[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.67,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.67,0:00:04.04,Default,,0000,0000,0000,,This triangle that we have right\Nover here is a right triangle.
Dialogue: 0,0:00:04.04,0:00:06.60,Default,,0000,0000,0000,,And it's a right triangle\Nbecause it has a 90 degree
Dialogue: 0,0:00:06.60,0:00:09.24,Default,,0000,0000,0000,,angle, or has a\Nright angle in it.
Dialogue: 0,0:00:09.24,0:00:12.52,Default,,0000,0000,0000,,Now, we call the longest\Nside of a right triangle,
Dialogue: 0,0:00:12.52,0:00:14.60,Default,,0000,0000,0000,,we call that side,\Nand you could either
Dialogue: 0,0:00:14.60,0:00:17.14,Default,,0000,0000,0000,,view it as the longest side of\Nthe right triangle or the side
Dialogue: 0,0:00:17.14,0:00:20.98,Default,,0000,0000,0000,,opposite the 90 degree angle,\Nit is called a hypotenuse.
Dialogue: 0,0:00:20.98,0:00:23.74,Default,,0000,0000,0000,,It's a very fancy word\Nfor a fairly simple idea,
Dialogue: 0,0:00:23.74,0:00:26.14,Default,,0000,0000,0000,,just the longest side of a\Nright triangle or the side
Dialogue: 0,0:00:26.14,0:00:27.54,Default,,0000,0000,0000,,opposite the 90 degree angle.
Dialogue: 0,0:00:27.54,0:00:29.50,Default,,0000,0000,0000,,And it's just good to\Nknow that because someone
Dialogue: 0,0:00:29.50,0:00:30.09,Default,,0000,0000,0000,,might say hypotenuse.
Dialogue: 0,0:00:30.09,0:00:32.55,Default,,0000,0000,0000,,You're like, oh, they're just\Ntalking about this side right
Dialogue: 0,0:00:32.55,0:00:36.58,Default,,0000,0000,0000,,here, the side longest, the side\Nopposite the 90 degree angle.
Dialogue: 0,0:00:36.58,0:00:38.86,Default,,0000,0000,0000,,Now, what I want\Nto do in this video
Dialogue: 0,0:00:38.86,0:00:42.17,Default,,0000,0000,0000,,is prove a relationship, a\Nvery famous relationship.
Dialogue: 0,0:00:42.17,0:00:43.75,Default,,0000,0000,0000,,And you might see\Nwhere this is going.
Dialogue: 0,0:00:43.75,0:00:46.37,Default,,0000,0000,0000,,A very famous relationship\Nbetween the lengths
Dialogue: 0,0:00:46.37,0:00:48.84,Default,,0000,0000,0000,,of the sides of\Na right triangle.
Dialogue: 0,0:00:48.84,0:00:53.21,Default,,0000,0000,0000,,So let's say that the length of\NAC, so uppercase A, uppercase
Dialogue: 0,0:00:53.21,0:00:55.93,Default,,0000,0000,0000,,C, let's call that\Nlength lowercase a.
Dialogue: 0,0:00:55.93,0:01:00.04,Default,,0000,0000,0000,,Let's call the length of BC\Nlowercase b right over here.
Dialogue: 0,0:01:00.04,0:01:03.42,Default,,0000,0000,0000,,I'll use uppercases for\Npoints, lowercases for lengths.
Dialogue: 0,0:01:03.42,0:01:06.63,Default,,0000,0000,0000,,And let's call the length of the\Nhypotenuse, the length of AB,
Dialogue: 0,0:01:06.63,0:01:07.82,Default,,0000,0000,0000,,let's call that c.
Dialogue: 0,0:01:07.82,0:01:10.03,Default,,0000,0000,0000,,And let's see if we can come\Nup with the relationship
Dialogue: 0,0:01:10.03,0:01:12.79,Default,,0000,0000,0000,,between a, b, and c.
Dialogue: 0,0:01:12.79,0:01:14.78,Default,,0000,0000,0000,,And to do that I'm\Nfirst going to construct
Dialogue: 0,0:01:14.78,0:01:16.41,Default,,0000,0000,0000,,another line or\Nanother segment, I
Dialogue: 0,0:01:16.41,0:01:19.52,Default,,0000,0000,0000,,should say, between\Nc and the hypotenuse.
