WEBVTT 00:00:00.000 --> 00:00:00.780 00:00:00.780 --> 00:00:11.240 Let's say we've got a rectangle and we have two diagonals 00:00:11.240 --> 00:00:15.140 across the rectangle-- that's one of them, and then we have 00:00:15.140 --> 00:00:19.910 the other diagonal --and this rectangle has a height of h-- 00:00:19.910 --> 00:00:25.080 so that distance right there is h --and it has a width of w. 00:00:25.080 --> 00:00:29.200 00:00:29.200 --> 00:00:32.050 What we're going to show in this video is that all of 00:00:32.050 --> 00:00:35.890 these four triangles have the same area. 00:00:35.890 --> 00:00:38.595 Now right when you look at it, it might be reasonably obvious 00:00:38.595 --> 00:00:47.060 that this bottom triangle will have the same area as the top 00:00:47.060 --> 00:00:50.440 triangle, as this top kind of upside down triangle. 00:00:50.440 --> 00:00:53.710 That these to have the same area, that might be reasonably 00:00:53.710 --> 00:00:57.110 obvious. they have the same dimension for their base, this 00:00:57.110 --> 00:00:59.860 width, and they have the same height because this distance 00:00:59.860 --> 00:01:04.840 right here is exactly half of the height of the rectangle. 00:01:04.840 --> 00:01:07.270 They are symmetric; they are equal triangles. 00:01:07.270 --> 00:01:09.800 They have the same proportions. 00:01:09.800 --> 00:01:13.520 Now it's probably equally obvious that this triangle on 00:01:13.520 --> 00:01:21.340 the left has the same area as this triangle on the right. 00:01:21.340 --> 00:01:23.200 That's probably equally obvious. 00:01:23.200 --> 00:01:26.960 What is not obvious is that these orange triangles angles 00:01:26.960 --> 00:01:31.020 have the same area as these green, blue triangles. 00:01:31.020 --> 00:01:32.320 And that's what we're going to show right here. 00:01:32.320 --> 00:01:34.930 So all we have to do is really calculate the areas of 00:01:34.930 --> 00:01:35.690 the different triangles. 00:01:35.690 --> 00:01:38.070 So let's do the orange triangles first. and before 00:01:38.070 --> 00:01:40.310 doing that let's just remind ourselves what the 00:01:40.310 --> 00:01:42.060 area of a triangle is. 00:01:42.060 --> 00:01:47.200 Area of a triangle is equal to 1/2 times the base of 00:01:47.200 --> 00:01:50.030 the triangle times the height of the triangle. 00:01:50.030 --> 00:01:52.260 That's just basic geometry. 00:01:52.260 --> 00:01:54.530 Not with that said, let's figure out the area of 00:01:54.530 --> 00:01:55.660 the orange triangle. 00:01:55.660 --> 00:01:58.250 00:01:58.250 --> 00:02:01.690 It's going to be 1/2 times the base. 00:02:01.690 --> 00:02:04.620 So the base of the orange triangle is this distance 00:02:04.620 --> 00:02:07.400 right here: it is w. 00:02:07.400 --> 00:02:09.734 So 1/2 times w. 00:02:09.734 --> 00:02:12.250 00:02:12.250 --> 00:02:14.800 I want to do that in a different color; the 00:02:14.800 --> 00:02:17.770 color I wrote the w in. 00:02:17.770 --> 00:02:19.040 Now what's the height here? 00:02:19.040 --> 00:02:21.820 00:02:21.820 --> 00:02:25.580 Well we already talked about it: it's exactly half way up 00:02:25.580 --> 00:02:28.300 the height of the rectangle. 00:02:28.300 --> 00:02:32.815 So times 1/2 times the height of the rectangle. 00:02:32.815 --> 00:02:36.270 00:02:36.270 --> 00:02:37.700 So what's that going to be? 00:02:37.700 --> 00:02:43.295 You have 1/2 times 1/2 is 1/4 times width times height. 00:02:43.295 --> 00:02:46.280 00:02:46.280 --> 00:02:49.820 So the area of that triangle is 1/4 width height. 00:02:49.820 --> 00:02:50.810 So is that one. 00:02:50.810 --> 00:02:53.790 Same exact argument; they have equal area. 00:02:53.790 --> 00:02:56.300 Now what's the area of these green or these 00:02:56.300 --> 00:02:58.020 green/blue triangles? 00:02:58.020 --> 00:03:04.440 Well once again-- we'll do this in a green color --area 00:03:04.440 --> 00:03:06.980 is equal to 1/2 base. 00:03:06.980 --> 00:03:08.680 So these guys are turned on their side. 00:03:08.680 --> 00:03:13.550 The best base I can think of is this distance right here. 00:03:13.550 --> 00:03:16.110 Or if you look at this triangle it's this distance right here; 00:03:16.110 --> 00:03:19.900 it is the height of the rectangle So now we're dealing 00:03:19.900 --> 00:03:25.080 with, the base in this case is the height of the rectangle. 00:03:25.080 --> 00:03:26.890 Don't want you to get too confused. 00:03:26.890 --> 00:03:31.210 The height is now going to be what? 00:03:31.210 --> 00:03:32.950 So these triangles are turned on the side, so what is 00:03:32.950 --> 00:03:36.170 this distance right here? 00:03:36.170 --> 00:03:39.640 Well it is exactly half of the width, right? 00:03:39.640 --> 00:03:42.190 We're going exactly half of this distance right here. 00:03:42.190 --> 00:03:44.740 This point right here is exactly halfway between 00:03:44.740 --> 00:03:47.570 these two sides and halfway between those two sides. 00:03:47.570 --> 00:03:50.880 So this distance right here is 1/2 the width. 00:03:50.880 --> 00:03:55.400 Or the height of these sideways triangles are 1/2 of the width. 00:03:55.400 --> 00:03:59.410 00:03:59.410 --> 00:04:01.900 Little confusing: the base is equal to the height of the 00:04:01.900 --> 00:04:05.960 rectangle, the height is equal to 1/2 of the width. but if you 00:04:05.960 --> 00:04:09.870 do the math here, area is equal to 1/2 times 1/2, which is 00:04:09.870 --> 00:04:12.450 1/4, height times width. 00:04:12.450 --> 00:04:16.560 Or you can just write that as 1/4 width times height, which 00:04:16.560 --> 00:04:18.040 is the exact same area. 00:04:18.040 --> 00:04:25.110 So the area here is 1/4 width times height, which is the 00:04:25.110 --> 00:04:27.870 exact same area as each of these orange triangles. 00:04:27.870 --> 00:04:31.570 And it makes sense because each of them are exactly 1/4 00:04:31.570 --> 00:04:33.460 the area of the rectangle. 00:04:33.460 --> 00:04:35.540 Hopefully you enjoyed that.