[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.57,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.57,0:00:06.93,Default,,0000,0000,0000,,We're told the triangle ABC\Nhas perimeter P and inradius r
Dialogue: 0,0:00:06.93,0:00:09.59,Default,,0000,0000,0000,,and then they want us\Nto find the area of ABC
Dialogue: 0,0:00:09.59,0:00:11.13,Default,,0000,0000,0000,,in terms of P and r.
Dialogue: 0,0:00:11.13,0:00:12.67,Default,,0000,0000,0000,,So we know that the\Nperimeter is just
Dialogue: 0,0:00:12.67,0:00:14.90,Default,,0000,0000,0000,,the sum of the sides\Nof the triangle,
Dialogue: 0,0:00:14.90,0:00:16.43,Default,,0000,0000,0000,,or how long a fence\Nwould have to be
Dialogue: 0,0:00:16.43,0:00:18.38,Default,,0000,0000,0000,,if you wanted to go\Naround the triangle.
Dialogue: 0,0:00:18.38,0:00:21.03,Default,,0000,0000,0000,,And let's just remind\Nourselves what the inradius is.
Dialogue: 0,0:00:21.03,0:00:26.23,Default,,0000,0000,0000,,If we take the angle\Nbisectors of each of these
Dialogue: 0,0:00:26.23,0:00:28.62,Default,,0000,0000,0000,,vertices-- each of these\Nangles right over here.
Dialogue: 0,0:00:28.62,0:00:31.85,Default,,0000,0000,0000,,So bisect that right over\Nthere and then bisect
Dialogue: 0,0:00:31.85,0:00:33.37,Default,,0000,0000,0000,,that right over there.
Dialogue: 0,0:00:33.37,0:00:35.91,Default,,0000,0000,0000,,This angle is going to\Nbe equal to that angle.
Dialogue: 0,0:00:35.91,0:00:38.53,Default,,0000,0000,0000,,This angle is going to\Nbe equal to that angle
Dialogue: 0,0:00:38.53,0:00:42.86,Default,,0000,0000,0000,,and then this angle is going to\Nbe equal to that angle there.
Dialogue: 0,0:00:42.86,0:00:47.47,Default,,0000,0000,0000,,And the point where those angle\Nbisectors intersect, that right
Dialogue: 0,0:00:47.47,0:00:50.21,Default,,0000,0000,0000,,over there, is our\Nincenter and it
Dialogue: 0,0:00:50.21,0:00:53.18,Default,,0000,0000,0000,,is equidistant from\Nall of the three sides.
Dialogue: 0,0:00:53.18,0:00:57.21,Default,,0000,0000,0000,,And the distance from those\Nsides, that's the inradius.
Dialogue: 0,0:00:57.21,0:00:58.84,Default,,0000,0000,0000,,So let me draw the inradius.
Dialogue: 0,0:00:58.84,0:01:01.21,Default,,0000,0000,0000,,So when you find the distance\Nbetween a point and a line,
Dialogue: 0,0:01:01.21,0:01:02.58,Default,,0000,0000,0000,,you want to drop\Na perpendicular.
Dialogue: 0,0:01:02.58,0:01:05.05,Default,,0000,0000,0000,,So this length right over\Nhere is the inradius.
Dialogue: 0,0:01:05.05,0:01:08.37,Default,,0000,0000,0000,,This length right over\Nhere is the inradius
Dialogue: 0,0:01:08.37,0:01:11.92,Default,,0000,0000,0000,,and this length right\Nover here is the inradius.
Dialogue: 0,0:01:11.92,0:01:14.70,Default,,0000,0000,0000,,And if you want, you could\Ndraw an incircle here
Dialogue: 0,0:01:14.70,0:01:18.33,Default,,0000,0000,0000,,with the center at the\Nincenter and with the radius r
Dialogue: 0,0:01:18.33,0:01:20.45,Default,,0000,0000,0000,,and that circle would\Nlook something like this.
Dialogue: 0,0:01:20.45,0:01:22.92,Default,,0000,0000,0000,,We don't have to necessarily\Ndraw it for this problem.