Dialogue: 0,0:01:19.52,0:01:21.60,Default,,0000,0000,0000,,And I'm going to\Nconstruct it so that they
Dialogue: 0,0:01:21.60,0:01:23.88,Default,,0000,0000,0000,,intersect at a right angle.
Dialogue: 0,0:01:23.88,0:01:25.01,Default,,0000,0000,0000,,And you can always do that.
Dialogue: 0,0:01:25.01,0:01:27.00,Default,,0000,0000,0000,,And we'll call this point\Nright over here we'll.
Dialogue: 0,0:01:27.00,0:01:28.12,Default,,0000,0000,0000,,Call this point capital D.
Dialogue: 0,0:01:28.12,0:01:31.01,Default,,0000,0000,0000,,And if you're wondering,\Nhow can you always do that?
Dialogue: 0,0:01:31.01,0:01:33.63,Default,,0000,0000,0000,,You could imagine rotating\Nthis entire triangle like this.
Dialogue: 0,0:01:33.63,0:01:36.05,Default,,0000,0000,0000,,This isn't a rigorous proof,\Nbut it just kind of gives you
Dialogue: 0,0:01:36.05,0:01:38.10,Default,,0000,0000,0000,,the general idea of\Nhow you can always
Dialogue: 0,0:01:38.10,0:01:39.81,Default,,0000,0000,0000,,construct a point like this.
Dialogue: 0,0:01:39.81,0:01:41.26,Default,,0000,0000,0000,,So if I've rotated it around.
Dialogue: 0,0:01:41.26,0:01:44.75,Default,,0000,0000,0000,,So now our hypotenuse, we're\Nnow sitting on our hypotenuse.
Dialogue: 0,0:01:44.75,0:01:48.41,Default,,0000,0000,0000,,This is now point\NB, this is point A.
Dialogue: 0,0:01:48.41,0:01:50.58,Default,,0000,0000,0000,,So we've rotated the whole\Nthing all the way around.
Dialogue: 0,0:01:50.58,0:01:52.71,Default,,0000,0000,0000,,This is point C. You\Ncould imagine just
Dialogue: 0,0:01:52.71,0:01:55.82,Default,,0000,0000,0000,,dropping a rock from point C,\Nmaybe with a string attached,
Dialogue: 0,0:01:55.82,0:01:59.46,Default,,0000,0000,0000,,and it would hit the\Nhypotenuse at a right angle.
Dialogue: 0,0:01:59.46,0:02:02.98,Default,,0000,0000,0000,,So that's all we did here to\Nestablish segment CD into where
Dialogue: 0,0:02:02.98,0:02:05.57,Default,,0000,0000,0000,,we put our point D\Nright over there.
Dialogue: 0,0:02:05.57,0:02:07.22,Default,,0000,0000,0000,,And the reason why\Ndid that is now we
Dialogue: 0,0:02:07.22,0:02:09.29,Default,,0000,0000,0000,,can do all sorts of\Ninteresting relationships
Dialogue: 0,0:02:09.29,0:02:10.49,Default,,0000,0000,0000,,between similar triangles.
Dialogue: 0,0:02:10.49,0:02:12.18,Default,,0000,0000,0000,,Because we have\Nthree triangles here.
Dialogue: 0,0:02:12.18,0:02:15.60,Default,,0000,0000,0000,,We have triangle ADC,\Nwe have triangle DBC,
Dialogue: 0,0:02:15.60,0:02:17.52,Default,,0000,0000,0000,,and then we have the\Nlarger original triangle.
Dialogue: 0,0:02:17.52,0:02:19.89,Default,,0000,0000,0000,,And we can hopefully\Nestablish similarity
Dialogue: 0,0:02:19.89,0:02:21.98,Default,,0000,0000,0000,,between those triangles.
Dialogue: 0,0:02:21.98,0:02:27.59,Default,,0000,0000,0000,,And first I'll show you that ADC\Nis similar to the larger one.
Dialogue: 0,0:02:27.59,0:02:29.71,Default,,0000,0000,0000,,Because both of them\Nhave a right angle.
Dialogue: 0,0:02:29.71,0:02:32.07,Default,,0000,0000,0000,,ADC has a right angle\Nright over here.
