0:00:01.090,0:00:02.690
I promised you that I'd give[br]you some more Pythagorean

0:00:02.690,0:00:05.720
theorem problems, so I will[br]now give you more Pythagorean

0:00:05.720,0:00:06.780
theorem problems.

0:00:06.780,0:00:09.790


0:00:09.790,0:00:12.382
And once again, this is[br]all about practice.

0:00:12.382,0:00:28.020
Let's say I had a triangle--[br]that's an ugly looking right

0:00:28.020,0:00:35.030
triangle, let me draw another[br]one --and if I were to tell

0:00:35.030,0:00:40.750
you that that side is 7, the[br]side is 6, and I want to

0:00:40.750,0:00:42.250
figure out this side.

0:00:42.250,0:00:45.510
Well, we learned in the last[br]presentation: which of these

0:00:45.510,0:00:46.990
sides is the hypotenuse?

0:00:46.990,0:00:49.470
Well, here's the right angle,[br]so the side opposite the right

0:00:49.470,0:00:51.600
angle is the hypotenuse.

0:00:51.600,0:00:53.120
So what we want to do[br]is actually figure

0:00:53.120,0:00:54.730
out the hypotenuse.

0:00:54.730,0:01:00.730
So we know that 6 squared[br]plus 7 squared is equal to

0:01:00.730,0:01:01.700
the hypotenuse squared.

0:01:01.700,0:01:03.800
And in the Pythagorean theorem[br]they use C to represent the

0:01:03.800,0:01:05.470
hypotenuse, so we'll[br]use C here as well.

0:01:05.470,0:01:10.930


0:01:10.930,0:01:16.030
And 36 plus 49 is[br]equal to C squared.

0:01:16.030,0:01:21.150


0:01:21.150,0:01:25.510
85 is equal to C squared.

0:01:25.510,0:01:30.760
Or C is equal to the[br]square root of 85.

0:01:30.760,0:01:32.490
And this is the part that most[br]people have trouble with, is

0:01:32.490,0:01:34.650
actually simplifying[br]the radical.

0:01:34.650,0:01:40.290
So the square root of 85: can I[br]factor 85 so it's a product of

0:01:40.290,0:01:42.820
a perfect square and[br]another number?

0:01:42.820,0:01:45.920
85 isn't divisible by 4.

0:01:45.920,0:01:48.350
So it won't be divisible by 16[br]or any of the multiples of 4.

0:01:48.350,0:01:52.400


0:01:52.400,0:01:55.940
5 goes into 85 how many times?

0:01:55.940,0:01:58.340
No, that's not perfect[br]square, either.

0:01:58.340,0:02:02.030
I don't think 85 can be[br]factored further as a

0:02:02.030,0:02:04.230
product of a perfect[br]square and another number.

0:02:04.230,0:02:06.980
So you might correct[br]me; I might be wrong.

0:02:06.980,0:02:09.570
This might be good exercise for[br]you to do later, but as far as

0:02:09.570,0:02:12.670
I can tell we have[br]gotten our answer.

0:02:12.670,0:02:15.070
The answer here is the[br]square root of 85.

0:02:15.070,0:02:17.250
And if we actually wanted to[br]estimate what that is, let's

0:02:17.250,0:02:21.810
think about it: the square root[br]of 81 is 9, and the square root

0:02:21.810,0:02:25.010
of 100 is 10 , so it's some[br]place in between 9 and 10, and

0:02:25.010,0:02:26.445
it's probably a little[br]bit closer to 9.

0:02:26.445,0:02:28.245
So it's 9 point something,[br]something, something.

0:02:28.245,0:02:30.260
And that's a good reality[br]check; that makes sense.

0:02:30.260,0:02:33.080
If this side is 6, this side[br]is 7, 9 point something,

0:02:33.080,0:02:36.270
something, something makes[br]sense for that length.

0:02:36.270,0:02:37.260
Let me give you[br]another problem.

0:02:37.260,0:02:44.790
[DRAWING]

0:02:44.790,0:02:49.250
Let's say that this is 10 .

0:02:49.250,0:02:51.300
This is 3.

0:02:51.300,0:02:53.090
What is this side?

0:02:53.090,0:02:55.060
First, let's identify[br]our hypotenuse.

0:02:55.060,0:02:57.680
We have our right angle here,[br]so the side opposite the right

0:02:57.680,0:03:00.230
angle is the hypotenuse and[br]it's also the longest side.

0:03:00.230,0:03:01.116
So it's 10.

0:03:01.116,0:03:05.390
So 10 squared is equal to[br]the sum of the squares

0:03:05.390,0:03:06.640
of the other two sides.

0:03:06.640,0:03:10.256
This is equal to 3 squared--[br]let's call this A.

0:03:10.256,0:03:11.890
Pick it arbitrarily.

0:03:11.890,0:03:14.380
--plus A squared.

0:03:14.380,0:03:23.860
Well, this is 100, is equal to[br]9 plus A squared, or A squared

0:03:23.860,0:03:29.720
is equal to 100 minus 9.

