[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.61,0:00:02.23,Default,,0000,0000,0000,,- [Tutor] Pause this video\Nand see if you can find
Dialogue: 0,0:00:02.23,0:00:04.63,Default,,0000,0000,0000,,the area of this triangle,
Dialogue: 0,0:00:04.63,0:00:06.58,Default,,0000,0000,0000,,and I'll give you two hints.
Dialogue: 0,0:00:06.58,0:00:09.38,Default,,0000,0000,0000,,Recognize, this is an isosceles triangle,
Dialogue: 0,0:00:09.38,0:00:12.03,Default,,0000,0000,0000,,and another hint is that\Nthe Pythagorean Theorem
Dialogue: 0,0:00:12.03,0:00:13.37,Default,,0000,0000,0000,,might be useful.
Dialogue: 0,0:00:14.25,0:00:16.76,Default,,0000,0000,0000,,Alright, now let's work\Nthrough this together.
Dialogue: 0,0:00:16.76,0:00:20.04,Default,,0000,0000,0000,,So, we might all remember\Nthat the area of a triangle
Dialogue: 0,0:00:20.04,0:00:24.70,Default,,0000,0000,0000,,is equal to one half times\Nour base times our height.
Dialogue: 0,0:00:24.70,0:00:25.94,Default,,0000,0000,0000,,They give us our base.
Dialogue: 0,0:00:25.94,0:00:28.43,Default,,0000,0000,0000,,Our base right over here is,
Dialogue: 0,0:00:28.43,0:00:29.68,Default,,0000,0000,0000,,our base is 10.
Dialogue: 0,0:00:31.21,0:00:32.86,Default,,0000,0000,0000,,But what is our height?
Dialogue: 0,0:00:32.86,0:00:34.20,Default,,0000,0000,0000,,Our height would be,
Dialogue: 0,0:00:34.20,0:00:35.90,Default,,0000,0000,0000,,let me do this in another color,
Dialogue: 0,0:00:35.90,0:00:40.01,Default,,0000,0000,0000,,our height would be the length\Nof this line right over here.
Dialogue: 0,0:00:40.01,0:00:41.72,Default,,0000,0000,0000,,So, if we can figure that out,
Dialogue: 0,0:00:41.72,0:00:44.86,Default,,0000,0000,0000,,then we can calculate what\None half times the base 10
Dialogue: 0,0:00:44.86,0:00:46.50,Default,,0000,0000,0000,,times the height is.
Dialogue: 0,0:00:46.50,0:00:49.15,Default,,0000,0000,0000,,But how do we figure out this height?
Dialogue: 0,0:00:49.15,0:00:51.48,Default,,0000,0000,0000,,Well, this is where\Nit's useful to recognize
Dialogue: 0,0:00:51.48,0:00:53.100,Default,,0000,0000,0000,,that this is an isosceles triangle.
Dialogue: 0,0:00:53.100,0:00:57.52,Default,,0000,0000,0000,,An isosceles triangle has\Ntwo sides that are the same.
Dialogue: 0,0:00:57.52,0:01:01.69,Default,,0000,0000,0000,,And so, these base angles are\Nalso going to be congruent.
Dialogue: 0,0:01:02.53,0:01:06.17,Default,,0000,0000,0000,,And so, and if we drop an\Naltitude right over here
Dialogue: 0,0:01:06.17,0:01:08.20,Default,,0000,0000,0000,,which is the whole\Npoint, that's the height,
Dialogue: 0,0:01:08.20,0:01:12.09,Default,,0000,0000,0000,,we know that this is, these\Nare going to be right angles.
Dialogue: 0,0:01:12.09,0:01:14.13,Default,,0000,0000,0000,,And so, if we have two triangles
Dialogue: 0,0:01:14.13,0:01:15.81,Default,,0000,0000,0000,,where two of the angles are the same,
Dialogue: 0,0:01:15.81,0:01:18.17,Default,,0000,0000,0000,,we know that the third angle\Nis going to be the same.
Dialogue: 0,0:01:18.17,0:01:21.20,Default,,0000,0000,0000,,So, that is going to be congruent to that.
Dialogue: 0,0:01:21.20,0:01:23.66,Default,,0000,0000,0000,,And so, if you have two triangles,
Dialogue: 0,0:01:23.66,0:01:26.62,Default,,0000,0000,0000,,and this might be obvious\Nalready to you intuitively,
Dialogue: 0,0:01:26.62,0:01:29.01,Default,,0000,0000,0000,,where look, I have two angles in common
Dialogue: 0,0:01:29.01,0:01:31.57,Default,,0000,0000,0000,,and the side in between them is common,
Dialogue: 0,0:01:31.57,0:01:33.70,Default,,0000,0000,0000,,it's the same length,
Dialogue: 0,0:01:33.70,0:01:35.73,Default,,0000,0000,0000,,well that means that these are going to be
Dialogue: 0,0:01:35.73,0:01:37.83,Default,,0000,0000,0000,,congruent triangles.