Dialogue: 0,0:02:32.07,0:02:33.57,Default,,0000,0000,0000,,Clearly if this\Nangle is 90 degrees,
Dialogue: 0,0:02:33.57,0:02:35.65,Default,,0000,0000,0000,,then this angle is going\Nto be 90 degrees as well.
Dialogue: 0,0:02:35.65,0:02:36.66,Default,,0000,0000,0000,,They are supplementary.
Dialogue: 0,0:02:36.66,0:02:38.51,Default,,0000,0000,0000,,They have to add up to 180.
Dialogue: 0,0:02:38.51,0:02:40.44,Default,,0000,0000,0000,,And so they both have\Na right angle in them.
Dialogue: 0,0:02:40.44,0:02:42.06,Default,,0000,0000,0000,,So the smaller one\Nhas a right angle.
Dialogue: 0,0:02:42.06,0:02:43.59,Default,,0000,0000,0000,,The larger one clearly\Nhas a right angle.
Dialogue: 0,0:02:43.59,0:02:44.84,Default,,0000,0000,0000,,That's where we started from.
Dialogue: 0,0:02:44.84,0:02:48.69,Default,,0000,0000,0000,,And they also both\Nshare this angle right
Dialogue: 0,0:02:48.69,0:02:52.15,Default,,0000,0000,0000,,over here, angle\NDAC or BAC, however
Dialogue: 0,0:02:52.15,0:02:53.58,Default,,0000,0000,0000,,you want to refer to it.
Dialogue: 0,0:02:53.58,0:02:56.72,Default,,0000,0000,0000,,So we can actually write\Ndown that triangle.
Dialogue: 0,0:02:56.72,0:03:00.29,Default,,0000,0000,0000,,I'm going to start with\Nthe smaller one, ADC.
Dialogue: 0,0:03:00.29,0:03:02.19,Default,,0000,0000,0000,,And maybe I'll shade\Nit in right over here.
Dialogue: 0,0:03:02.19,0:03:04.02,Default,,0000,0000,0000,,So this is the triangle\Nwe're talking about.
Dialogue: 0,0:03:04.02,0:03:05.43,Default,,0000,0000,0000,,Triangle ADC.
Dialogue: 0,0:03:05.43,0:03:07.47,Default,,0000,0000,0000,,And I went from the blue\Nangle to the right angle
Dialogue: 0,0:03:07.47,0:03:10.62,Default,,0000,0000,0000,,to the unlabeled angle from the\Npoint of view of triangle ADC.
Dialogue: 0,0:03:10.62,0:03:13.86,Default,,0000,0000,0000,,This right angle isn't applying\Nto that right over there.
Dialogue: 0,0:03:13.86,0:03:15.82,Default,,0000,0000,0000,,It's applying to\Nthe larger triangle.
Dialogue: 0,0:03:15.82,0:03:24.82,Default,,0000,0000,0000,,So we could say triangle\NADC is similar to triangle--
Dialogue: 0,0:03:24.82,0:03:27.13,Default,,0000,0000,0000,,once again, you want to\Nstart at the blue angle.
Dialogue: 0,0:03:27.13,0:03:29.50,Default,,0000,0000,0000,,A. Then we went to\Nthe right angle.
Dialogue: 0,0:03:29.50,0:03:32.22,Default,,0000,0000,0000,,So we have to go to\Nthe right angle again.
Dialogue: 0,0:03:32.22,0:03:32.83,Default,,0000,0000,0000,,So it's ACB.
Dialogue: 0,0:03:32.83,0:03:37.19,Default,,0000,0000,0000,,
Dialogue: 0,0:03:37.19,0:03:39.27,Default,,0000,0000,0000,,And because they're\Nsimilar, we can set up
Dialogue: 0,0:03:39.27,0:03:42.22,Default,,0000,0000,0000,,a relationship between\Nthe ratios of their sides.
Dialogue: 0,0:03:42.22,0:03:44.70,Default,,0000,0000,0000,,For example, we know the\Nratio of corresponding sides
Dialogue: 0,0:03:44.70,0:03:47.08,Default,,0000,0000,0000,,are going to do, well, in\Ngeneral for a similar triangle,
Dialogue: 0,0:03:47.08,0:03:48.64,Default,,0000,0000,0000,,we know the ratio of\Nthe corresponding sides
Dialogue: 0,0:03:48.64,0:03:49.89,Default,,0000,0000,0000,,are going to be a constant.
