a over b minus 1.

00:11:45.570 --> 00:11:47.350
I just rewrote this right there.

00:11:47.350 --> 00:11:51.100
And that's just going to
be equal to 1 over phi.

00:11:51.100 --> 00:11:55.930
The ratio of a to b, we said,
by definition was phi minus 1.

00:11:55.930 --> 00:11:57.820
But what is phi minus 1?

00:11:57.820 --> 00:11:59.830
Well phi minus 1 is 1 over phi.

00:11:59.830 --> 00:12:00.870
It's this cool number.

00:12:00.870 --> 00:12:05.780
So it's equal to 1 over 1 over
1 over phi, which is once again,

00:12:05.780 --> 00:12:07.380
just equal to phi.

00:12:07.380 --> 00:12:10.270
So once again, the ratio
of this smaller rectangle,

00:12:10.270 --> 00:12:12.990
of its height to its
width, is once again

00:12:12.990 --> 00:12:16.010
this golden ratio, this
number that keeps showing up.

00:12:16.010 --> 00:12:17.760
And then we could do
the same thing again.

00:12:17.760 --> 00:12:23.030
We could separate this into an
a minus b by a minus b square.

00:12:23.030 --> 00:12:24.140
Just like that.

00:12:24.140 --> 00:12:26.090
And then we'll have
another golden rectangle,

00:12:26.090 --> 00:12:27.756
sometimes it's called,
right over there.

00:12:27.756 --> 00:12:30.150
And then we could separate
that into a square

00:12:30.150 --> 00:12:32.280
and another golden rectangle.

00:12:32.280 --> 00:12:34.570
Then we could separate
that into a square and then

00:12:34.570 --> 00:12:36.481
another golden rectangle.

00:12:36.481 --> 00:12:37.730
Then another golden rectangle.

00:12:37.730 --> 00:12:39.063
Actually let me do it like this.

00:12:39.063 --> 00:12:40.940
This would be better.

00:12:40.940 --> 00:12:42.940
So let me separate.

00:12:42.940 --> 00:12:45.890
Let me do the square up here.

00:12:45.890 --> 00:12:49.460
So this is an a minus
b by a minus b square

00:12:49.460 --> 00:12:52.470
and then we have another golden
rectangle right over here.

00:12:52.470 --> 00:12:54.650
I could put a square
right in there.

00:12:54.650 --> 00:12:56.520
Then we'll have another
golden rectangle.

00:12:56.520 --> 00:12:58.604
Then we could put another
square right over there,

00:12:58.604 --> 00:13:00.020
you have another
golden rectangle.

00:13:00.020 --> 00:13:01.530
I think you see
where this is going.

00:13:01.530 --> 00:13:03.420
Another square, another
golden rectangle,

00:13:03.420 --> 00:13:06.200
which by itself starts to create
a cool design that we can keep

00:13:06.200 --> 00:13:08.230
kind of circling
in and in and in.

00:13:08.230 --> 00:13:10.170
And then, if we actually
draw an arc here,

00:13:10.170 --> 00:13:12.000
something kind of cool happens.

00:13:12.000 --> 00:13:15.369
If we have an arc that
traces these things out,

00:13:15.369 --> 00:13:16.910
we have something,
a pattern that you

00:13:16.910 --> 00:13:19.640
might have seen
many times before.

00:13:19.640 --> 00:13:22.140
And that pattern does not
look too different than what

00:13:22.140 --> 00:13:25.476
you might see in something
like a nautilus shell.

00:13:25.476 --> 00:13:27.350
And it shows up all over
the place in nature.

00:13:27.350 --> 00:13:29.220
And that makes sense
because just the way

00:13:29.220 --> 00:13:31.085
cells construct themselves.

00:13:31.085 --> 00:13:33.460
It kind of makes sense to be
the same at different scales

00:13:33.460 --> 00:13:35.126
and the ratio from
one scale to the next

00:13:35.126 --> 00:13:38.490
is maybe the same as
the constituent ratios.

00:13:38.490 --> 00:13:40.530
This right here,
and it shows up all

00:13:40.530 --> 00:13:42.490
over a lot of Leonardo
DaVinci's paintings.

00:13:42.490 --> 00:13:45.530
He never explicitly
stated it, but there's

00:13:45.530 --> 00:13:47.030
a lot of interesting
ratios in them.

