[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.71,0:00:05.42,Default,,0000,0000,0000,,Welcome to the presentation\Non 45-45-90 triangles.
Dialogue: 0,0:00:05.42,0:00:07.20,Default,,0000,0000,0000,,Let me write that down.
Dialogue: 0,0:00:07.20,0:00:08.30,Default,,0000,0000,0000,,How come the pen--\Noh, there you go.
Dialogue: 0,0:00:08.30,0:00:15.77,Default,,0000,0000,0000,,45-45-90 triangles.
Dialogue: 0,0:00:15.77,0:00:19.05,Default,,0000,0000,0000,,Or we could say 45-45-90 right\Ntriangles, but that might be
Dialogue: 0,0:00:19.05,0:00:21.63,Default,,0000,0000,0000,,redundant, because we know any\Nangle that has a 90 degree
Dialogue: 0,0:00:21.63,0:00:24.11,Default,,0000,0000,0000,,measure in it is a\Nright triangle.
Dialogue: 0,0:00:24.11,0:00:27.79,Default,,0000,0000,0000,,And as you can imagine, the\N45-45-90, these are actually
Dialogue: 0,0:00:27.79,0:00:30.91,Default,,0000,0000,0000,,the degrees of the\Nangles of the triangle.
Dialogue: 0,0:00:30.91,0:00:33.22,Default,,0000,0000,0000,,So why are these\Ntriangles special?
Dialogue: 0,0:00:33.22,0:00:35.72,Default,,0000,0000,0000,,Well, if you saw the last\Npresentation I gave you a
Dialogue: 0,0:00:35.72,0:00:43.95,Default,,0000,0000,0000,,little theorem that told you\Nthat if two of the base angles
Dialogue: 0,0:00:43.95,0:00:49.00,Default,,0000,0000,0000,,of a triangle are equal-- and\Nit's I guess only a base angle
Dialogue: 0,0:00:49.00,0:00:49.80,Default,,0000,0000,0000,,if you draw it like this.
Dialogue: 0,0:00:49.80,0:00:51.83,Default,,0000,0000,0000,,You could draw it like this, in\Nwhich case it's maybe not so
Dialogue: 0,0:00:51.83,0:00:55.41,Default,,0000,0000,0000,,obviously a base angle, but\Nit would still be true.
Dialogue: 0,0:00:55.41,0:00:58.52,Default,,0000,0000,0000,,If these two angles are equal,\Nthen the sides that they don't
Dialogue: 0,0:00:58.52,0:01:02.00,Default,,0000,0000,0000,,share-- so this side and this\Nside in this example, or this
Dialogue: 0,0:01:02.00,0:01:05.28,Default,,0000,0000,0000,,side and this side in this\Nexample-- then the two sides
Dialogue: 0,0:01:05.28,0:01:07.05,Default,,0000,0000,0000,,are going to be equal.
Dialogue: 0,0:01:07.05,0:01:11.14,Default,,0000,0000,0000,,So what's interesting about\Na 45-45-90 triangle is that
Dialogue: 0,0:01:11.14,0:01:13.90,Default,,0000,0000,0000,,it is a right triangle\Nthat has this property.
Dialogue: 0,0:01:13.90,0:01:16.40,Default,,0000,0000,0000,,And how do we know that it's\Nthe only right triangle
Dialogue: 0,0:01:16.40,0:01:17.69,Default,,0000,0000,0000,,that has this property?
Dialogue: 0,0:01:17.69,0:01:20.79,Default,,0000,0000,0000,,Well, you could imagine a\Nworld where I told you that
Dialogue: 0,0:01:20.79,0:01:24.14,Default,,0000,0000,0000,,this is a right triangle.
Dialogue: 0,0:01:24.14,0:01:28.03,Default,,0000,0000,0000,,This is 90 degrees, so\Nthis is the hypotenuse.
Dialogue: 0,0:01:28.03,0:01:32.14,Default,,0000,0000,0000,,Right, it's the side opposite\Nthe 90 degree angle.