Dialogue: 0,0:03:49.89,0:03:54.10,Default,,0000,0000,0000,,So we could take the ratio of\Nthe hypotenuse of the smaller
Dialogue: 0,0:03:54.10,0:03:54.96,Default,,0000,0000,0000,,triangle.
Dialogue: 0,0:03:54.96,0:03:57.35,Default,,0000,0000,0000,,So the hypotenuse is AC.
Dialogue: 0,0:03:57.35,0:04:00.71,Default,,0000,0000,0000,,So AC over the hypotenuse\Nover the larger one, which
Dialogue: 0,0:04:00.71,0:04:10.48,Default,,0000,0000,0000,,is a AB, AC over AB is going\Nto be the same thing as AD
Dialogue: 0,0:04:10.48,0:04:14.18,Default,,0000,0000,0000,,as one of the legs, AD.
Dialogue: 0,0:04:14.18,0:04:16.96,Default,,0000,0000,0000,,And just to show that, I'm just\Ntaking corresponding points
Dialogue: 0,0:04:16.96,0:04:23.79,Default,,0000,0000,0000,,on both similar triangles,\Nthis is AD over AC.
Dialogue: 0,0:04:23.79,0:04:25.96,Default,,0000,0000,0000,,You could look at these\Ntriangles yourself and show,
Dialogue: 0,0:04:25.96,0:04:29.93,Default,,0000,0000,0000,,look, AD, point AD, is\Nbetween the blue angle
Dialogue: 0,0:04:29.93,0:04:31.41,Default,,0000,0000,0000,,and the right angle.
Dialogue: 0,0:04:31.41,0:04:34.76,Default,,0000,0000,0000,,Sorry, side AD is between the\Nblue angle and the right angle.
Dialogue: 0,0:04:34.76,0:04:38.02,Default,,0000,0000,0000,,Side AC is between the blue\Nangle and the right angle
Dialogue: 0,0:04:38.02,0:04:39.01,Default,,0000,0000,0000,,on the larger triangle.
Dialogue: 0,0:04:39.01,0:04:40.95,Default,,0000,0000,0000,,So both of these are\Nfrom the larger triangle.
Dialogue: 0,0:04:40.95,0:04:43.66,Default,,0000,0000,0000,,These are the corresponding\Nsides on the smaller triangle.
Dialogue: 0,0:04:43.66,0:04:46.99,Default,,0000,0000,0000,,And if that is confusing\Nlooking at them visually,
Dialogue: 0,0:04:46.99,0:04:50.20,Default,,0000,0000,0000,,as long as we wrote our\Nsimilarity statement correctly,
Dialogue: 0,0:04:50.20,0:04:51.99,Default,,0000,0000,0000,,you can just find the\Ncorresponding points.
Dialogue: 0,0:04:51.99,0:04:56.59,Default,,0000,0000,0000,,AC corresponds to AB\Non the larger triangle,
Dialogue: 0,0:04:56.59,0:04:58.84,Default,,0000,0000,0000,,AD on the smaller\Ntriangle corresponds
Dialogue: 0,0:04:58.84,0:05:02.33,Default,,0000,0000,0000,,to AC on the larger triangle.
Dialogue: 0,0:05:02.33,0:05:06.92,Default,,0000,0000,0000,,And we know that AC, we can\Nrewrite that as lowercase a.
Dialogue: 0,0:05:06.92,0:05:10.86,Default,,0000,0000,0000,,AC is lowercase a.
Dialogue: 0,0:05:10.86,0:05:16.81,Default,,0000,0000,0000,,We don't have any\Nlabel for AD or for AB.
Dialogue: 0,0:05:16.81,0:05:18.90,Default,,0000,0000,0000,,Sorry, we do have\Na label for AB.
Dialogue: 0,0:05:18.90,0:05:20.59,Default,,0000,0000,0000,,That is c right over here.
Dialogue: 0,0:05:20.59,0:05:23.79,Default,,0000,0000,0000,,We don't have a label for AD.
Dialogue: 0,0:05:23.79,0:05:26.84,Default,,0000,0000,0000,,So AD, let's just\Ncall that lowercase d.
Dialogue: 0,0:05:26.84,0:05:30.40,Default,,0000,0000,0000,,So lowercase d applies to\Nthat part right over there.
Dialogue: 0,0:05:30.40,0:05:33.56,Default,,0000,0000,0000,,c applies to that entire\Npart right over there.