00:13:47.030 --> 00:13:49.550
But Salvador Dali, this painting
right here, The Sacrament

00:13:49.550 --> 00:13:51.450
of the Last Supper,
he explicitly

00:13:51.450 --> 00:13:53.170
used the golden ratio.

00:13:53.170 --> 00:13:56.740
So the actual ratio of
the width to the height

00:13:56.740 --> 00:13:59.361
is the golden ratio, so
this is a golden rectangle.

00:13:59.361 --> 00:14:00.860
And also there's
all sorts of ratios

00:14:00.860 --> 00:14:02.600
and I'll invite
you to explore it.

00:14:02.600 --> 00:14:04.696
The ratio of the different
parts of the tables

00:14:04.696 --> 00:14:06.070
to where it sits
in the painting.

00:14:06.070 --> 00:14:08.460
The golden ratio shows up
a ton in this painting.

00:14:08.460 --> 00:14:11.670
And then he does have
the pentagons over here.

00:14:11.670 --> 00:14:14.110
And we know that the ratio of
the diagonal of the pentagon

00:14:14.110 --> 00:14:17.227
to the sides of the pentagon
are also the golden ratio.

00:14:17.227 --> 00:14:19.310
And so he just thought it
was a really cool thing.

00:14:19.310 --> 00:14:20.684
There's all sorts
of neat things.

00:14:20.684 --> 00:14:24.720
That if you find where these
two guys are bowing down are,

00:14:24.720 --> 00:14:27.840
if you draw that line right over
here, this is the golden ratio.

00:14:27.840 --> 00:14:30.050
The ratio of this
length right over here

00:14:30.050 --> 00:14:31.292
to that length over there.

00:14:31.292 --> 00:14:32.500
Once again, the golden ratio.

00:14:32.500 --> 00:14:34.259
It just keeps showing
up in this painting.

00:14:34.259 --> 00:14:36.050
So it's a really,
really, really cool thing

00:14:36.050 --> 00:14:38.540
and I really encourage
you to explore this

00:14:38.540 --> 00:14:41.760
further because it's
kind of exciting.


45-45-90 triangle side ratios

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:02.530
In the last video, we
showed that the ratios

00:00:02.530 --> 00:00:05.080
of the sides of a
30-60-90 triangle

00:00:05.080 --> 00:00:06.675
are-- if we assume
the longest side is

00:00:06.675 --> 00:00:08.350
x, if the hypotenuse is x.

00:00:08.350 --> 00:00:11.480
Then the shortest side is
x/2 and the side in between,

00:00:11.480 --> 00:00:13.690
the side that's opposite
the 60 degree side,

00:00:13.690 --> 00:00:14.890
is square root of 3x/2.

00:00:14.890 --> 00:00:19.125
Or another way to think about it
is if the shortest side is 1--

00:00:19.125 --> 00:00:21.540
Now I'll do the shortest side,
then the medium size, then

00:00:21.540 --> 00:00:22.390
the longest side.

00:00:22.390 --> 00:00:24.500
So if the side opposite
the 30 degree side is 1,

00:00:24.500 --> 00:00:27.400
then the side opposite
the 60 degree side

00:00:27.400 --> 00:00:29.340
is square root of 3 times that.

00:00:29.340 --> 00:00:31.110
So it's going to be
square root of 3.

00:00:31.110 --> 00:00:33.850
And then the hypotenuse
is going to be twice that.

00:00:33.850 --> 00:00:35.480
In the last video,
we started with x

00:00:35.480 --> 00:00:37.460
and we said that the
30 degree side is x/2.

00:00:37.460 --> 00:00:40.110
But if the 30 degree
side is 1, then this

00:00:40.110 --> 00:00:41.270
is going to be twice that.

00:00:41.270 --> 00:00:42.440
So it's going to be 2.

00:00:42.440 --> 00:00:46.120
This right here is the side
opposite the 30 degree side,

00:00:46.120 --> 00:00:49.440
opposite the 60 degree side,
and then the hypotenuse opposite

00:00:49.440 --> 00:00:51.030
the 90 degree side.

00:00:51.030 --> 00:00:53.800
And so, in general, if
you see any triangle that

00:00:53.800 --> 00:00:56.570
has those ratios, you say hey,
that's a 30-60-90 triangle.

