0:00:00.000,0:00:00.730


0:00:00.730,0:00:03.550
Let's say I've got a triangle.

0:00:03.550,0:00:05.900
There is my triangle[br]right there.

0:00:05.900,0:00:08.550
And I only know the lengths of[br]the sides of the triangle.

0:00:08.550,0:00:12.490
This side has length a, this[br]side has length b, and

0:00:12.490,0:00:14.360
that side has length c.

0:00:14.360,0:00:17.480
And I'm asked to find the[br]area of that triangle.

0:00:17.480,0:00:21.870
So far all I'm equipped with is[br]the idea that the area, the

0:00:21.870,0:00:26.930
area of a triangle is equal[br]to 1/2 times the base of

0:00:26.930,0:00:30.380
the triangle times the[br]height of the triangle.

0:00:30.380,0:00:33.570
So the way I've drawn this[br]triangle, the base of this

0:00:33.570,0:00:39.060
triangle, would be side c, but[br]the height we don't know.

0:00:39.060,0:00:41.900
The height would be that h[br]right there and we don't

0:00:41.900,0:00:43.750
even know what that h is.

0:00:43.750,0:00:45.200
So this would be the h.

0:00:45.200,0:00:48.360
So the question is how do[br]we figure out the area

0:00:48.360,0:00:49.660
of this triangle?

0:00:49.660,0:00:51.150
If you watched the last[br]video you know that you

0:00:51.150,0:00:52.440
use Heron's formula.

0:00:52.440,0:00:55.910
But the idea here is to try[br]to prove Heron's formula.

0:00:55.910,0:00:59.530
So let's just try to figure[br]out h from just using

0:00:59.530,0:01:00.970
the Pythagorean theorem.

0:01:00.970,0:01:04.190
And from there, once we know h,[br]we can apply this formula and

0:01:04.190,0:01:07.100
figure out the area[br]of this triangle.

0:01:07.100,0:01:11.370
So we already[br]labeled this as h.

0:01:11.370,0:01:13.120
Let me define another[br]variable here.

0:01:13.120,0:01:15.880


0:01:15.880,0:01:19.270
This is a trick you'll see[br]pretty often in geometry.

0:01:19.270,0:01:24.990
Let me define this is x, and if[br]this is x in magenta, then in

0:01:24.990,0:01:29.990
this bluish-purplish color,[br]that would be c minus x, right?

0:01:29.990,0:01:33.710
This whole length is c[br]-- the whole base is c.

0:01:33.710,0:01:37.560
So if this part is x, then[br]this part is c minus x.

0:01:37.560,0:01:41.060
What I could do now, since[br]these are both right angles,

0:01:41.060,0:01:44.250
and I know that because this is[br]the height, I can set up two

0:01:44.250,0:01:46.600
Pythagorean theorem equations.

0:01:46.600,0:01:50.890
First, I could do this left[br]hand side and I can write that

0:01:50.890,0:01:57.880
x squared plus h squared[br]is equal to a squared.

0:01:57.880,0:02:00.690
That's what I get from[br]this left hand triangle.

0:02:00.690,0:02:05.000
Then from this right hand[br]triangle, I get c minus x

0:02:05.000,0:02:14.030
squared plus h squared[br]is equal to b squared.

0:02:14.030,0:02:17.760
So I'm assuming I know a, b and[br]c, so I have two equations

0:02:17.760,0:02:18.950
with two unknowns.

0:02:18.950,0:02:22.290
The unknowns are x and h.

0:02:22.290,0:02:24.220
And remember, h is what[br]we're trying to figure out

0:02:24.220,0:02:25.270
because we already know c.

0:02:25.270,0:02:27.540
If we know h, we can[br]apply the area formula.

0:02:27.540,0:02:28.900
So how can we do that?

0:02:28.900,0:02:32.200
Well, let's substitute[br]for h to figure out x.

0:02:32.200,0:02:36.360
When I say that I mean let's[br]solve for h squared here.

0:02:36.360,0:02:38.890
If we solve for h squared[br]here we just subtract x

0:02:38.890,0:02:40.320
squared from both sides.

0:02:40.320,0:02:44.540
We can write that x squared --[br]sorry, we could write that

0:02:44.540,0:02:51.720
h squared is equal to a[br]squared minus x squared.

0:02:51.720,0:02:53.770
Then we could take this[br]information and substitute

0:02:53.770,0:02:56.640
it over here for h squared.

0:02:56.640,0:03:02.030
So this bottom equation[br]becomes c minus x

0:03:02.030,0:03:04.990
squared plus h squared.

0:03:04.990,0:03:08.610
h squared we know from this[br]left hand side equation.

0:03:08.610,0:03:11.620
h squared is going to be equal[br]to -- so plus, I'll do it in

0:03:11.620,0:03:19.160
that color -- a squared minus x[br]squared is equal to b squared.

0:03:19.160,0:03:21.650
I just substituted the[br]value of that in here, the

0:03:21.650,0:03:23.280
value of that in there.

