[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.60,0:00:02.53,Default,,0000,0000,0000,,In the last video, we\Nshowed that the ratios
Dialogue: 0,0:00:02.53,0:00:05.08,Default,,0000,0000,0000,,of the sides of a\N30-60-90 triangle
Dialogue: 0,0:00:05.08,0:00:06.68,Default,,0000,0000,0000,,are-- if we assume\Nthe longest side is
Dialogue: 0,0:00:06.68,0:00:08.35,Default,,0000,0000,0000,,x, if the hypotenuse is x.
Dialogue: 0,0:00:08.35,0:00:11.48,Default,,0000,0000,0000,,Then the shortest side is\Nx/2 and the side in between,
Dialogue: 0,0:00:11.48,0:00:13.69,Default,,0000,0000,0000,,the side that's opposite\Nthe 60 degree side,
Dialogue: 0,0:00:13.69,0:00:14.89,Default,,0000,0000,0000,,is square root of 3x/2.
Dialogue: 0,0:00:14.89,0:00:19.12,Default,,0000,0000,0000,,Or another way to think about it\Nis if the shortest side is 1--
Dialogue: 0,0:00:19.12,0:00:21.54,Default,,0000,0000,0000,,Now I'll do the shortest side,\Nthen the medium size, then
Dialogue: 0,0:00:21.54,0:00:22.39,Default,,0000,0000,0000,,the longest side.
Dialogue: 0,0:00:22.39,0:00:24.50,Default,,0000,0000,0000,,So if the side opposite\Nthe 30 degree side is 1,
Dialogue: 0,0:00:24.50,0:00:27.40,Default,,0000,0000,0000,,then the side opposite\Nthe 60 degree side
Dialogue: 0,0:00:27.40,0:00:29.34,Default,,0000,0000,0000,,is square root of 3 times that.
Dialogue: 0,0:00:29.34,0:00:31.11,Default,,0000,0000,0000,,So it's going to be\Nsquare root of 3.
Dialogue: 0,0:00:31.11,0:00:33.85,Default,,0000,0000,0000,,And then the hypotenuse\Nis going to be twice that.
Dialogue: 0,0:00:33.85,0:00:35.48,Default,,0000,0000,0000,,In the last video,\Nwe started with x
Dialogue: 0,0:00:35.48,0:00:37.46,Default,,0000,0000,0000,,and we said that the\N30 degree side is x/2.
Dialogue: 0,0:00:37.46,0:00:40.11,Default,,0000,0000,0000,,But if the 30 degree\Nside is 1, then this
Dialogue: 0,0:00:40.11,0:00:41.27,Default,,0000,0000,0000,,is going to be twice that.
Dialogue: 0,0:00:41.27,0:00:42.44,Default,,0000,0000,0000,,So it's going to be 2.
Dialogue: 0,0:00:42.44,0:00:46.12,Default,,0000,0000,0000,,This right here is the side\Nopposite the 30 degree side,
Dialogue: 0,0:00:46.12,0:00:49.44,Default,,0000,0000,0000,,opposite the 60 degree side,\Nand then the hypotenuse opposite
Dialogue: 0,0:00:49.44,0:00:51.03,Default,,0000,0000,0000,,the 90 degree side.
Dialogue: 0,0:00:51.03,0:00:53.80,Default,,0000,0000,0000,,And so, in general, if\Nyou see any triangle that
Dialogue: 0,0:00:53.80,0:00:56.57,Default,,0000,0000,0000,,has those ratios, you say hey,\Nthat's a 30-60-90 triangle.
Dialogue: 0,0:00:56.57,0:00:58.08,Default,,0000,0000,0000,,Or if you see a\Ntriangle that you
Dialogue: 0,0:00:58.08,0:01:02.29,Default,,0000,0000,0000,,know is a 30-60-90 triangle,\Nyou could say, hey,
Dialogue: 0,0:01:02.29,0:01:05.10,Default,,0000,0000,0000,,I know how to figure out\None of the sides based
Dialogue: 0,0:01:05.10,0:01:06.66,Default,,0000,0000,0000,,on this ratio right over here.
Dialogue: 0,0:01:06.66,0:01:08.93,Default,,0000,0000,0000,,Just an example, if\Nyou see a triangle that
Dialogue: 0,0:01:08.93,0:01:14.51,Default,,0000,0000,0000,,looks like this, where the\Nsides are 2, 2 square root of 3,
Dialogue: 0,0:01:14.51,0:01:15.48,Default,,0000,0000,0000,,and 4.