Dialogue: 0,0:01:22.92,0:01:25.34,Default,,0000,0000,0000,,So you could draw a circle\Nthat looks something like that.
Dialogue: 0,0:01:25.34,0:01:27.93,Default,,0000,0000,0000,,And then we'd call\Nthat the incircle.
Dialogue: 0,0:01:27.93,0:01:30.85,Default,,0000,0000,0000,,So let's think about how we can\Nfind the area here, especially
Dialogue: 0,0:01:30.85,0:01:32.89,Default,,0000,0000,0000,,in terms of this inradius.
Dialogue: 0,0:01:32.89,0:01:34.51,Default,,0000,0000,0000,,Well, the cool thing\Nabout the inradius
Dialogue: 0,0:01:34.51,0:01:37.11,Default,,0000,0000,0000,,is it looks like the\Naltitude-- or this looks
Dialogue: 0,0:01:37.11,0:01:39.28,Default,,0000,0000,0000,,like the altitude for this\Ntriangle right over here,
Dialogue: 0,0:01:39.28,0:01:42.44,Default,,0000,0000,0000,,triangle A. Let's\Nlabel the center.
Dialogue: 0,0:01:42.44,0:01:46.09,Default,,0000,0000,0000,,Let's call it I for incenter.
Dialogue: 0,0:01:46.09,0:01:49.36,Default,,0000,0000,0000,,This r right over here is\Nthe altitude of triangle AIC.
Dialogue: 0,0:01:49.36,0:01:52.71,Default,,0000,0000,0000,,This r is the altitude\Nof triangle BIC.
Dialogue: 0,0:01:52.71,0:01:56.02,Default,,0000,0000,0000,,And this r, which we didn't\Nlabel, that r right over there
Dialogue: 0,0:01:56.02,0:01:59.17,Default,,0000,0000,0000,,is the altitude of triangle AIB.
Dialogue: 0,0:01:59.17,0:02:01.43,Default,,0000,0000,0000,,And we know-- and\Nso we could find
Dialogue: 0,0:02:01.43,0:02:02.89,Default,,0000,0000,0000,,the area of each\Nof those triangles
Dialogue: 0,0:02:02.89,0:02:05.21,Default,,0000,0000,0000,,in terms of both\Nr and their bases
Dialogue: 0,0:02:05.21,0:02:08.03,Default,,0000,0000,0000,,and maybe if we sum up the\Narea of all the triangles,
Dialogue: 0,0:02:08.03,0:02:11.38,Default,,0000,0000,0000,,we can get something in terms of\Nour perimeter and our inradius.
Dialogue: 0,0:02:11.38,0:02:13.05,Default,,0000,0000,0000,,So let's just try to do this.
Dialogue: 0,0:02:13.05,0:02:17.44,Default,,0000,0000,0000,,So the area of the entire\Ntriangle, the area of ABC,
Dialogue: 0,0:02:17.44,0:02:19.14,Default,,0000,0000,0000,,is going to be\Nequal to-- and I'll
Dialogue: 0,0:02:19.14,0:02:24.10,Default,,0000,0000,0000,,color code this-- is going to\Nbe equal to the area of AIC.
Dialogue: 0,0:02:24.10,0:02:27.89,Default,,0000,0000,0000,,So that's what I'm\Nshading here in magenta.
Dialogue: 0,0:02:27.89,0:02:34.75,Default,,0000,0000,0000,,It's going to be equal to\Nthe area of AIC plus the area
Dialogue: 0,0:02:34.75,0:02:37.55,Default,,0000,0000,0000,,of BIC, which is this\Ntriangle right over here.
Dialogue: 0,0:02:37.55,0:02:39.42,Default,,0000,0000,0000,,Actually let me do that\Nin a different color.
Dialogue: 0,0:02:39.42,0:02:41.88,Default,,0000,0000,0000,,I've already used the blue.
Dialogue: 0,0:02:41.88,0:02:44.43,Default,,0000,0000,0000,,So let me do that in orange.
Dialogue: 0,0:02:44.43,0:02:47.89,Default,,0000,0000,0000,,Plus the area of BIC.