00:00:56.570 --> 00:00:58.075
Or if you see a
triangle that you

00:00:58.075 --> 00:01:02.290
know is a 30-60-90 triangle,
you could say, hey,

00:01:02.290 --> 00:01:05.099
I know how to figure out
one of the sides based

00:01:05.099 --> 00:01:06.660
on this ratio right over here.

00:01:06.660 --> 00:01:08.930
Just an example, if
you see a triangle that

00:01:08.930 --> 00:01:14.510
looks like this, where the
sides are 2, 2 square root of 3,

00:01:14.510 --> 00:01:15.480
and 4.

00:01:15.480 --> 00:01:17.790
Once again, the ratio of
2 to 2 square root of 3

00:01:17.790 --> 00:01:19.440
is 1 to square root of 3.

00:01:19.440 --> 00:01:22.340
The ratio of 2 to 4 is
the same thing as 1 to 2.

00:01:22.340 --> 00:01:25.440
This right here must
be a 30-60-90 triangle.

00:01:25.440 --> 00:01:27.380
What I want to introduce
you to in this video

00:01:27.380 --> 00:01:29.570
is another important
type of triangle

00:01:29.570 --> 00:01:32.980
that shows up a lot in geometry
and a lot in trigonometry.

00:01:32.980 --> 00:01:36.649
And this is a 45-45-90 triangle.

00:01:36.649 --> 00:01:38.190
Or another way to
think about is if I

00:01:38.190 --> 00:01:40.155
have a right triangle
that is also isosceles.

00:01:44.250 --> 00:01:47.140
You obviously can't have a right
triangle that is equilateral,

00:01:47.140 --> 00:01:49.770
because an equilateral triangle
has all of their angles

00:01:49.770 --> 00:01:51.030
have to be 60 degrees.

00:01:51.030 --> 00:01:52.870
But you can have
a right angle, you

00:01:52.870 --> 00:01:55.260
can have a right triangle,
that is isosceles.

00:01:55.260 --> 00:01:56.790
And isosceles--
let me write this--

00:01:56.790 --> 00:02:03.470
this is a right
isosceles triangle.

00:02:03.470 --> 00:02:05.830
And if it's isosceles,
that means two of the sides

00:02:05.830 --> 00:02:06.500
are equal.

00:02:06.500 --> 00:02:09.851
So these are the two
sides that are equal.

00:02:09.851 --> 00:02:11.350
And then if the two
sides are equal,

00:02:11.350 --> 00:02:14.990
we have proved to ourselves
that the base angles are equal.

00:02:14.990 --> 00:02:17.440
And if we called the measure
of these base angles x,

00:02:17.440 --> 00:02:25.440
then we know that x plus x plus
90 have to be equal to 180.

00:02:25.440 --> 00:02:27.610
Or if we subtract
90 from both sides,

00:02:27.610 --> 00:02:32.060
you get x plus x is equal
to 90 or 2x is equal to 90.

00:02:32.060 --> 00:02:33.780
Or if you divide
both sides by 2,

00:02:33.780 --> 00:02:38.750
you get x is equal
to 45 degrees.

00:02:38.750 --> 00:02:41.850
So a right isosceles
triangle can also be called--

00:02:41.850 --> 00:02:44.010
and this is the more
typical name for it--

00:02:44.010 --> 00:02:50.140
it can also be called
a 45-45-90 triangle.

00:02:54.239 --> 00:02:56.030
And what I want to do
this video is come up

00:02:56.030 --> 00:02:59.180
with the ratios for the
sides of a 45-45-90 triangle,

00:02:59.180 --> 00:03:01.270
just like we did for
a 30-60-90 triangle.

00:03:01.270 --> 00:03:03.180
And this one's actually
more straightforward.

00:03:03.180 --> 00:03:08.950
Because in a 45-45-90 triangle,
if we call one of the legs x,

00:03:08.950 --> 00:03:10.819
the other leg is
also going to be x.

00:03:10.819 --> 00:03:12.610
And then we can use
the Pythagorean Theorem

00:03:12.610 --> 00:03:14.770
to figure out the length
of the hypotenuse.

00:03:14.770 --> 00:03:18.090
So the length of the
hypotenuse, let's call that c.