Dialogue: 0,0:05:33.56,0:05:35.90,Default,,0000,0000,0000,,And then we'll call DB,\Nlet's call that length e.
Dialogue: 0,0:05:35.90,0:05:38.70,Default,,0000,0000,0000,,That'll just make things a\Nlittle bit simpler for us.
Dialogue: 0,0:05:38.70,0:05:41.76,Default,,0000,0000,0000,,So AD we'll just call d.
Dialogue: 0,0:05:41.76,0:05:43.85,Default,,0000,0000,0000,,And so we have a over\Nc is equal to d over a.
Dialogue: 0,0:05:43.85,0:05:47.83,Default,,0000,0000,0000,,If we cross multiply, you have\Na times a, which is a squared,
Dialogue: 0,0:05:47.83,0:05:50.79,Default,,0000,0000,0000,,is equal to c times\Nd, which is cd.
Dialogue: 0,0:05:50.79,0:05:52.79,Default,,0000,0000,0000,,So that's a little bit\Nof an interesting result.
Dialogue: 0,0:05:52.79,0:05:54.79,Default,,0000,0000,0000,,Let's see what we can do\Nwith the other triangle
Dialogue: 0,0:05:54.79,0:05:55.93,Default,,0000,0000,0000,,right over here.
Dialogue: 0,0:05:55.93,0:05:57.94,Default,,0000,0000,0000,,So this triangle\Nright over here.
Dialogue: 0,0:05:57.94,0:05:59.49,Default,,0000,0000,0000,,So once again, it\Nhas a right angle.
Dialogue: 0,0:05:59.49,0:06:00.86,Default,,0000,0000,0000,,The larger one\Nhas a right angle.
Dialogue: 0,0:06:00.86,0:06:04.27,Default,,0000,0000,0000,,And they both share this\Nangle right over here.
Dialogue: 0,0:06:04.27,0:06:07.07,Default,,0000,0000,0000,,So by angle, angle\Nsimilarity, the two triangles
Dialogue: 0,0:06:07.07,0:06:08.21,Default,,0000,0000,0000,,are going to be similar.
Dialogue: 0,0:06:08.21,0:06:11.04,Default,,0000,0000,0000,,So we could say triangle\NBDC, we went from pink
Dialogue: 0,0:06:11.04,0:06:12.97,Default,,0000,0000,0000,,to right to not labeled.
Dialogue: 0,0:06:12.97,0:06:20.35,Default,,0000,0000,0000,,So triangle BDC is\Nsimilar to triangle.
Dialogue: 0,0:06:20.35,0:06:22.31,Default,,0000,0000,0000,,Now we're going to look\Nat the larger triangle,
Dialogue: 0,0:06:22.31,0:06:23.43,Default,,0000,0000,0000,,we're going to start\Nat the pink angle.
Dialogue: 0,0:06:23.43,0:06:25.57,Default,,0000,0000,0000,,B. Now we're going to\Ngo to the right angle.
Dialogue: 0,0:06:25.57,0:06:26.07,Default,,0000,0000,0000,,CA.
Dialogue: 0,0:06:26.07,0:06:29.19,Default,,0000,0000,0000,,
Dialogue: 0,0:06:29.19,0:06:31.68,Default,,0000,0000,0000,,BCA.
Dialogue: 0,0:06:31.68,0:06:34.98,Default,,0000,0000,0000,,From pink angle to right\Nangle to non-labeled angle,
Dialogue: 0,0:06:34.98,0:06:36.52,Default,,0000,0000,0000,,at least from the\Npoint of view here.
Dialogue: 0,0:06:36.52,0:06:38.42,Default,,0000,0000,0000,,We labeled it before\Nwith that blue.
Dialogue: 0,0:06:38.42,0:06:40.62,Default,,0000,0000,0000,,So now let's set up some\Ntype of relationship here.
Dialogue: 0,0:06:40.62,0:06:45.04,Default,,0000,0000,0000,,We can say that the ratio on\Nthe smaller triangle, BC, side
Dialogue: 0,0:06:45.04,0:06:50.13,Default,,0000,0000,0000,,BC over BA, BC over\NBA, once again,
Dialogue: 0,0:06:50.13,0:06:53.23,Default,,0000,0000,0000,,we're taking the\Nhypotenuses of both of them.