Dialogue: 0,0:02:47.89,0:02:50.16,Default,,0000,0000,0000,,So that's this area\Nright over here.
Dialogue: 0,0:02:50.16,0:02:55.16,Default,,0000,0000,0000,,
Dialogue: 0,0:02:55.16,0:02:59.96,Default,,0000,0000,0000,,And then finally plus the area--\NI'll do this in a, let's see,
Dialogue: 0,0:02:59.96,0:03:04.22,Default,,0000,0000,0000,,I'll use this pink color--\Nplus the area of AIB.
Dialogue: 0,0:03:04.22,0:03:07.04,Default,,0000,0000,0000,,
Dialogue: 0,0:03:07.04,0:03:11.24,Default,,0000,0000,0000,,That is the area AIB.
Dialogue: 0,0:03:11.24,0:03:13.28,Default,,0000,0000,0000,,Take the sum of the areas\Nof these two triangles,
Dialogue: 0,0:03:13.28,0:03:15.59,Default,,0000,0000,0000,,you got the area of\Nthe larger triangle.
Dialogue: 0,0:03:15.59,0:03:19.03,Default,,0000,0000,0000,,Now AIC, the area\Nof AIC, is going
Dialogue: 0,0:03:19.03,0:03:21.70,Default,,0000,0000,0000,,to be equal to 1/2\Nbase times height.
Dialogue: 0,0:03:21.70,0:03:23.76,Default,,0000,0000,0000,,So this is going to be 1/2.
Dialogue: 0,0:03:23.76,0:03:27.84,Default,,0000,0000,0000,,The base is the length\Nof AC, 1/2 AC times
Dialogue: 0,0:03:27.84,0:03:30.03,Default,,0000,0000,0000,,the height-- times this\Naltitude right over here,
Dialogue: 0,0:03:30.03,0:03:32.41,Default,,0000,0000,0000,,which is just going\Nto be r-- times r.
Dialogue: 0,0:03:32.41,0:03:34.39,Default,,0000,0000,0000,,That's the area of AIC.
Dialogue: 0,0:03:34.39,0:03:41.28,Default,,0000,0000,0000,,And then the area of BIC is\Ngoing to be 1/2 the base,
Dialogue: 0,0:03:41.28,0:03:45.54,Default,,0000,0000,0000,,which is BC, times the\Nheight, which is r.
Dialogue: 0,0:03:45.54,0:03:49.26,Default,,0000,0000,0000,,And then plus the area of\NAIB, this one over here,
Dialogue: 0,0:03:49.26,0:03:51.62,Default,,0000,0000,0000,,is going to be 1/2\Nthe base, which
Dialogue: 0,0:03:51.62,0:03:56.58,Default,,0000,0000,0000,,is the length of this side\NAB, times the height, which
Dialogue: 0,0:03:56.58,0:04:00.09,Default,,0000,0000,0000,,is once again r.
Dialogue: 0,0:04:00.09,0:04:03.90,Default,,0000,0000,0000,,And over here, we can factor out\Na 1/2 r from all of these terms
Dialogue: 0,0:04:03.90,0:04:16.23,Default,,0000,0000,0000,,and you get 1/2 r times\NAC plus BC plus AB.
Dialogue: 0,0:04:16.23,0:04:17.90,Default,,0000,0000,0000,,And I think you see\Nwhere this is going.
Dialogue: 0,0:04:17.90,0:04:21.32,Default,,0000,0000,0000,,Plus-- now that's a different\Nshade of pink-- plus AB.
Dialogue: 0,0:04:21.32,0:04:24.68,Default,,0000,0000,0000,,
Dialogue: 0,0:04:24.68,0:04:28.56,Default,,0000,0000,0000,,Now what is AC plus BC plus AB?
Dialogue: 0,0:04:28.56,0:04:32.76,Default,,0000,0000,0000,,
Dialogue: 0,0:04:32.76,0:04:37.61,Default,,0000,0000,0000,,Well that's going to\Nbe the perimeter, P,
Dialogue: 0,0:04:37.61,0:04:39.19,Default,,0000,0000,0000,,if you just take the\Nsum of the sides.