00:03:18.090 --> 00:03:22.740
So we get x squared
plus x squared.

00:03:22.740 --> 00:03:26.429
That's the square of
length of both of the legs.

00:03:26.429 --> 00:03:27.970
So when we sum those
up, that's going

00:03:27.970 --> 00:03:29.740
to have to be
equal to c squared.

00:03:29.740 --> 00:03:32.310
This is just straight out
of the Pythagorean theorem.

00:03:32.310 --> 00:03:37.490
So we get 2x squared
is equal to c squared.

00:03:37.490 --> 00:03:41.566
We can take the principal
root of both sides of that.

00:03:41.566 --> 00:03:45.930
I wanted to just
change it to yellow.

00:03:45.930 --> 00:03:48.230
Last, take the principal
root of both sides of that.

00:03:51.290 --> 00:03:53.420
The left-hand side you
get, principal root of 2

00:03:53.420 --> 00:03:54.860
is just square
root of 2, and then

00:03:54.860 --> 00:03:57.790
the principal root of x
squared is just going to be x.

00:03:57.790 --> 00:04:01.290
So you're going to have x
times the square root of 2

00:04:01.290 --> 00:04:04.690
is equal to c.

00:04:04.690 --> 00:04:07.790
So if you have a right isosceles
triangle, whatever the two

00:04:07.790 --> 00:04:09.790
legs are, they're going
to have the same length.

00:04:09.790 --> 00:04:11.200
That's why it's isosceles.

00:04:11.200 --> 00:04:13.980
The hypotenuse is going to be
square root of 2 times that.

00:04:13.980 --> 00:04:18.230
So c is equal to x times
the square root of 2.

00:04:18.230 --> 00:04:22.130
So for example, if you have a
triangle that looks like this.

00:04:22.130 --> 00:04:23.940
Let me draw it a
slightly different way.

00:04:23.940 --> 00:04:26.650
It's good to have to orient
ourselves in different ways

00:04:26.650 --> 00:04:27.750
every time.

00:04:27.750 --> 00:04:30.690
So if we see a triangle
that's 90 degrees,

00:04:30.690 --> 00:04:33.640
45 and 45 like this,
and you really just

00:04:33.640 --> 00:04:35.900
have to know two of
these angles to know

00:04:35.900 --> 00:04:37.510
what the other one
is going to be,

00:04:37.510 --> 00:04:39.790
and if I tell you that
this side right over here

00:04:39.790 --> 00:04:41.909
is 3-- I actually don't
even have to tell you

00:04:41.909 --> 00:04:43.450
that this other
side's going to be 3.

00:04:43.450 --> 00:04:45.970
This is an isosceles
triangle, so those two legs

00:04:45.970 --> 00:04:47.210
are going to be the same.

00:04:47.210 --> 00:04:49.050
And you won't even have to
apply the Pythagorean theorem

00:04:49.050 --> 00:04:50.370
if you know this--
and this is a good one

00:04:50.370 --> 00:04:53.109
to know-- that the hypotenuse
here, the side opposite the 90

00:04:53.109 --> 00:04:55.150
degree side, is just going
to be square root of 2

00:04:55.150 --> 00:04:57.920
times the length of
either of the legs.

00:04:57.920 --> 00:05:01.400
So it's going to be 3
times the square root of 2.

00:05:01.400 --> 00:05:03.790
So the ratio of the
size of the hypotenuse

00:05:03.790 --> 00:05:09.190
in a 45-45-90 triangle or
a right isosceles triangle,

00:05:09.190 --> 00:05:12.342
the ratio of the sides are
one of the legs can be 1.

00:05:12.342 --> 00:05:14.550
Then the other leg is going
to have the same measure,

00:05:14.550 --> 00:05:16.820
the same length, and then
the hypotenuse is going

00:05:16.820 --> 00:05:19.100
to be square root of 2
times either of those.

00:05:19.100 --> 00:05:21.690
1 to 1, 2 square root of 2.

00:05:21.690 --> 00:05:22.690
So this is 45-45-90.

00:05:28.740 --> 00:05:30.010
That's the ratios.

00:05:30.010 --> 00:05:33.680
And just as a review,
if you have a 30-60-90,

00:05:33.680 --> 00:05:38.800
the ratios were 1 to
square root of 3 to 2.