Dialogue: 0,0:06:53.23,0:07:00.59,Default,,0000,0000,0000,,So BC over BA is going\Nto be equal to BD.
Dialogue: 0,0:07:00.59,0:07:02.59,Default,,0000,0000,0000,,Let me do this in another color.
Dialogue: 0,0:07:02.59,0:07:03.45,Default,,0000,0000,0000,,BD.
Dialogue: 0,0:07:03.45,0:07:04.89,Default,,0000,0000,0000,,So this is one of the legs.
Dialogue: 0,0:07:04.89,0:07:05.57,Default,,0000,0000,0000,,BD.
Dialogue: 0,0:07:05.57,0:07:07.43,Default,,0000,0000,0000,,The way I drew it\Nis the shorter legs.
Dialogue: 0,0:07:07.43,0:07:10.37,Default,,0000,0000,0000,,BD over BC.
Dialogue: 0,0:07:10.37,0:07:12.77,Default,,0000,0000,0000,,I'm just taking the\Ncorresponding vertices.
Dialogue: 0,0:07:12.77,0:07:14.60,Default,,0000,0000,0000,,Over BC.
Dialogue: 0,0:07:14.60,0:07:18.20,Default,,0000,0000,0000,,And once again, we know BC is\Nthe same thing as lowercase b.
Dialogue: 0,0:07:18.20,0:07:20.32,Default,,0000,0000,0000,,BC is lowercase b.
Dialogue: 0,0:07:20.32,0:07:22.93,Default,,0000,0000,0000,,BA is lowercase c.
Dialogue: 0,0:07:22.93,0:07:25.57,Default,,0000,0000,0000,,
Dialogue: 0,0:07:25.57,0:07:29.74,Default,,0000,0000,0000,,And then BD we defined\Nas lowercase e.
Dialogue: 0,0:07:29.74,0:07:31.26,Default,,0000,0000,0000,,So this is lowercase e.
Dialogue: 0,0:07:31.26,0:07:33.21,Default,,0000,0000,0000,,We can cross\Nmultiply here and we
Dialogue: 0,0:07:33.21,0:07:37.83,Default,,0000,0000,0000,,get b times b, which, and I've\Nmentioned this in many videos,
Dialogue: 0,0:07:37.83,0:07:40.31,Default,,0000,0000,0000,,cross multiplying is really\Nthe same thing as multiplying
Dialogue: 0,0:07:40.31,0:07:42.68,Default,,0000,0000,0000,,both sides by both denominators.
Dialogue: 0,0:07:42.68,0:07:47.96,Default,,0000,0000,0000,,b times b is b squared\Nis equal to ce.
Dialogue: 0,0:07:47.96,0:07:50.01,Default,,0000,0000,0000,,And now we can do something\Nkind of interesting.
Dialogue: 0,0:07:50.01,0:07:51.41,Default,,0000,0000,0000,,We can add these two statements.
Dialogue: 0,0:07:51.41,0:07:53.03,Default,,0000,0000,0000,,Let me rewrite the\Nstatement down here.
Dialogue: 0,0:07:53.03,0:07:56.10,Default,,0000,0000,0000,,So b squared is equal to ce.
Dialogue: 0,0:07:56.10,0:07:58.31,Default,,0000,0000,0000,,So if we add the\Nleft hand sides,
Dialogue: 0,0:07:58.31,0:08:02.12,Default,,0000,0000,0000,,we get a squared plus b squared.
Dialogue: 0,0:08:02.12,0:08:09.42,Default,,0000,0000,0000,,a squared plus b squared\Nis equal to cd plus ce.
Dialogue: 0,0:08:09.42,0:08:12.60,Default,,0000,0000,0000,,
Dialogue: 0,0:08:12.60,0:08:14.92,Default,,0000,0000,0000,,And then we have a c\Nboth of these terms,
Dialogue: 0,0:08:14.92,0:08:16.00,Default,,0000,0000,0000,,so we could factor it out.
Dialogue: 0,0:08:16.00,0:08:19.88,Default,,0000,0000,0000,,So this is going to be equal\Nto-- we can factor out the c.
Dialogue: 0,0:08:19.88,0:08:22.95,Default,,0000,0000,0000,,It's going to be equal\Nto c times d plus e.
Dialogue: 0,0:08:22.95,0:08:29.79,Default,,0000,0000,0000,,c times d plus e and\Nclose the parentheses.