Dialogue: 0,0:01:15.48,0:01:17.79,Default,,0000,0000,0000,,Once again, the ratio of\N2 to 2 square root of 3
Dialogue: 0,0:01:17.79,0:01:19.44,Default,,0000,0000,0000,,is 1 to square root of 3.
Dialogue: 0,0:01:19.44,0:01:22.34,Default,,0000,0000,0000,,The ratio of 2 to 4 is\Nthe same thing as 1 to 2.
Dialogue: 0,0:01:22.34,0:01:25.44,Default,,0000,0000,0000,,This right here must\Nbe a 30-60-90 triangle.
Dialogue: 0,0:01:25.44,0:01:27.38,Default,,0000,0000,0000,,What I want to introduce\Nyou to in this video
Dialogue: 0,0:01:27.38,0:01:29.57,Default,,0000,0000,0000,,is another important\Ntype of triangle
Dialogue: 0,0:01:29.57,0:01:32.98,Default,,0000,0000,0000,,that shows up a lot in geometry\Nand a lot in trigonometry.
Dialogue: 0,0:01:32.98,0:01:36.65,Default,,0000,0000,0000,,And this is a 45-45-90 triangle.
Dialogue: 0,0:01:36.65,0:01:38.19,Default,,0000,0000,0000,,Or another way to\Nthink about is if I
Dialogue: 0,0:01:38.19,0:01:40.16,Default,,0000,0000,0000,,have a right triangle\Nthat is also isosceles.
Dialogue: 0,0:01:44.25,0:01:47.14,Default,,0000,0000,0000,,You obviously can't have a right\Ntriangle that is equilateral,
Dialogue: 0,0:01:47.14,0:01:49.77,Default,,0000,0000,0000,,because an equilateral triangle\Nhas all of their angles
Dialogue: 0,0:01:49.77,0:01:51.03,Default,,0000,0000,0000,,have to be 60 degrees.
Dialogue: 0,0:01:51.03,0:01:52.87,Default,,0000,0000,0000,,But you can have\Na right angle, you
Dialogue: 0,0:01:52.87,0:01:55.26,Default,,0000,0000,0000,,can have a right triangle,\Nthat is isosceles.
Dialogue: 0,0:01:55.26,0:01:56.79,Default,,0000,0000,0000,,And isosceles--\Nlet me write this--
Dialogue: 0,0:01:56.79,0:02:03.47,Default,,0000,0000,0000,,this is a right\Nisosceles triangle.
Dialogue: 0,0:02:03.47,0:02:05.83,Default,,0000,0000,0000,,And if it's isosceles,\Nthat means two of the sides
Dialogue: 0,0:02:05.83,0:02:06.50,Default,,0000,0000,0000,,are equal.
Dialogue: 0,0:02:06.50,0:02:09.85,Default,,0000,0000,0000,,So these are the two\Nsides that are equal.
Dialogue: 0,0:02:09.85,0:02:11.35,Default,,0000,0000,0000,,And then if the two\Nsides are equal,
Dialogue: 0,0:02:11.35,0:02:14.99,Default,,0000,0000,0000,,we have proved to ourselves\Nthat the base angles are equal.
Dialogue: 0,0:02:14.99,0:02:17.44,Default,,0000,0000,0000,,And if we called the measure\Nof these base angles x,
Dialogue: 0,0:02:17.44,0:02:25.44,Default,,0000,0000,0000,,then we know that x plus x plus\N90 have to be equal to 180.
Dialogue: 0,0:02:25.44,0:02:27.61,Default,,0000,0000,0000,,Or if we subtract\N90 from both sides,
Dialogue: 0,0:02:27.61,0:02:32.06,Default,,0000,0000,0000,,you get x plus x is equal\Nto 90 or 2x is equal to 90.
Dialogue: 0,0:02:32.06,0:02:33.78,Default,,0000,0000,0000,,Or if you divide\Nboth sides by 2,
Dialogue: 0,0:02:33.78,0:02:38.75,Default,,0000,0000,0000,,you get x is equal\Nto 45 degrees.