Dialogue: 0,0:04:39.19,0:04:42.11,Default,,0000,0000,0000,,So that is the perimeter of P\Nand it looks like we're done.
Dialogue: 0,0:04:42.11,0:04:51.68,Default,,0000,0000,0000,,The area of our triangle ABC\Nis equal to 1/2 times r times
Dialogue: 0,0:04:51.68,0:04:54.75,Default,,0000,0000,0000,,the perimeter, which is\Nkind of a neat result.
Dialogue: 0,0:04:54.75,0:04:59.64,Default,,0000,0000,0000,,1/2 times the inradius times\Nthe perimeter of the triangle.
Dialogue: 0,0:04:59.64,0:05:01.51,Default,,0000,0000,0000,,Or sometimes you'll see\Nit written like this.
Dialogue: 0,0:05:01.51,0:05:07.68,Default,,0000,0000,0000,,It's equal to r times P\Nover s-- sorry, P over 2.
Dialogue: 0,0:05:07.68,0:05:10.29,Default,,0000,0000,0000,,And this term right over here,\Nthe perimeter divided by 2,
Dialogue: 0,0:05:10.29,0:05:11.87,Default,,0000,0000,0000,,is sometimes called\Nthe semiperimeter.
Dialogue: 0,0:05:11.87,0:05:17.12,Default,,0000,0000,0000,,
Dialogue: 0,0:05:17.12,0:05:19.94,Default,,0000,0000,0000,,And sometimes it's denoted\Nby s so sometimes you'll
Dialogue: 0,0:05:19.94,0:05:22.76,Default,,0000,0000,0000,,see the area is\Nequal to r times s,
Dialogue: 0,0:05:22.76,0:05:24.60,Default,,0000,0000,0000,,where s is the semiperimeter.
Dialogue: 0,0:05:24.60,0:05:27.00,Default,,0000,0000,0000,,It's the perimeter divided by 2.
Dialogue: 0,0:05:27.00,0:05:28.75,Default,,0000,0000,0000,,I personally like it\Nthis way a little bit
Dialogue: 0,0:05:28.75,0:05:31.06,Default,,0000,0000,0000,,more because I remember\Nthat P is perimeter.
Dialogue: 0,0:05:31.06,0:05:33.43,Default,,0000,0000,0000,,This is useful because obviously\Nnow if someone gives you
Dialogue: 0,0:05:33.43,0:05:35.40,Default,,0000,0000,0000,,an inradius and a perimeter,\Nyou can figure out
Dialogue: 0,0:05:35.40,0:05:36.64,Default,,0000,0000,0000,,the area of a triangle.
Dialogue: 0,0:05:36.64,0:05:38.68,Default,,0000,0000,0000,,Or if someone gives you\Nthe area of the triangle
Dialogue: 0,0:05:38.68,0:05:40.51,Default,,0000,0000,0000,,and the perimeter, you\Ncan get the inradius.
Dialogue: 0,0:05:40.51,0:05:42.63,Default,,0000,0000,0000,,So if they give you\Ntwo of these variables,
Dialogue: 0,0:05:42.63,0:05:44.14,Default,,0000,0000,0000,,you can always get the third.
Dialogue: 0,0:05:44.14,0:05:47.94,Default,,0000,0000,0000,,So for example, if this was\Na triangle right over here,
Dialogue: 0,0:05:47.94,0:05:50.75,Default,,0000,0000,0000,,this is maybe the most famous\Nof the right triangles.
Dialogue: 0,0:05:50.75,0:05:55.24,Default,,0000,0000,0000,,If I have a triangle that\Nhas lengths 3, 4, and 5,
Dialogue: 0,0:05:55.24,0:05:56.61,Default,,0000,0000,0000,,we know this is\Na right triangle.
Dialogue: 0,0:05:56.61,0:05:58.67,Default,,0000,0000,0000,,You can verify this from\Nthe Pythagorean theorem.
