[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.78,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.78,0:00:11.24,Default,,0000,0000,0000,,Let's say we've got a rectangle\Nand we have two diagonals
Dialogue: 0,0:00:11.24,0:00:15.14,Default,,0000,0000,0000,,across the rectangle-- that's\None of them, and then we have
Dialogue: 0,0:00:15.14,0:00:19.91,Default,,0000,0000,0000,,the other diagonal --and this\Nrectangle has a height of h--
Dialogue: 0,0:00:19.91,0:00:25.08,Default,,0000,0000,0000,,so that distance right there is\Nh --and it has a width of w.
Dialogue: 0,0:00:25.08,0:00:29.20,Default,,0000,0000,0000,,
Dialogue: 0,0:00:29.20,0:00:32.05,Default,,0000,0000,0000,,What we're going to show in\Nthis video is that all of
Dialogue: 0,0:00:32.05,0:00:35.89,Default,,0000,0000,0000,,these four triangles\Nhave the same area.
Dialogue: 0,0:00:35.89,0:00:38.60,Default,,0000,0000,0000,,Now right when you look at it,\Nit might be reasonably obvious
Dialogue: 0,0:00:38.60,0:00:47.06,Default,,0000,0000,0000,,that this bottom triangle will\Nhave the same area as the top
Dialogue: 0,0:00:47.06,0:00:50.44,Default,,0000,0000,0000,,triangle, as this top kind\Nof upside down triangle.
Dialogue: 0,0:00:50.44,0:00:53.71,Default,,0000,0000,0000,,That these to have the same\Narea, that might be reasonably
Dialogue: 0,0:00:53.71,0:00:57.11,Default,,0000,0000,0000,,obvious. they have the same\Ndimension for their base, this
Dialogue: 0,0:00:57.11,0:00:59.86,Default,,0000,0000,0000,,width, and they have the same\Nheight because this distance
Dialogue: 0,0:00:59.86,0:01:04.84,Default,,0000,0000,0000,,right here is exactly half of\Nthe height of the rectangle.
Dialogue: 0,0:01:04.84,0:01:07.27,Default,,0000,0000,0000,,They are symmetric; they\Nare equal triangles.
Dialogue: 0,0:01:07.27,0:01:09.80,Default,,0000,0000,0000,,They have the same proportions.
Dialogue: 0,0:01:09.80,0:01:13.52,Default,,0000,0000,0000,,Now it's probably equally\Nobvious that this triangle on
Dialogue: 0,0:01:13.52,0:01:21.34,Default,,0000,0000,0000,,the left has the same area as\Nthis triangle on the right.
Dialogue: 0,0:01:21.34,0:01:23.20,Default,,0000,0000,0000,,That's probably\Nequally obvious.
Dialogue: 0,0:01:23.20,0:01:26.96,Default,,0000,0000,0000,,What is not obvious is that\Nthese orange triangles angles
Dialogue: 0,0:01:26.96,0:01:31.02,Default,,0000,0000,0000,,have the same area as these\Ngreen, blue triangles.
Dialogue: 0,0:01:31.02,0:01:32.32,Default,,0000,0000,0000,,And that's what we're\Ngoing to show right here.
Dialogue: 0,0:01:32.32,0:01:34.93,Default,,0000,0000,0000,,So all we have to do is really\Ncalculate the areas of
Dialogue: 0,0:01:34.93,0:01:35.69,Default,,0000,0000,0000,,the different triangles.
Dialogue: 0,0:01:35.69,0:01:38.07,Default,,0000,0000,0000,,So let's do the orange\Ntriangles first. and before
Dialogue: 0,0:01:38.07,0:01:40.31,Default,,0000,0000,0000,,doing that let's just\Nremind ourselves what the
Dialogue: 0,0:01:40.31,0:01:42.06,Default,,0000,0000,0000,,area of a triangle is.
Dialogue: 0,0:01:42.06,0:01:47.20,Default,,0000,0000,0000,,Area of a triangle is equal\Nto 1/2 times the base of
Dialogue: 0,0:01:47.20,0:01:50.03,Default,,0000,0000,0000,,the triangle times the\Nheight of the triangle.
Dialogue: 0,0:01:50.03,0:01:52.26,Default,,0000,0000,0000,,That's just basic geometry.
Dialogue: 0,0:01:52.26,0:01:54.53,Default,,0000,0000,0000,,Not with that said, let's\Nfigure out the area of
Dialogue: 0,0:01:54.53,0:01:55.66,Default,,0000,0000,0000,,the orange triangle.
Dialogue: 0,0:01:55.66,0:01:58.25,Default,,0000,0000,0000,,
Dialogue: 0,0:01:58.25,0:02:01.69,Default,,0000,0000,0000,,It's going to be 1/2\Ntimes the base.
Dialogue: 0,0:02:01.69,0:02:04.62,Default,,0000,0000,0000,,So the base of the orange\Ntriangle is this distance
Dialogue: 0,0:02:04.62,0:02:07.40,Default,,0000,0000,0000,,right here: it is w.
Dialogue: 0,0:02:07.40,0:02:09.73,Default,,0000,0000,0000,,So 1/2 times w.
Dialogue: 0,0:02:09.73,0:02:12.25,Default,,0000,0000,0000,,
Dialogue: 0,0:02:12.25,0:02:14.80,Default,,0000,0000,0000,,I want to do that in a\Ndifferent color; the
Dialogue: 0,0:02:14.80,0:02:17.77,Default,,0000,0000,0000,,color I wrote the w in.
Dialogue: 0,0:02:17.77,0:02:19.04,Default,,0000,0000,0000,,Now what's the height here?