Dialogue: 0,0:02:38.75,0:02:41.85,Default,,0000,0000,0000,,So a right isosceles\Ntriangle can also be called--
Dialogue: 0,0:02:41.85,0:02:44.01,Default,,0000,0000,0000,,and this is the more\Ntypical name for it--
Dialogue: 0,0:02:44.01,0:02:50.14,Default,,0000,0000,0000,,it can also be called\Na 45-45-90 triangle.
Dialogue: 0,0:02:54.24,0:02:56.03,Default,,0000,0000,0000,,And what I want to do\Nthis video is come up
Dialogue: 0,0:02:56.03,0:02:59.18,Default,,0000,0000,0000,,with the ratios for the\Nsides of a 45-45-90 triangle,
Dialogue: 0,0:02:59.18,0:03:01.27,Default,,0000,0000,0000,,just like we did for\Na 30-60-90 triangle.
Dialogue: 0,0:03:01.27,0:03:03.18,Default,,0000,0000,0000,,And this one's actually\Nmore straightforward.
Dialogue: 0,0:03:03.18,0:03:08.95,Default,,0000,0000,0000,,Because in a 45-45-90 triangle,\Nif we call one of the legs x,
Dialogue: 0,0:03:08.95,0:03:10.82,Default,,0000,0000,0000,,the other leg is\Nalso going to be x.
Dialogue: 0,0:03:10.82,0:03:12.61,Default,,0000,0000,0000,,And then we can use\Nthe Pythagorean Theorem
Dialogue: 0,0:03:12.61,0:03:14.77,Default,,0000,0000,0000,,to figure out the length\Nof the hypotenuse.
Dialogue: 0,0:03:14.77,0:03:18.09,Default,,0000,0000,0000,,So the length of the\Nhypotenuse, let's call that c.
Dialogue: 0,0:03:18.09,0:03:22.74,Default,,0000,0000,0000,,So we get x squared\Nplus x squared.
Dialogue: 0,0:03:22.74,0:03:26.43,Default,,0000,0000,0000,,That's the square of\Nlength of both of the legs.
Dialogue: 0,0:03:26.43,0:03:27.97,Default,,0000,0000,0000,,So when we sum those\Nup, that's going
Dialogue: 0,0:03:27.97,0:03:29.74,Default,,0000,0000,0000,,to have to be\Nequal to c squared.
Dialogue: 0,0:03:29.74,0:03:32.31,Default,,0000,0000,0000,,This is just straight out\Nof the Pythagorean theorem.
Dialogue: 0,0:03:32.31,0:03:37.49,Default,,0000,0000,0000,,So we get 2x squared\Nis equal to c squared.
Dialogue: 0,0:03:37.49,0:03:41.57,Default,,0000,0000,0000,,We can take the principal\Nroot of both sides of that.
Dialogue: 0,0:03:41.57,0:03:45.93,Default,,0000,0000,0000,,I wanted to just\Nchange it to yellow.
Dialogue: 0,0:03:45.93,0:03:48.23,Default,,0000,0000,0000,,Last, take the principal\Nroot of both sides of that.
Dialogue: 0,0:03:51.29,0:03:53.42,Default,,0000,0000,0000,,The left-hand side you\Nget, principal root of 2
Dialogue: 0,0:03:53.42,0:03:54.86,Default,,0000,0000,0000,,is just square\Nroot of 2, and then
Dialogue: 0,0:03:54.86,0:03:57.79,Default,,0000,0000,0000,,the principal root of x\Nsquared is just going to be x.
Dialogue: 0,0:03:57.79,0:04:01.29,Default,,0000,0000,0000,,So you're going to have x\Ntimes the square root of 2
Dialogue: 0,0:04:01.29,0:04:04.69,Default,,0000,0000,0000,,is equal to c.
Dialogue: 0,0:04:04.69,0:04:07.79,Default,,0000,0000,0000,,So if you have a right isosceles\Ntriangle, whatever the two
Dialogue: 0,0:04:07.79,0:04:09.79,Default,,0000,0000,0000,,legs are, they're going\Nto have the same length.
Dialogue: 0,0:04:09.79,0:04:11.20,Default,,0000,0000,0000,,That's why it's isosceles.
Dialogue: 0,0:04:11.20,0:04:13.98,Default,,0000,0000,0000,,The hypotenuse is going to be\Nsquare root of 2 times that.
Dialogue: 0,0:04:13.98,0:04:18.23,Default,,0000,0000,0000,,So c is equal to x times\Nthe square root of 2.