Dialogue: 0,0:08:29.79,0:08:31.46,Default,,0000,0000,0000,,Now what is d plus e?
Dialogue: 0,0:08:31.46,0:08:34.16,Default,,0000,0000,0000,,d is this length,\Ne is this length.
Dialogue: 0,0:08:34.16,0:08:37.17,Default,,0000,0000,0000,,So d plus e is actually\Ngoing to be c as well.
Dialogue: 0,0:08:37.17,0:08:38.50,Default,,0000,0000,0000,,So this is going to be c.
Dialogue: 0,0:08:38.50,0:08:41.04,Default,,0000,0000,0000,,So you have c times c,\Nwhich is just the same thing
Dialogue: 0,0:08:41.04,0:08:43.03,Default,,0000,0000,0000,,as c squared.
Dialogue: 0,0:08:43.03,0:08:45.70,Default,,0000,0000,0000,,So now we have an\Ninteresting relationship.
Dialogue: 0,0:08:45.70,0:08:51.15,Default,,0000,0000,0000,,We have that a squared plus b\Nsquared is equal to c squared.
Dialogue: 0,0:08:51.15,0:08:52.58,Default,,0000,0000,0000,,Let me rewrite that.
Dialogue: 0,0:08:52.58,0:08:54.30,Default,,0000,0000,0000,,a squared.
Dialogue: 0,0:08:54.30,0:08:58.62,Default,,0000,0000,0000,,Well, let me just do\Nan arbitrary new color.
Dialogue: 0,0:08:58.62,0:09:02.38,Default,,0000,0000,0000,,I deleted that by accident,\Nso let me rewrite it.
Dialogue: 0,0:09:02.38,0:09:07.39,Default,,0000,0000,0000,,So we've just established\Nthat a squared plus b squared
Dialogue: 0,0:09:07.39,0:09:09.40,Default,,0000,0000,0000,,is equal to c squared.
Dialogue: 0,0:09:09.40,0:09:11.32,Default,,0000,0000,0000,,And this is just an\Narbitrary right triangle.
Dialogue: 0,0:09:11.32,0:09:13.59,Default,,0000,0000,0000,,This is true for any\Ntwo right triangles.
Dialogue: 0,0:09:13.59,0:09:17.12,Default,,0000,0000,0000,,We've just established that\Nthe sum of the squares of each
Dialogue: 0,0:09:17.12,0:09:20.06,Default,,0000,0000,0000,,of the legs is equal to the\Nsquare of the hypotenuse.
Dialogue: 0,0:09:20.06,0:09:22.55,Default,,0000,0000,0000,,And this is probably\Nwhat's easily
Dialogue: 0,0:09:22.55,0:09:26.22,Default,,0000,0000,0000,,one of the most famous\Ntheorem in mathematics, named
Dialogue: 0,0:09:26.22,0:09:27.36,Default,,0000,0000,0000,,for Pythagoras.
Dialogue: 0,0:09:27.36,0:09:30.37,Default,,0000,0000,0000,,Not clear if he's the first\Nperson to establish this,
Dialogue: 0,0:09:30.37,0:09:32.31,Default,,0000,0000,0000,,but it's called the\NPythagorean Theorem.
Dialogue: 0,0:09:32.31,0:09:38.29,Default,,0000,0000,0000,,
Dialogue: 0,0:09:38.29,0:09:41.47,Default,,0000,0000,0000,,And it's really the basis of,\Nwell, all not all of geometry,
Dialogue: 0,0:09:41.47,0:09:43.51,Default,,0000,0000,0000,,but a lot of the geometry\Nthat we're going to do.
Dialogue: 0,0:09:43.51,0:09:45.88,Default,,0000,0000,0000,,And it forms the basis of a\Nlot of the trigonometry we're
Dialogue: 0,0:09:45.88,0:09:46.23,Default,,0000,0000,0000,,going to do.
Dialogue: 0,0:09:46.23,0:09:47.55,Default,,0000,0000,0000,,And it's a really\Nuseful way, if you
Dialogue: 0,0:09:47.55,0:09:49.30,Default,,0000,0000,0000,,know two of the sides\Nof a right triangle,
Dialogue: 0,0:09:49.30,0:09:51.89,Default,,0000,0000,0000,,you can always find the third.