Dialogue: 0,0:02:19.04,0:02:21.82,Default,,0000,0000,0000,,
Dialogue: 0,0:02:21.82,0:02:25.58,Default,,0000,0000,0000,,Well we already talked about\Nit: it's exactly half way up
Dialogue: 0,0:02:25.58,0:02:28.30,Default,,0000,0000,0000,,the height of the rectangle.
Dialogue: 0,0:02:28.30,0:02:32.82,Default,,0000,0000,0000,,So times 1/2 times the\Nheight of the rectangle.
Dialogue: 0,0:02:32.82,0:02:36.27,Default,,0000,0000,0000,,
Dialogue: 0,0:02:36.27,0:02:37.70,Default,,0000,0000,0000,,So what's that going to be?
Dialogue: 0,0:02:37.70,0:02:43.30,Default,,0000,0000,0000,,You have 1/2 times 1/2 is 1/4\Ntimes width times height.
Dialogue: 0,0:02:43.30,0:02:46.28,Default,,0000,0000,0000,,
Dialogue: 0,0:02:46.28,0:02:49.82,Default,,0000,0000,0000,,So the area of that triangle\Nis 1/4 width height.
Dialogue: 0,0:02:49.82,0:02:50.81,Default,,0000,0000,0000,,So is that one.
Dialogue: 0,0:02:50.81,0:02:53.79,Default,,0000,0000,0000,,Same exact argument;\Nthey have equal area.
Dialogue: 0,0:02:53.79,0:02:56.30,Default,,0000,0000,0000,,Now what's the area of\Nthese green or these
Dialogue: 0,0:02:56.30,0:02:58.02,Default,,0000,0000,0000,,green/blue triangles?
Dialogue: 0,0:02:58.02,0:03:04.44,Default,,0000,0000,0000,,Well once again-- we'll do\Nthis in a green color --area
Dialogue: 0,0:03:04.44,0:03:06.98,Default,,0000,0000,0000,,is equal to 1/2 base.
Dialogue: 0,0:03:06.98,0:03:08.68,Default,,0000,0000,0000,,So these guys are\Nturned on their side.
Dialogue: 0,0:03:08.68,0:03:13.55,Default,,0000,0000,0000,,The best base I can think of\Nis this distance right here.
Dialogue: 0,0:03:13.55,0:03:16.11,Default,,0000,0000,0000,,Or if you look at this triangle\Nit's this distance right here;
Dialogue: 0,0:03:16.11,0:03:19.90,Default,,0000,0000,0000,,it is the height of the\Nrectangle So now we're dealing
Dialogue: 0,0:03:19.90,0:03:25.08,Default,,0000,0000,0000,,with, the base in this case is\Nthe height of the rectangle.
Dialogue: 0,0:03:25.08,0:03:26.89,Default,,0000,0000,0000,,Don't want you to\Nget too confused.
Dialogue: 0,0:03:26.89,0:03:31.21,Default,,0000,0000,0000,,The height is now\Ngoing to be what?
Dialogue: 0,0:03:31.21,0:03:32.95,Default,,0000,0000,0000,,So these triangles are turned\Non the side, so what is
Dialogue: 0,0:03:32.95,0:03:36.17,Default,,0000,0000,0000,,this distance right here?
Dialogue: 0,0:03:36.17,0:03:39.64,Default,,0000,0000,0000,,Well it is exactly half\Nof the width, right?
Dialogue: 0,0:03:39.64,0:03:42.19,Default,,0000,0000,0000,,We're going exactly half of\Nthis distance right here.
Dialogue: 0,0:03:42.19,0:03:44.74,Default,,0000,0000,0000,,This point right here is\Nexactly halfway between
Dialogue: 0,0:03:44.74,0:03:47.57,Default,,0000,0000,0000,,these two sides and halfway\Nbetween those two sides.
Dialogue: 0,0:03:47.57,0:03:50.88,Default,,0000,0000,0000,,So this distance right\Nhere is 1/2 the width.
Dialogue: 0,0:03:50.88,0:03:55.40,Default,,0000,0000,0000,,Or the height of these sideways\Ntriangles are 1/2 of the width.
Dialogue: 0,0:03:55.40,0:03:59.41,Default,,0000,0000,0000,,
Dialogue: 0,0:03:59.41,0:04:01.90,Default,,0000,0000,0000,,Little confusing: the base is\Nequal to the height of the
Dialogue: 0,0:04:01.90,0:04:05.96,Default,,0000,0000,0000,,rectangle, the height is equal\Nto 1/2 of the width. but if you
Dialogue: 0,0:04:05.96,0:04:09.87,Default,,0000,0000,0000,,do the math here, area is equal\Nto 1/2 times 1/2, which is
Dialogue: 0,0:04:09.87,0:04:12.45,Default,,0000,0000,0000,,1/4, height times width.
Dialogue: 0,0:04:12.45,0:04:16.56,Default,,0000,0000,0000,,Or you can just write that as\N1/4 width times height, which
Dialogue: 0,0:04:16.56,0:04:18.04,Default,,0000,0000,0000,,is the exact same area.
Dialogue: 0,0:04:18.04,0:04:25.11,Default,,0000,0000,0000,,So the area here is 1/4 width\Ntimes height, which is the
Dialogue: 0,0:04:25.11,0:04:27.87,Default,,0000,0000,0000,,exact same area as each of\Nthese orange triangles.
Dialogue: 0,0:04:27.87,0:04:31.57,Default,,0000,0000,0000,,And it makes sense because\Neach of them are exactly 1/4
Dialogue: 0,0:04:31.57,0:04:33.46,Default,,0000,0000,0000,,the area of the rectangle.
Dialogue: 0,0:04:33.46,0:04:35.54,Default,,0000,0000,0000,,Hopefully you enjoyed that.