00:05:38.800 --> 00:05:41.820
And now we'll apply this
in a bunch of problems.


30-60-90 triangle side ratios proof

WEBVTT
Kind: captions
Language: en

00:00:00.494 --> 00:00:02.910
What I want to do in this video
is discuss a special class

00:00:02.910 --> 00:00:06.096
of triangles called
30-60-90 triangles.

00:00:06.096 --> 00:00:07.970
And I think you know
why they're called this.

00:00:07.970 --> 00:00:11.580
The measures of its angles
are 30 degrees, 60 degrees,

00:00:11.580 --> 00:00:12.567
and 90 degrees.

00:00:12.567 --> 00:00:14.400
And what we're going
to prove in this video,

00:00:14.400 --> 00:00:16.680
and this tends to be a very
useful result, at least

00:00:16.680 --> 00:00:19.100
for a lot of what you see
in a geometry class and then

00:00:19.100 --> 00:00:21.970
later on in trigonometry
class, is the ratios

00:00:21.970 --> 00:00:24.530
between the sides of
a 30-60-90 triangle.

00:00:28.820 --> 00:00:31.970
Remember, the hypotenuse is
opposite the 90-degree side.

00:00:31.970 --> 00:00:33.470
If the hypotenuse
has length x, what

00:00:33.470 --> 00:00:35.930
we're going to prove is that
the shortest side, which

00:00:35.930 --> 00:00:39.920
is opposite the 30-degree
side, has length x/2,

00:00:39.920 --> 00:00:44.880
and that the 60 degree
side, or the side that's

00:00:44.880 --> 00:00:47.060
opposite the 60-degree
angle, I should say,

00:00:47.060 --> 00:00:50.360
is going to be square root
of 3 times the shortest side.

00:00:50.360 --> 00:00:53.915
So square root of 3 times x/2,
that's going to be its length.

00:00:53.915 --> 00:00:56.040
So that's where we're going
to prove in this video.

00:00:56.040 --> 00:00:57.510
And then in other videos,
we're just going to apply this.

00:00:57.510 --> 00:00:59.218
We're going to show
that this is actually

00:00:59.218 --> 00:01:00.510
a pretty useful result.

00:01:00.510 --> 00:01:03.730
Now, let's start with a triangle
that we're very familiar with.

00:01:03.730 --> 00:01:08.410
So let me draw ourselves
an equilateral triangle.

00:01:08.410 --> 00:01:12.220
So drawing the triangles
is always the hard part.

00:01:12.220 --> 00:01:16.760
This is my best shot at
a equilateral triangle.

00:01:16.760 --> 00:01:18.600
So let's call this ABC.

00:01:21.104 --> 00:01:22.520
I'm just going to
assume that I've

00:01:22.520 --> 00:01:24.080
constructed an
equilateral triangle.

00:01:24.080 --> 00:01:28.460
So triangle ABC is equilateral.

00:01:28.460 --> 00:01:31.070
And if it's equilateral,
that means all of its sides

00:01:31.070 --> 00:01:33.530
are equal.

00:01:33.530 --> 00:01:41.010
And let's say equilateral
with sides of length x.

00:01:41.010 --> 00:01:44.100
So this is going to be
x, this is going to be x,

00:01:44.100 --> 00:01:45.560
and this is going to be x.

00:01:45.560 --> 00:01:47.580
We also know,
based on what we've

00:01:47.580 --> 00:01:49.730
seen from equilateral
triangles before,

00:01:49.730 --> 00:01:51.630
that the measures of
all of these angles

00:01:51.630 --> 00:01:53.980
are going to be 60 degrees.

00:01:53.980 --> 00:01:56.410
So this is going
to be 60 degrees,

00:01:56.410 --> 00:01:58.820
this is going to be
60 degrees, and then

00:01:58.820 --> 00:02:00.187
this is going to be 60 degrees.

00:02:00.187 --> 00:02:01.770
Now, what I'm going
to do is I'm going

00:02:01.770 --> 00:02:05.230
to drop an altitude from this
top point right over here.

00:02:05.230 --> 00:02:07.230
So I'm going to drop
an altitude right down,

00:02:07.230 --> 00:02:09.550
and by definition, when I'm
constructing an altitude,

00:02:09.550 --> 00:02:12.751
it's going to intersect the base
right here at a right angle.