0:03:29.720,0:03:32.560
A squared is equal to 91.

0:03:32.560,0:03:38.390


0:03:38.390,0:03:40.390
I don't think that can be[br]simplified further, either.

0:03:40.390,0:03:41.710
3 doesn't go into it.

0:03:41.710,0:03:43.950
I wonder, is 91 a prime number?

0:03:43.950,0:03:44.880
I'm not sure.

0:03:44.880,0:03:49.200
As far as I know, we're[br]done with this problem.

0:03:49.200,0:03:51.890
Let me give you another[br]problem, And actually, this

0:03:51.890,0:03:56.500
time I'm going to include one[br]extra step just to confuse you

0:03:56.500,0:04:00.240
because I think you're getting[br]this a little bit too easily.

0:04:00.240,0:04:01.805
Let's say I have a triangle.

0:04:01.805,0:04:05.130


0:04:05.130,0:04:07.990
And once again, we're dealing[br]all with right triangles now.

0:04:07.990,0:04:10.130
And never are you going to[br]attempt to use the Pythagorean

0:04:10.130,0:04:12.780
theorem unless you know for a[br]fact that's all right triangle.

0:04:12.780,0:04:16.130


0:04:16.130,0:04:19.810
But this example, we know[br]that this is right triangle.

0:04:19.810,0:04:25.050
If I would tell you the length[br]of this side is 5, and if our

0:04:25.050,0:04:32.810
tell you that this angle is 45[br]degrees, can we figure out the

0:04:32.810,0:04:36.410
other two sides of[br]this triangle?

0:04:36.410,0:04:38.220
Well, we can't use the[br]Pythagorean theorem directly

0:04:38.220,0:04:40.830
because the Pythagorean theorem[br]tells us that if have a right

0:04:40.830,0:04:43.750
triangle and we know two of the[br]sides that we can figure

0:04:43.750,0:04:45.140
out the third side.

0:04:45.140,0:04:47.320
Here we have a right[br]triangle and we only

0:04:47.320,0:04:48.870
know one of the sides.

0:04:48.870,0:04:51.080
So we can't figure out[br]the other two just yet.

0:04:51.080,0:04:54.330
But maybe we can use this extra[br]information right here, this 45

0:04:54.330,0:04:57.120
degrees, to figure out another[br]side, and then we'd be able

0:04:57.120,0:04:59.280
use the Pythagorean theorem.

0:04:59.280,0:05:01.810
Well, we know that the[br]angles in a triangle

0:05:01.810,0:05:03.860
add up to 180 degrees.

0:05:03.860,0:05:05.610
Well, hopefully you know[br]the angles in a triangle

0:05:05.610,0:05:06.630
add up to 180 degrees.

0:05:06.630,0:05:08.320
If you don't it's my fault[br]because I haven't taught

0:05:08.320,0:05:09.720
you that already.

0:05:09.720,0:05:14.310
So let's figure out what[br]the angles of this

0:05:14.310,0:05:15.080
triangle add up to.

0:05:15.080,0:05:17.410
Well, I mean we know they add[br]up to 180, but using that

0:05:17.410,0:05:20.790
information, we could figure[br]out what this angle is.

0:05:20.790,0:05:23.590
Because we know that this angle[br]is 90, this angle is 45.

0:05:23.590,0:05:30.340
So we say 45-- lets call this[br]angle x; I'm trying to make it

0:05:30.340,0:05:35.870
messy --45 plus 90--[br]this just symbolizes

0:05:35.870,0:05:40.720
a 90 degree angle --plus x[br]is equal to 180 degrees.

0:05:40.720,0:05:43.520
And that's because the[br]angles in a triangle always

0:05:43.520,0:05:46.740
add up to 180 degrees.

0:05:46.740,0:05:55.970
So if we just solve for x, we[br]get 135 plus x is equal to 180.

0:05:55.970,0:05:57.550
Subtract 135 from both sides.

0:05:57.550,0:06:01.190
We get x is equal[br]to 45 degrees.

0:06:01.190,0:06:02.680
Interesting.

0:06:02.680,0:06:06.800
x is also 45 degrees.

0:06:06.800,0:06:11.380
So we have a 90 degree angle[br]and two 45 degree angles.

0:06:11.380,0:06:13.710
Now I'm going to give you[br]another theorem that's not

0:06:13.710,0:06:16.920
named after the head[br]of a religion or the

0:06:16.920,0:06:17.560
founder of religion.

0:06:17.560,0:06:19.730
I actually don't think this[br]theorem doesn't have a name at.

0:06:19.730,0:06:26.920
All It's the fact that if I[br]have another triangle --I'm

0:06:26.920,0:06:31.980
going to draw another triangle[br]out here --where two of the

0:06:31.980,0:06:34.840
base angles are the same-- and[br]when I say base angle, I just

0:06:34.840,0:06:39.890
mean if these two angles are[br]the same, let's call it a.

0:06:39.890,0:06:44.770
They're both a --then the sides[br]that they don't share-- these

0:06:44.770,0:06:46.610
angles share this side, right?