0:03:23.280,0:03:25.860
Now let's expand this[br]expression out.

0:03:25.860,0:03:29.750
c minus x squared, that[br]is c squared minus

0:03:29.750,0:03:34.320
2cx plus x squared.

0:03:34.320,0:03:38.200
Then we have the minus --[br]sorry, we have the plus a

0:03:38.200,0:03:44.280
squared minus x squared[br]equals b squared.

0:03:44.280,0:03:47.660


0:03:47.660,0:03:50.060
We have an x squared and[br]a minus x squared there,

0:03:50.060,0:03:51.610
so those cancel out.

0:03:51.610,0:03:54.680


0:03:54.680,0:03:58.790
Let's add the 2cx to both[br]sides of this equation.

0:03:58.790,0:04:01.930
So now our equation[br]would become c squared

0:04:01.930,0:04:04.720
plus a squared.

0:04:04.720,0:04:06.490
I'm adding 2cx to both sides.

0:04:06.490,0:04:10.440
So you add 2cx to this,[br]you get 0 is equal to

0:04:10.440,0:04:13.580
b squared plus 2cx.

0:04:13.580,0:04:16.370
All I did here is I canceled[br]out the x squared and then I

0:04:16.370,0:04:19.600
added 2cx to both sides[br]of this equation.

0:04:19.600,0:04:22.130
My goal here is to solve for x.

0:04:22.130,0:04:24.580
Once I solve for x, then[br]I can solve for h and

0:04:24.580,0:04:26.350
apply that formula.

0:04:26.350,0:04:29.090
Now to solve for x, let's[br]subtract b squared

0:04:29.090,0:04:30.040
from both sides.

0:04:30.040,0:04:36.200
So we'll get c squared[br]plus a squared minus b

0:04:36.200,0:04:41.020
squared is equal to 2cx.

0:04:41.020,0:04:46.160
Then if we divide both sides by[br]2c, we get c squared plus a

0:04:46.160,0:04:52.600
squared minus b squared[br]over 2c is equal to x.

0:04:52.600,0:04:54.880
We've just solved for x here.

0:04:54.880,0:04:57.290
Now, our goal is to solve[br]for the height, so that

0:04:57.290,0:04:59.930
we can apply 1/2 times[br]base times height.

0:04:59.930,0:05:04.120
So to do that, we go back to[br]this equation right here

0:05:04.120,0:05:07.040
and solve for our height.

0:05:07.040,0:05:10.800
Let me scroll down[br]a little bit.

0:05:10.800,0:05:16.290
We know that our height[br]squared is equal to a

0:05:16.290,0:05:20.520
squared minus x squared.

0:05:20.520,0:05:23.330
Instead of just writing x[br]squared let's substitute here.

0:05:23.330,0:05:27.430
So it's minus x squared -- x[br]is this thing right here.

0:05:27.430,0:05:32.880
So c squared plus a[br]squared minus b squared

0:05:32.880,0:05:36.670
over 2c, squared.

0:05:36.670,0:05:39.320
This is the same[br]thing as x squared.

0:05:39.320,0:05:41.090
We just solved for that.

0:05:41.090,0:05:47.950
So h is going to be equal to[br]the square root of all this

0:05:47.950,0:05:51.610
business in there -- I'll[br]switch the colors -- of a

0:05:51.610,0:06:00.070
squared minus c squared plus[br]a squared minus b squared

0:06:00.070,0:06:02.150
-- all of that squared.

0:06:02.150,0:06:04.800
Let me make it a little bit[br]neater than that because

0:06:04.800,0:06:06.720
I don't want to--.

0:06:06.720,0:06:13.980
The square root -- make sure I[br]have enough space -- of a

0:06:13.980,0:06:20.130
squared minus all of this stuff[br]squared -- we have c squared

0:06:20.130,0:06:25.800
plus a squared minus b[br]squared, all of that over 2c.

0:06:25.800,0:06:27.670
That is the height[br]of our triangle.

0:06:27.670,0:06:30.310
The triangle that we[br]started off with up here.

0:06:30.310,0:06:33.360
Let me copy and paste that[br]just so that we can remember

0:06:33.360,0:06:36.070
what we're dealing with.

0:06:36.070,0:06:41.600
Copy it and then let me[br]paste it down here.

0:06:41.600,0:06:43.300
So I've pasted it down here.

0:06:43.300,0:06:45.210
So we know what the height[br]is -- it's this big

0:06:45.210,0:06:46.830
convoluted formula.

0:06:46.830,0:06:51.180
The height in terms of a, b[br]and c is this right here.

0:06:51.180,0:06:54.570
So if we wanted to figure out[br]the area -- the area of our

0:06:54.570,0:06:58.270
triangle -- let me[br]do it in pink.

0:06:58.270,0:07:03.772
The area of our triangle is[br]going to be 1/2 times our base

0:07:03.772,0:07:09.850
-- our base is this entire[br]length, c -- times c times our

0:07:09.850,0:07:13.260
height, which is this[br]expression right here.