00:02:12.751 --> 00:02:14.250
So that's going to
be a right angle,

00:02:14.250 --> 00:02:16.245
and then this is going
to be a right angle.

00:02:16.245 --> 00:02:17.870
And it's a pretty
straightforward proof

00:02:17.870 --> 00:02:20.610
to show that not only
is this an altitude,

00:02:20.610 --> 00:02:22.361
not only is it
perpendicular to this base,

00:02:22.361 --> 00:02:23.985
but it's a pretty
straightforward proof

00:02:23.985 --> 00:02:25.430
to show that it
bisects the base.

00:02:25.430 --> 00:02:28.162
And you could pause it, if you
like and prove it yourself.

00:02:28.162 --> 00:02:29.620
But it really comes
out of the fact

00:02:29.620 --> 00:02:31.786
that it's easy to prove
that these two triangles are

00:02:31.786 --> 00:02:32.510
congruent.

00:02:32.510 --> 00:02:34.620
So let me prove it for you.

00:02:34.620 --> 00:02:37.620
So let's call this
point D right over here.

00:02:37.620 --> 00:02:43.890
So triangles ABD and BDC, they
clearly both share this side.

00:02:43.890 --> 00:02:49.740
So this side is common to
both of them right over here.

00:02:49.740 --> 00:02:53.010
This angle right over here
is congruent to this angle

00:02:53.010 --> 00:02:54.040
over there.

00:02:54.040 --> 00:02:56.680
This angle right over here
is congruent to this angle

00:02:56.680 --> 00:02:57.700
over here.

00:02:57.700 --> 00:03:01.040
And so if these two are
congruent to each other,

00:03:01.040 --> 00:03:04.030
then the third angle has to
be congruent to each other.

00:03:04.030 --> 00:03:05.560
So this angle right
over here needs

00:03:05.560 --> 00:03:08.170
to be congruent to that
angle right over there.

00:03:08.170 --> 00:03:09.680
So these two are congruent.

00:03:09.680 --> 00:03:12.140
And so you can use
actually a variety

00:03:12.140 --> 00:03:15.040
of our congruence postulates.

00:03:15.040 --> 00:03:20.610
We could say,
side-angle-side congruence.

00:03:20.610 --> 00:03:23.640
We could use
angle-side-angle, any of those

00:03:23.640 --> 00:03:32.550
to show that triangle ABD is
congruent to triangle CBD.

00:03:35.680 --> 00:03:38.290
And what that does for us,
and we could use, as I said,

00:03:38.290 --> 00:03:40.960
we could use angle-side-angle
or side-angle-side,

00:03:40.960 --> 00:03:42.790
whatever we like
to use for this.

00:03:42.790 --> 00:03:44.430
What that does for
us is it tells us

00:03:44.430 --> 00:03:47.150
that the corresponding
sides of these triangles

00:03:47.150 --> 00:03:48.160
are going to be equal.

00:03:51.900 --> 00:03:54.832
In particular, AD is
going to be equal to CD.

00:03:57.500 --> 00:03:59.500
These are corresponding sides.

00:03:59.500 --> 00:04:01.750
So these are going to
be equal to each other.

00:04:01.750 --> 00:04:03.310
And if we know that they're
equal to each other,

00:04:03.310 --> 00:04:04.851
and they add up to
x-- remember, this

00:04:04.851 --> 00:04:06.880
was an equilateral
triangle of length x--

00:04:06.880 --> 00:04:10.477
we know that this side right
over here, is going to be x/2.

00:04:10.477 --> 00:04:11.810
We know this is going to be x/2.

00:04:11.810 --> 00:04:14.410
Not only do we know
that, but we also

00:04:14.410 --> 00:04:16.579
knew when we dropped
this altitude,

00:04:16.579 --> 00:04:19.630
we showed that this angle has
to be congruent to that angle,

00:04:19.630 --> 00:04:21.550
and their measures
have to add up to 60.

00:04:21.550 --> 00:04:23.860
So if two things are the
same and they add up to 60,

00:04:23.860 --> 00:04:26.010
this is going to be
30 degrees, and this

00:04:26.010 --> 00:04:27.530
is going to be 30 degrees.