Dialogue: 0,0:04:18.23,0:04:22.13,Default,,0000,0000,0000,,So for example, if you have a\Ntriangle that looks like this.
Dialogue: 0,0:04:22.13,0:04:23.94,Default,,0000,0000,0000,,Let me draw it a\Nslightly different way.
Dialogue: 0,0:04:23.94,0:04:26.65,Default,,0000,0000,0000,,It's good to have to orient\Nourselves in different ways
Dialogue: 0,0:04:26.65,0:04:27.75,Default,,0000,0000,0000,,every time.
Dialogue: 0,0:04:27.75,0:04:30.69,Default,,0000,0000,0000,,So if we see a triangle\Nthat's 90 degrees,
Dialogue: 0,0:04:30.69,0:04:33.64,Default,,0000,0000,0000,,45 and 45 like this,\Nand you really just
Dialogue: 0,0:04:33.64,0:04:35.90,Default,,0000,0000,0000,,have to know two of\Nthese angles to know
Dialogue: 0,0:04:35.90,0:04:37.51,Default,,0000,0000,0000,,what the other one\Nis going to be,
Dialogue: 0,0:04:37.51,0:04:39.79,Default,,0000,0000,0000,,and if I tell you that\Nthis side right over here
Dialogue: 0,0:04:39.79,0:04:41.91,Default,,0000,0000,0000,,is 3-- I actually don't\Neven have to tell you
Dialogue: 0,0:04:41.91,0:04:43.45,Default,,0000,0000,0000,,that this other\Nside's going to be 3.
Dialogue: 0,0:04:43.45,0:04:45.97,Default,,0000,0000,0000,,This is an isosceles\Ntriangle, so those two legs
Dialogue: 0,0:04:45.97,0:04:47.21,Default,,0000,0000,0000,,are going to be the same.
Dialogue: 0,0:04:47.21,0:04:49.05,Default,,0000,0000,0000,,And you won't even have to\Napply the Pythagorean theorem
Dialogue: 0,0:04:49.05,0:04:50.37,Default,,0000,0000,0000,,if you know this--\Nand this is a good one
Dialogue: 0,0:04:50.37,0:04:53.11,Default,,0000,0000,0000,,to know-- that the hypotenuse\Nhere, the side opposite the 90
Dialogue: 0,0:04:53.11,0:04:55.15,Default,,0000,0000,0000,,degree side, is just going\Nto be square root of 2
Dialogue: 0,0:04:55.15,0:04:57.92,Default,,0000,0000,0000,,times the length of\Neither of the legs.
Dialogue: 0,0:04:57.92,0:05:01.40,Default,,0000,0000,0000,,So it's going to be 3\Ntimes the square root of 2.
Dialogue: 0,0:05:01.40,0:05:03.79,Default,,0000,0000,0000,,So the ratio of the\Nsize of the hypotenuse
Dialogue: 0,0:05:03.79,0:05:09.19,Default,,0000,0000,0000,,in a 45-45-90 triangle or\Na right isosceles triangle,
Dialogue: 0,0:05:09.19,0:05:12.34,Default,,0000,0000,0000,,the ratio of the sides are\None of the legs can be 1.
Dialogue: 0,0:05:12.34,0:05:14.55,Default,,0000,0000,0000,,Then the other leg is going\Nto have the same measure,
Dialogue: 0,0:05:14.55,0:05:16.82,Default,,0000,0000,0000,,the same length, and then\Nthe hypotenuse is going
Dialogue: 0,0:05:16.82,0:05:19.10,Default,,0000,0000,0000,,to be square root of 2\Ntimes either of those.
Dialogue: 0,0:05:19.10,0:05:21.69,Default,,0000,0000,0000,,1 to 1, 2 square root of 2.
Dialogue: 0,0:05:21.69,0:05:22.69,Default,,0000,0000,0000,,So this is 45-45-90.
Dialogue: 0,0:05:28.74,0:05:30.01,Default,,0000,0000,0000,,That's the ratios.
Dialogue: 0,0:05:30.01,0:05:33.68,Default,,0000,0000,0000,,And just as a review,\Nif you have a 30-60-90,
Dialogue: 0,0:05:33.68,0:05:38.80,Default,,0000,0000,0000,,the ratios were 1 to\Nsquare root of 3 to 2.
Dialogue: 0,0:05:38.80,0:05:41.82,Default,,0000,0000,0000,,And now we'll apply this\Nin a bunch of problems.