Dialogue: 0,0:01:32.14,0:01:36.78,Default,,0000,0000,0000,,And if I were to tell you that\Nthese two angles are equal to
Dialogue: 0,0:01:36.78,0:01:39.64,Default,,0000,0000,0000,,each other, what do those\Ntwo angles have to be?
Dialogue: 0,0:01:39.64,0:01:42.84,Default,,0000,0000,0000,,Well if we call these two\Nangles x, we know that the
Dialogue: 0,0:01:42.84,0:01:44.41,Default,,0000,0000,0000,,angles in a triangle\Nadd up to 180.
Dialogue: 0,0:01:44.41,0:01:49.22,Default,,0000,0000,0000,,So we'd say x plus x\Nplus-- this is 90-- plus
Dialogue: 0,0:01:49.22,0:01:52.65,Default,,0000,0000,0000,,90 is equal to 180.
Dialogue: 0,0:01:52.65,0:01:57.95,Default,,0000,0000,0000,,Or 2x plus 90 is equal to 180.
Dialogue: 0,0:01:57.95,0:02:01.26,Default,,0000,0000,0000,,Or 2x is equal to 90.
Dialogue: 0,0:02:01.26,0:02:05.50,Default,,0000,0000,0000,,Or x is equal to 45 degrees.
Dialogue: 0,0:02:05.50,0:02:10.18,Default,,0000,0000,0000,,So the only right triangle in\Nwhich the other two angles are
Dialogue: 0,0:02:10.18,0:02:17.99,Default,,0000,0000,0000,,equal is a 45-45-90 triangle.
Dialogue: 0,0:02:17.99,0:02:22.68,Default,,0000,0000,0000,,So what's interesting about\Na 45-45-90 triangle?
Dialogue: 0,0:02:22.68,0:02:27.16,Default,,0000,0000,0000,,Well other than what I just\Ntold you-- let me redraw it.
Dialogue: 0,0:02:27.16,0:02:29.18,Default,,0000,0000,0000,,I'll redraw it like this.
Dialogue: 0,0:02:29.18,0:02:35.19,Default,,0000,0000,0000,,So we already know this is 90\Ndegrees, this is 45 degrees,
Dialogue: 0,0:02:35.19,0:02:37.32,Default,,0000,0000,0000,,this is 45 degrees.
Dialogue: 0,0:02:37.32,0:02:40.37,Default,,0000,0000,0000,,And based on what I just told\Nyou, we also know that the
Dialogue: 0,0:02:40.37,0:02:45.85,Default,,0000,0000,0000,,sides that the 45 degree\Nangles don't share are equal.
Dialogue: 0,0:02:45.85,0:02:49.56,Default,,0000,0000,0000,,So this side is\Nequal to this side.
Dialogue: 0,0:02:49.56,0:02:52.08,Default,,0000,0000,0000,,And if we're viewing it from a\NPythagorean theorem point of
Dialogue: 0,0:02:52.08,0:02:55.24,Default,,0000,0000,0000,,view, this tells us that the\Ntwo sides that are not the
Dialogue: 0,0:02:55.24,0:02:57.71,Default,,0000,0000,0000,,hypotenuse are equal.
Dialogue: 0,0:02:57.71,0:02:58.40,Default,,0000,0000,0000,,So this is a hypotenuse.
Dialogue: 0,0:03:03.66,0:03:09.50,Default,,0000,0000,0000,,So let's call this side\NA and this side B.
Dialogue: 0,0:03:09.50,0:03:11.36,Default,,0000,0000,0000,,We know from the Pythagorean\Ntheorem-- let's say the
Dialogue: 0,0:03:11.36,0:03:14.88,Default,,0000,0000,0000,,hypotenuse is equal to C-- the\NPythagorean theorem tells us
Dialogue: 0,0:03:14.88,0:03:21.38,Default,,0000,0000,0000,,that A squared plus B squared\Nis equal to C squared.
Dialogue: 0,0:03:21.38,0:03:21.86,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:03:24.72,0:03:26.62,Default,,0000,0000,0000,,Well we know that A equals\NB, because this is a
Dialogue: 0,0:03:26.62,0:03:30.07,Default,,0000,0000,0000,,45-45-90 triangle.