0:07:13.260,0:07:15.680
Let me just copy and[br]paste this instead of--.

0:07:15.680,0:07:21.390
So let me copy and paste.

0:07:21.390,0:07:24.450
So times the height.

0:07:24.450,0:07:27.910
So this now is our[br]expression for the area.

0:07:27.910,0:07:29.810
Now you're immediately saying[br]gee, that doesn't look a lot

0:07:29.810,0:07:32.790
like Heron's formula,[br]and you're right.

0:07:32.790,0:07:35.360
It does not look a lot like[br]Heron's formula, but what I'm

0:07:35.360,0:07:37.820
going to show you in the next[br]video is that this essentially

0:07:37.820,0:07:39.230
is Heron's formula.

0:07:39.230,0:07:43.050
This is a harder to remember[br]version of Heron's formula.

0:07:43.050,0:07:46.000
I'm going to apply a lot of[br]algebra to essentially simplify

0:07:46.000,0:07:47.260
this to Heron's formula.

0:07:47.260,0:07:49.430
But this will work.

0:07:49.430,0:07:51.520
If you could memorize this,[br]I think Heron's a lot

0:07:51.520,0:07:53.050
easier to memorize.

0:07:53.050,0:07:56.300
But if you can memorize this[br]and you just know a, b and

0:07:56.300,0:08:00.700
c, you apply this formula[br]right here and you will get

0:08:00.700,0:08:04.940
the area of a triangle.

0:08:04.940,0:08:07.290
Well, actually let's just apply[br]this just to show that this at

0:08:07.290,0:08:09.710
least gives the same[br]number as Heron's.

0:08:09.710,0:08:15.920
So in the last video we had a[br]triangle that had sides 9, 11

0:08:15.920,0:08:22.350
and 16, and its area using[br]Heron's was equal to 18

0:08:22.350,0:08:26.290
times the square root of 7.

0:08:26.290,0:08:29.780
Let's see what we get when we[br]applied this formula here.

0:08:29.780,0:08:36.260
So we get the area is equal[br]to 1/2 times 16 times the

0:08:36.260,0:08:40.300
square root of a squared.

0:08:40.300,0:08:49.330
That is 81 minus -- let's see,[br]c squared is 16, so that's 256.

0:08:49.330,0:08:58.020
256 plus a squared, that's[br]at 81 minus b squared,

0:08:58.020,0:09:02.250
so minus 121.

0:09:02.250,0:09:04.120
All of this stuff is squared.

0:09:04.120,0:09:09.530
All of that over 2 times c[br]-- all of that over 32.

0:09:09.530,0:09:12.150
So let's see if we can[br]simplify this a little bit.

0:09:12.150,0:09:15.770
81 minus 121, that is minus 40.

0:09:15.770,0:09:18.790
So this becomes 216 over 32.

0:09:18.790,0:09:22.470
So area is equal to[br]1/2 times 8 is 8.

0:09:22.470,0:09:24.530
Let me switch colors.

0:09:24.530,0:09:38.820
1/2 times 16 is 8 times the[br]square root of 81 minus 256.

0:09:38.820,0:09:41.370
81 minus 121, that's minus 40.

0:09:41.370,0:09:43.370
256 minus 40 is 216.

0:09:43.370,0:09:48.270
216 over 32 squared.

0:09:48.270,0:09:50.630
Now, this is a lot of[br]math to do so let me

0:09:50.630,0:09:51.870
get out a calculator.

0:09:51.870,0:09:54.140
I'm really just trying to show[br]you that these two numbers

0:09:54.140,0:09:57.440
should give us our same number.

0:09:57.440,0:10:01.290
So if we turn on[br]our calculator--.

0:10:01.290,0:10:02.440
First of all, let's just[br]figure out what 18

0:10:02.440,0:10:03.420
square root of 7 are.

0:10:03.420,0:10:07.590
18 times the square root[br]of 7 -- this is what

0:10:07.590,0:10:08.580
we got using Heron's.

0:10:08.580,0:10:11.100
We got 47.62.

0:10:11.100,0:10:13.160
Let's see if this is 47.62.

0:10:13.160,0:10:26.700
So we have 8 times the square[br]root of 81 minus 216 divided

0:10:26.700,0:10:35.140
by 32 squared, and then we[br]close our square roots.

0:10:35.140,0:10:37.990
And we get the[br]exact same number.

0:10:37.990,0:10:39.890
I was worried -- I actually[br]didn't do this calculation

0:10:39.890,0:10:41.580
ahead of time so I might have[br]made a careless mistake.

0:10:41.580,0:10:43.310
But there you go, you get[br]the exact same number.

0:10:43.310,0:10:47.170
So our formula just now gave[br]us the exact same value

0:10:47.170,0:10:48.350
as Heron's formula.

0:10:48.350,0:10:54.030
But what I'm going to do in the[br]next video is prove to you that

0:10:54.030,0:10:57.690
this can actually be reduced[br]algebraically to Heron's.

0:10:57.690,0:10:58.990