0:06:46.610,0:06:49.560
--but if we look at the sides[br]that they don't share, we know

0:06:49.560,0:06:53.240
that these sides are equal.

0:06:53.240,0:06:54.810
I forgot what we call[br]this in geometry class.

0:06:54.810,0:06:57.270
Maybe I'll look it up in[br]another presentation;

0:06:57.270,0:06:57.960
I'll let you know.

0:06:57.960,0:07:00.040
But I got this far without[br]knowing what the name

0:07:00.040,0:07:01.370
of the theorem is.

0:07:01.370,0:07:04.170
And it makes sense; you don't[br]even need me to tell you that.

0:07:04.170,0:07:07.080


0:07:07.080,0:07:10.480
If I were to change one of[br]these angles, the length

0:07:10.480,0:07:11.660
would also change.

0:07:11.660,0:07:14.310
Or another way to think about[br]it, the only way-- no, I

0:07:14.310,0:07:15.350
don't confuse you too much.

0:07:15.350,0:07:18.820
But you can visually see that[br]if these two sides are the

0:07:18.820,0:07:21.670
same, then these two angles[br]are going to be the same.

0:07:21.670,0:07:25.430
If you changed one of these[br]sides' lengths, then the angles

0:07:25.430,0:07:28.660
will also change, or the angles[br]will not be equal anymore.

0:07:28.660,0:07:31.120
But I'll leave that for[br]you to think about.

0:07:31.120,0:07:34.320
But just take my word for it[br]right now that if two angles in

0:07:34.320,0:07:39.400
a triangle are equivalent, then[br]the sides that they don't share

0:07:39.400,0:07:41.690
are also equal in length.

0:07:41.690,0:07:43.820
Make sure you remember: not the[br]side that they share-- because

0:07:43.820,0:07:46.920
that can't be equal to anything[br]--it's the side that they don't

0:07:46.920,0:07:49.410
share are equal in length.

0:07:49.410,0:07:52.990
So here we have an example[br]where we have to equal angles.

0:07:52.990,0:07:55.020
They're both 45 degrees.

0:07:55.020,0:07:58.910
So that means that the sides[br]that they don't share-- this is

0:07:58.910,0:08:00.230
the side they share, right?

0:08:00.230,0:08:03.210
Both angle share this side --so[br]that means that the side that

0:08:03.210,0:08:05.080
they don't share are equal.

0:08:05.080,0:08:08.460
So this side is[br]equal to this side.

0:08:08.460,0:08:10.520
And I think you might be[br]experiencing an ah-hah

0:08:10.520,0:08:12.020
moment that right now.

0:08:12.020,0:08:15.380
Well this side is equal to this[br]side-- I gave you at the

0:08:15.380,0:08:18.050
beginning of this problem that[br]this side is equal to 5 --so

0:08:18.050,0:08:20.320
then we know that this[br]side is equal to 5.

0:08:20.320,0:08:23.920
And now we can do the[br]Pythagorean theorem.

0:08:23.920,0:08:25.750
We know this is the[br]hypotenuse, right?

0:08:25.750,0:08:28.940


0:08:28.940,0:08:35.180
So we can say 5 squared plus 5[br]squared is equal to-- let's say

0:08:35.180,0:08:38.950
C squared, where C is the[br]length of the hypotenuse --5

0:08:38.950,0:08:42.010
squared plus 5 squared-- that's[br]just 50 --is equal

0:08:42.010,0:08:44.110
to C squared.

0:08:44.110,0:08:48.370
And then we get C is equal[br]to the square root of 50.

0:08:48.370,0:08:56.250
And 50 is 2 times 25, so C is[br]equal to 5 square roots of 2.

0:08:56.250,0:08:57.220
Interesting.

0:08:57.220,0:09:00.110
So I think I might have given[br]you a lot of information there.

0:09:00.110,0:09:02.840
If you get confused, maybe you[br]want to re-watch this video.

0:09:02.840,0:09:05.630
But on the next video I'm[br]actually going to give you more

0:09:05.630,0:09:08.095
information about this type of[br]triangle, which is actually a

0:09:08.095,0:09:11.550
very common type of triangle[br]you'll see in geometry and

0:09:11.550,0:09:14.470
trigonometry 45,[br]45, 90 triangle.

0:09:14.470,0:09:15.930
And it makes sense why it's[br]called that because it has

0:09:15.930,0:09:19.930
45 degrees, 45 degrees,[br]and a 90 degree angle.

0:09:19.930,0:09:22.460
And I'll actually show you[br]a quick way of using that

0:09:22.460,0:09:25.920
information that it is a 45,[br]45, 90 degree triangle to

0:09:25.920,0:09:29.520
figure out the size if you're[br]given even one of the sides.

0:09:29.520,0:09:31.870
I hope I haven't confused you[br]too much, and I look forward

0:09:31.870,0:09:33.195
to seeing you in the[br]next presentation.

0:09:33.195,0:09:35.120
See you later.