Dialogue: 0,0:03:30.07,0:03:32.01,Default,,0000,0000,0000,,So we could substitute\NA for B or B for A.
Dialogue: 0,0:03:32.01,0:03:34.58,Default,,0000,0000,0000,,But let's just\Nsubstitute B for A.
Dialogue: 0,0:03:34.58,0:03:38.96,Default,,0000,0000,0000,,So we could say B squared\Nplus B squared is
Dialogue: 0,0:03:38.96,0:03:41.53,Default,,0000,0000,0000,,equal to C squared.
Dialogue: 0,0:03:41.53,0:03:47.49,Default,,0000,0000,0000,,Or 2B squared is\Nequal to C squared.
Dialogue: 0,0:03:47.49,0:03:54.94,Default,,0000,0000,0000,,Or B squared is equal\Nto C squared over 2.
Dialogue: 0,0:03:54.94,0:04:03.64,Default,,0000,0000,0000,,Or B is equal to the square\Nroot of C squared over 2.
Dialogue: 0,0:04:03.64,0:04:06.53,Default,,0000,0000,0000,,Which is equal to C-- because\Nwe just took the square root of
Dialogue: 0,0:04:06.53,0:04:09.13,Default,,0000,0000,0000,,the numerator and the square\Nroot of the denominator-- C
Dialogue: 0,0:04:09.13,0:04:10.57,Default,,0000,0000,0000,,over the square root of 2.
Dialogue: 0,0:04:10.57,0:04:15.25,Default,,0000,0000,0000,,And actually, even though this\Nis a presentation on triangles,
Dialogue: 0,0:04:15.25,0:04:17.63,Default,,0000,0000,0000,,I'm going to give you a little\Nbit of extra information
Dialogue: 0,0:04:17.63,0:04:19.93,Default,,0000,0000,0000,,on something called\Nrationalizing denominators.
Dialogue: 0,0:04:19.93,0:04:21.27,Default,,0000,0000,0000,,So this is perfectly correct.
Dialogue: 0,0:04:21.27,0:04:25.95,Default,,0000,0000,0000,,We just arrived at B-- and we\Nalso know that A equals B-- but
Dialogue: 0,0:04:25.95,0:04:29.51,Default,,0000,0000,0000,,that B is equal to C divided\Nby the square root of 2.
Dialogue: 0,0:04:29.51,0:04:31.82,Default,,0000,0000,0000,,It turns out that in most of\Nmathematics, and I never
Dialogue: 0,0:04:31.82,0:04:34.78,Default,,0000,0000,0000,,understood quite exactly why\Nthis was the case, people
Dialogue: 0,0:04:34.78,0:04:37.87,Default,,0000,0000,0000,,don't like square root of\N2s in the denominator.
Dialogue: 0,0:04:37.87,0:04:40.72,Default,,0000,0000,0000,,Or in general they don't\Nlike irrational numbers
Dialogue: 0,0:04:40.72,0:04:41.14,Default,,0000,0000,0000,,in the denominator.
Dialogue: 0,0:04:41.14,0:04:45.03,Default,,0000,0000,0000,,Irrational numbers are numbers\Nthat have decimal places that
Dialogue: 0,0:04:45.03,0:04:46.92,Default,,0000,0000,0000,,never repeat and never end.
Dialogue: 0,0:04:46.92,0:04:49.87,Default,,0000,0000,0000,,So the way that they get rid\Nof irrational numbers in the
Dialogue: 0,0:04:49.87,0:04:52.23,Default,,0000,0000,0000,,denominator is that you do\Nsomething called rationalizing
Dialogue: 0,0:04:52.23,0:04:53.57,Default,,0000,0000,0000,,the denominator.
Dialogue: 0,0:04:53.57,0:04:55.46,Default,,0000,0000,0000,,And the way you rationalize\Na denominator-- let's take
Dialogue: 0,0:04:55.46,0:04:56.11,Default,,0000,0000,0000,,our example right now.