00:04:27.530 --> 00:04:30.140
So we've already shown one
of the interesting parts

00:04:30.140 --> 00:04:33.750
of a 30-60-90 triangle, that if
the hypotenuse-- notice, and I

00:04:33.750 --> 00:04:35.000
guess I didn't point this out.

00:04:35.000 --> 00:04:37.100
By dropping this
altitude, I've essentially

00:04:37.100 --> 00:04:43.410
split this equilateral triangle
into two 30-60-90 triangles.

00:04:43.410 --> 00:04:46.260
And so we've already shown
that if the side opposite

00:04:46.260 --> 00:04:48.780
the 90-degree side
is x, that the side

00:04:48.780 --> 00:04:51.270
opposite the 30-degree
side is going to be x/2.

00:04:51.270 --> 00:04:54.090
That's what we showed
right over here.

00:04:54.090 --> 00:04:56.850
Now we just have to come up with
the third side, the side that

00:04:56.850 --> 00:04:58.610
is opposite the 60-degree side.

00:05:06.664 --> 00:05:08.830
I'll just use the letters
that we already have here.

00:05:08.830 --> 00:05:10.280
This is BD.

00:05:10.280 --> 00:05:12.600
And we can just use the
Pythagorean theorem right here.

00:05:12.600 --> 00:05:15.530
BD squared plus this
length right over here

00:05:15.530 --> 00:05:17.860
squared plus x/2
squared is going

00:05:17.860 --> 00:05:20.040
to be equal to the
hypotenuse squared.

00:05:20.040 --> 00:05:25.669
So we get BD squared
plus x/2 squared--

00:05:25.669 --> 00:05:27.960
this is just straight out of
the Pythagorean theorem.--

00:05:27.960 --> 00:05:31.500
plus x/2 squared is going to
equal this hypotenuse squared.

00:05:31.500 --> 00:05:33.720
It's going to equal x squared.

00:05:33.720 --> 00:05:36.190
And just to be clear, I'm
looking at this triangle right

00:05:36.190 --> 00:05:36.690
here.

00:05:36.690 --> 00:05:39.106
I'm looking at this triangle
right over here on the right,

00:05:39.106 --> 00:05:41.190
and I'm just applying
the Pythagorean theorem.

00:05:41.190 --> 00:05:42.950
This side squared
plus this side squared

00:05:42.950 --> 00:05:45.110
is going to equal the
hypotenuse squared.

00:05:45.110 --> 00:05:47.140
And let's solve now for BD.

00:05:47.140 --> 00:05:52.312
You get BD squared
plus x squared over 4.

00:05:52.312 --> 00:05:57.360
x squared over 4 is
equal to x squared.

00:05:57.360 --> 00:05:59.242
You could view this
as 4x squared over 4.

00:05:59.242 --> 00:06:01.200
That's the same thing,
obviously, as x squared.

00:06:04.150 --> 00:06:06.460
If you subtract 1/4 x
squared from both sides,

00:06:06.460 --> 00:06:08.520
or x squared over
4 from both sides,

00:06:08.520 --> 00:06:12.950
you get BD squared is
equal to-- 4x squared

00:06:12.950 --> 00:06:16.780
over 4 minus x squared over 4 is
going to be 3x squared over 4.

00:06:16.780 --> 00:06:20.300
So it's just going to
be 3x squared over 4.

00:06:20.300 --> 00:06:22.390
Take the principal
root of both sides.

00:06:22.390 --> 00:06:30.556
You get BD is equal to the
square root of 3 times x.

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The principal root
of 3 is square root

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of 3. the principal root
of x squared is just x,

00:06:37.660 --> 00:06:40.710
over the principal
root of 4 which is 2.

00:06:40.710 --> 00:06:43.580
And BD is the side opposite
the 60-degree side.

00:06:43.580 --> 00:06:44.430
So we're done.

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If this hypotenuse
is x, the side

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opposite the 30-degree
side is going to be x/2,

00:06:48.450 --> 00:06:51.370
and the side opposite
the 60-degree side

00:06:51.370 --> 00:06:55.035
is going to be square
root of 3 over 2 times x,

00:06:55.035 --> 00:06:57.010
or the square root of
3x over 2, depending

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on how you want to view it.