Dialogue: 0,0:05:58.67,0:06:00.09,Default,,0000,0000,0000,,And if someone\Nwere to say what is
Dialogue: 0,0:06:00.09,0:06:03.42,Default,,0000,0000,0000,,the inradius of this\Ntriangle right over here?
Dialogue: 0,0:06:03.42,0:06:05.63,Default,,0000,0000,0000,,Well we can figure out\Nthe area pretty easily.
Dialogue: 0,0:06:05.63,0:06:07.00,Default,,0000,0000,0000,,We know this is\Na right triangle.
Dialogue: 0,0:06:07.00,0:06:09.80,Default,,0000,0000,0000,,3 squared plus 4 squared\Nis equal to 5 squared.
Dialogue: 0,0:06:09.80,0:06:16.37,Default,,0000,0000,0000,,So the area is going to be\Nequal to 3 times 4 times 1/2.
Dialogue: 0,0:06:16.37,0:06:19.04,Default,,0000,0000,0000,,So 3 times 4 times\N1/2 is 6 and then
Dialogue: 0,0:06:19.04,0:06:21.17,Default,,0000,0000,0000,,the perimeter here\Nis going to be
Dialogue: 0,0:06:21.17,0:06:26.69,Default,,0000,0000,0000,,equal to 3 plus 4, which\Nis 7, plus 5 is 12.
Dialogue: 0,0:06:26.69,0:06:29.67,Default,,0000,0000,0000,,And so we have the area.
Dialogue: 0,0:06:29.67,0:06:35.72,Default,,0000,0000,0000,,So let's write this area is\Nequal to 1/2 times the inradius
Dialogue: 0,0:06:35.72,0:06:37.51,Default,,0000,0000,0000,,times the perimeter.
Dialogue: 0,0:06:37.51,0:06:42.77,Default,,0000,0000,0000,,So here we have 12 is equal\Nto 1/2 times the inradius
Dialogue: 0,0:06:42.77,0:06:44.75,Default,,0000,0000,0000,,times the perimeter.
Dialogue: 0,0:06:44.75,0:06:46.75,Default,,0000,0000,0000,,So we have-- oh\Nsorry, we have 6.
Dialogue: 0,0:06:46.75,0:06:47.72,Default,,0000,0000,0000,,Let me write this in.
Dialogue: 0,0:06:47.72,0:06:49.62,Default,,0000,0000,0000,,The area is 6.
Dialogue: 0,0:06:49.62,0:06:55.20,Default,,0000,0000,0000,,We have 6 is equal to 1/2\Ntimes the inradius times 12.
Dialogue: 0,0:06:55.20,0:06:58.17,Default,,0000,0000,0000,,And so in this situation,\N1/2 times 12 is just 6.
Dialogue: 0,0:06:58.17,0:07:00.31,Default,,0000,0000,0000,,We have 6 is equal to 6r.
Dialogue: 0,0:07:00.31,0:07:03.56,Default,,0000,0000,0000,,Divide both sides by 6,\Nyou get r is equal to 1.
Dialogue: 0,0:07:03.56,0:07:06.44,Default,,0000,0000,0000,,So if you were to draw\Nthe inradius for this one,
Dialogue: 0,0:07:06.44,0:07:08.12,Default,,0000,0000,0000,,which is kind of a neat result.
Dialogue: 0,0:07:08.12,0:07:11.24,Default,,0000,0000,0000,,So let me draw some\Nangle bisectors here.
Dialogue: 0,0:07:11.24,0:07:13.83,Default,,0000,0000,0000,,
Dialogue: 0,0:07:13.83,0:07:17.57,Default,,0000,0000,0000,,This 3, 4, 5 right triangle\Nhas an inradius of 1.
Dialogue: 0,0:07:17.57,0:07:19.98,Default,,0000,0000,0000,,So this distance\Nequals this distance,
Dialogue: 0,0:07:19.98,0:07:22.53,Default,,0000,0000,0000,,which is equal to\Nthis distance, which
Dialogue: 0,0:07:22.53,0:07:28.29,Default,,0000,0000,0000,,is equal to 1, which is\Nkind of a neat result.