Dialogue: 0,0:04:56.11,0:05:00.64,Default,,0000,0000,0000,,If we had C over the square\Nroot of 2, we just multiply
Dialogue: 0,0:05:00.64,0:05:03.20,Default,,0000,0000,0000,,both the numerator and\Nthe denominator by the
Dialogue: 0,0:05:03.20,0:05:05.13,Default,,0000,0000,0000,,same number, right?
Dialogue: 0,0:05:05.13,0:05:08.12,Default,,0000,0000,0000,,Because when you multiply the\Nnumerator and the denominator
Dialogue: 0,0:05:08.12,0:05:11.28,Default,,0000,0000,0000,,by the same number, that's just\Nlike multiplying it by 1.
Dialogue: 0,0:05:11.28,0:05:13.68,Default,,0000,0000,0000,,The square root of 2 over\Nthe square root of 2 is 1.
Dialogue: 0,0:05:13.68,0:05:15.53,Default,,0000,0000,0000,,And as you see, the reason\Nwe're doing this is because
Dialogue: 0,0:05:15.53,0:05:17.02,Default,,0000,0000,0000,,square root of 2 times square\Nroot of 2, what's the
Dialogue: 0,0:05:17.02,0:05:19.04,Default,,0000,0000,0000,,square root of 2 times\Nsquare root of 2?
Dialogue: 0,0:05:19.04,0:05:20.22,Default,,0000,0000,0000,,Right, it's 2.
Dialogue: 0,0:05:20.22,0:05:21.03,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:05:21.03,0:05:23.93,Default,,0000,0000,0000,,We just said, something times\Nsomething is 2, well the square
Dialogue: 0,0:05:23.93,0:05:25.99,Default,,0000,0000,0000,,root of 2 times square root\Nof 2, that's going to be 2.
Dialogue: 0,0:05:25.99,0:05:31.01,Default,,0000,0000,0000,,And then the numerator is C\Ntimes the square root of 2.
Dialogue: 0,0:05:31.01,0:05:34.42,Default,,0000,0000,0000,,So notice, C times the square\Nroot of 2 over 2 is the same
Dialogue: 0,0:05:34.42,0:05:37.15,Default,,0000,0000,0000,,thing as C over the\Nsquare root of 2.
Dialogue: 0,0:05:37.15,0:05:39.52,Default,,0000,0000,0000,,And this is important to\Nrealize, because sometimes
Dialogue: 0,0:05:39.52,0:05:41.09,Default,,0000,0000,0000,,while you're taking a\Nstandardized test or you're
Dialogue: 0,0:05:41.09,0:05:44.19,Default,,0000,0000,0000,,doing a test in class, you\Nmight get an answer that looks
Dialogue: 0,0:05:44.19,0:05:46.32,Default,,0000,0000,0000,,like this, has a square root of\N2, or maybe even a square root
Dialogue: 0,0:05:46.32,0:05:49.55,Default,,0000,0000,0000,,of 3 or whatever, in\Nthe denominator.
Dialogue: 0,0:05:49.55,0:05:51.42,Default,,0000,0000,0000,,And you might not see your\Nanswer if it's a multiple
Dialogue: 0,0:05:51.42,0:05:52.75,Default,,0000,0000,0000,,choice question.
Dialogue: 0,0:05:52.75,0:05:55.71,Default,,0000,0000,0000,,What you ned to do in that case\Nis rationalize the denominator.
Dialogue: 0,0:05:55.71,0:05:57.99,Default,,0000,0000,0000,,So multiply the numerator and\Nthe denominator by square
Dialogue: 0,0:05:57.99,0:06:01.47,Default,,0000,0000,0000,,root of 2 and you'll get\Nsquare root of 2 over 2.
Dialogue: 0,0:06:01.47,0:06:03.25,Default,,0000,0000,0000,,But anyway, back\Nto the problem.
Dialogue: 0,0:06:03.25,0:06:04.45,Default,,0000,0000,0000,,So what did we learn?
Dialogue: 0,0:06:04.45,0:06:06.88,Default,,0000,0000,0000,,This is equal to B, right?
Dialogue: 0,0:06:06.88,0:06:11.24,Default,,0000,0000,0000,,So turns out that B is equal\Nto C times the square
Dialogue: 0,0:06:11.24,0:06:13.42,Default,,0000,0000,0000,,root of 2 over 2.
Dialogue: 0,0:06:13.42,0:06:14.41,Default,,0000,0000,0000,,So let me write that.
Dialogue: 0,0:06:14.41,0:06:18.76,Default,,0000,0000,0000,,So we know that A\Nequals B, right?
Dialogue: 0,0:06:18.76,0:06:27.61,Default,,0000,0000,0000,,And that equals the square\Nroot of 2 over 2 times C.
Dialogue: 0,0:06:27.61,0:06:29.68,Default,,0000,0000,0000,,Now you might want to memorize\Nthis, though you can always
Dialogue: 0,0:06:29.68,0:06:32.44,Default,,0000,0000,0000,,derive it if you use the\NPythagorean theorem and
Dialogue: 0,0:06:32.44,0:06:35.72,Default,,0000,0000,0000,,remember that the sides that\Naren't the hypotenuse in a
Dialogue: 0,0:06:35.72,0:06:40.11,Default,,0000,0000,0000,,45-45-90 triangle are\Nequal to each other.
Dialogue: 0,0:06:40.11,0:06:41.37,Default,,0000,0000,0000,,But this is very good to know.
Dialogue: 0,0:06:41.37,0:06:44.64,Default,,0000,0000,0000,,Because if, say, you're taking\Nthe SAT and you need to solve a
Dialogue: 0,0:06:44.64,0:06:48.18,Default,,0000,0000,0000,,problem really fast, and if you\Nhave this memorized and someone
Dialogue: 0,0:06:48.18,0:06:49.94,Default,,0000,0000,0000,,gives you the hypotenuse, you\Ncan figure out what are the
Dialogue: 0,0:06:49.94,0:06:51.89,Default,,0000,0000,0000,,sides very fast, or if someone\Ngives you one of the sides,
Dialogue: 0,0:06:51.89,0:06:54.10,Default,,0000,0000,0000,,you can figure out the\Nhypotenuse very fast.
Dialogue: 0,0:06:54.10,0:06:56.29,Default,,0000,0000,0000,,Let's try that out.
Dialogue: 0,0:06:56.29,0:06:59.25,Default,,0000,0000,0000,,I'm going to erase everything.
Dialogue: 0,0:06:59.25,0:07:06.06,Default,,0000,0000,0000,,So we learned just now that A\Nis equal to B is equal to the
Dialogue: 0,0:07:06.06,0:07:10.21,Default,,0000,0000,0000,,square root of 2\Nover 2 times C.
Dialogue: 0,0:07:10.21,0:07:16.22,Default,,0000,0000,0000,,So if I were to give you a\Nright triangle, and I were to
Dialogue: 0,0:07:16.22,0:07:23.76,Default,,0000,0000,0000,,tell you that this angle is 90\Nand this angle is 45, and that
Dialogue: 0,0:07:23.76,0:07:28.57,Default,,0000,0000,0000,,this side is, let's\Nsay this side is 8.
Dialogue: 0,0:07:28.57,0:07:32.67,Default,,0000,0000,0000,,I want to figure out\Nwhat this side is.
Dialogue: 0,0:07:32.67,0:07:34.59,Default,,0000,0000,0000,,Well first of all, let's\Nfigure out what side
Dialogue: 0,0:07:34.59,0:07:35.50,Default,,0000,0000,0000,,is the hypotenuse.
Dialogue: 0,0:07:35.50,0:07:39.62,Default,,0000,0000,0000,,Well the hypotenuse is the side\Nopposite the right angle.
Dialogue: 0,0:07:39.62,0:07:42.06,Default,,0000,0000,0000,,So we're trying to actually\Nfigure out the hypotenuse.
Dialogue: 0,0:07:42.06,0:07:44.64,Default,,0000,0000,0000,,Let's call the hypotenuse C.
Dialogue: 0,0:07:44.64,0:07:47.56,Default,,0000,0000,0000,,And we also know this is a\N45-45-90 triangle, right?
Dialogue: 0,0:07:47.56,0:07:50.18,Default,,0000,0000,0000,,Because this angle is 45, so\Nthis one also has to be 45,
Dialogue: 0,0:07:50.18,0:07:54.62,Default,,0000,0000,0000,,because 45 plus 90 plus\N90 is equal to 180.
Dialogue: 0,0:07:54.62,0:07:58.84,Default,,0000,0000,0000,,So this is a 45-45-90 triangle,\Nand we know one of the sides--
Dialogue: 0,0:07:58.84,0:08:05.88,Default,,0000,0000,0000,,this side could be A or B-- we\Nknow that 8 is equal to the
Dialogue: 0,0:08:05.88,0:08:10.03,Default,,0000,0000,0000,,square root of 2\Nover 2 times C.
Dialogue: 0,0:08:10.03,0:08:12.16,Default,,0000,0000,0000,,C is what we're trying\Nto figure out.
Dialogue: 0,0:08:12.16,0:08:16.40,Default,,0000,0000,0000,,So if we multiply both sides of\Nthis equation by 2 times the
Dialogue: 0,0:08:16.40,0:08:22.01,Default,,0000,0000,0000,,square root of 2-- I'm just\Nmultiplying it by the inverse
Dialogue: 0,0:08:22.01,0:08:23.60,Default,,0000,0000,0000,,of the coefficient on C.
Dialogue: 0,0:08:23.60,0:08:25.75,Default,,0000,0000,0000,,Because the square root of 2\Ncancels out that square root
Dialogue: 0,0:08:25.75,0:08:28.43,Default,,0000,0000,0000,,of 2, this 2 cancels\Nout with this 2.
Dialogue: 0,0:08:28.43,0:08:37.64,Default,,0000,0000,0000,,We get 2 times 8, 16 over the\Nsquare root of 2 equals C.
Dialogue: 0,0:08:37.64,0:08:40.20,Default,,0000,0000,0000,,Which would be correct, but as\NI just showed you, people don't
Dialogue: 0,0:08:40.20,0:08:42.12,Default,,0000,0000,0000,,like having radicals\Nin the denominator.
Dialogue: 0,0:08:42.12,0:08:46.25,Default,,0000,0000,0000,,So we can just say C is equal\Nto 16 over the square root of
Dialogue: 0,0:08:46.25,0:08:51.29,Default,,0000,0000,0000,,2 times the square root of 2\Nover the square root of 2.
Dialogue: 0,0:08:51.29,0:08:58.79,Default,,0000,0000,0000,,So this equals 16 square\Nroots of 2 over 2.
Dialogue: 0,0:08:58.79,0:09:04.33,Default,,0000,0000,0000,,Which is the same thing\Nas 8 square roots of 2.
Dialogue: 0,0:09:04.33,0:09:10.17,Default,,0000,0000,0000,,So C in this example is\N8 square roots of 2.
Dialogue: 0,0:09:10.17,0:09:13.79,Default,,0000,0000,0000,,And we also knows, since this\Nis a 45-45-90 triangle,
Dialogue: 0,0:09:13.79,0:09:16.70,Default,,0000,0000,0000,,that this side is 8.
Dialogue: 0,0:09:16.70,0:09:17.94,Default,,0000,0000,0000,,Hope that makes sense.
Dialogue: 0,0:09:17.94,0:09:19.74,Default,,0000,0000,0000,,In the next presentation\NI'm going to show you a
Dialogue: 0,0:09:19.74,0:09:20.68,Default,,0000,0000,0000,,different type of triangle.
Dialogue: 0,0:09:20.68,0:09:22.90,Default,,0000,0000,0000,,Actually, I might even start\Noff with a couple more examples
Dialogue: 0,0:09:22.90,0:09:25.08,Default,,0000,0000,0000,,of this, because I feel I\Nmight have rushed it a bit.
Dialogue: 0,0:09:25.08,0:09:28.45,Default,,0000,0000,0000,,But anyway, I'll see you soon\Nin the next presentation.