Dialogue: 0,0:01:37.83,0:01:39.67,Default,,0000,0000,0000,,Now, what's useful about\Nthat is if we recognize
Dialogue: 0,0:01:39.67,0:01:41.54,Default,,0000,0000,0000,,that these are congruent triangles,
Dialogue: 0,0:01:41.54,0:01:43.64,Default,,0000,0000,0000,,notice that they both have a side 13,
Dialogue: 0,0:01:43.64,0:01:46.40,Default,,0000,0000,0000,,they both have a side, whatever\Nthis length in blue is.
Dialogue: 0,0:01:46.40,0:01:49.23,Default,,0000,0000,0000,,And then, they're both\Ngoing to have a side length
Dialogue: 0,0:01:49.23,0:01:51.15,Default,,0000,0000,0000,,that's half of this 10.
Dialogue: 0,0:01:52.57,0:01:55.38,Default,,0000,0000,0000,,So, this is going to be five,\Nand this is going to be five.
Dialogue: 0,0:01:55.38,0:01:57.11,Default,,0000,0000,0000,,How was I able to deduce that?
Dialogue: 0,0:01:57.11,0:01:59.29,Default,,0000,0000,0000,,You might just say, oh that\Nfeels intuitively right.
Dialogue: 0,0:01:59.29,0:02:00.65,Default,,0000,0000,0000,,I was a little bit more rigorous here,
Dialogue: 0,0:02:00.65,0:02:03.42,Default,,0000,0000,0000,,where I said these are\Ntwo congruent triangles,
Dialogue: 0,0:02:03.42,0:02:06.14,Default,,0000,0000,0000,,then we're going to split this 10 in half
Dialogue: 0,0:02:06.14,0:02:07.74,Default,,0000,0000,0000,,because this is going to be equal to that
Dialogue: 0,0:02:07.74,0:02:09.52,Default,,0000,0000,0000,,and they add up to 10.
Dialogue: 0,0:02:09.52,0:02:12.28,Default,,0000,0000,0000,,Alright, now we can use\Nthe Pythagorean Theorem
Dialogue: 0,0:02:12.28,0:02:16.07,Default,,0000,0000,0000,,to figure out the length of\Nthis blue side or the height.
Dialogue: 0,0:02:16.07,0:02:19.66,Default,,0000,0000,0000,,If we call this h, the\NPythagorean Theorem tells us
Dialogue: 0,0:02:19.66,0:02:23.22,Default,,0000,0000,0000,,that h squared plus five\Nsquared is equal to 13 squared.
Dialogue: 0,0:02:23.22,0:02:25.55,Default,,0000,0000,0000,,H squared plus five squared,
Dialogue: 0,0:02:27.13,0:02:31.59,Default,,0000,0000,0000,,plus five squared is going\Nto be equal to 13 squared,
Dialogue: 0,0:02:31.59,0:02:33.19,Default,,0000,0000,0000,,is going to be equal to our longest side,
Dialogue: 0,0:02:33.19,0:02:35.47,Default,,0000,0000,0000,,our hypotenuse squared.
Dialogue: 0,0:02:35.47,0:02:36.39,Default,,0000,0000,0000,,And so, let's see.
Dialogue: 0,0:02:36.39,0:02:37.97,Default,,0000,0000,0000,,Five squared is 25.
Dialogue: 0,0:02:40.44,0:02:41.94,Default,,0000,0000,0000,,13 squared is 169.
Dialogue: 0,0:02:44.49,0:02:47.81,Default,,0000,0000,0000,,We can subtract 25 from both sides
Dialogue: 0,0:02:47.81,0:02:49.95,Default,,0000,0000,0000,,to isolate the h squared.
Dialogue: 0,0:02:49.95,0:02:51.60,Default,,0000,0000,0000,,So, let's do that.
Dialogue: 0,0:02:51.60,0:02:53.73,Default,,0000,0000,0000,,And what are we left with?
Dialogue: 0,0:02:53.73,0:02:57.42,Default,,0000,0000,0000,,We are left with h squared is equal to
Dialogue: 0,0:02:57.42,0:03:00.75,Default,,0000,0000,0000,,these canceled out, 169 minus 25 is 144.
Dialogue: 0,0:03:03.48,0:03:04.84,Default,,0000,0000,0000,,Now, if you're doing it\Npurely mathematically,
Dialogue: 0,0:03:04.84,0:03:07.02,Default,,0000,0000,0000,,you say, oh h could be plus or minus 12,
Dialogue: 0,0:03:07.02,0:03:08.32,Default,,0000,0000,0000,,but we're dealing with the distance,
Dialogue: 0,0:03:08.32,0:03:10.60,Default,,0000,0000,0000,,so we'll focus on the positive.
Dialogue: 0,0:03:10.60,0:03:15.30,Default,,0000,0000,0000,,So, h is going to be equal\Nto the principal root of 144.
Dialogue: 0,0:03:15.30,0:03:17.08,Default,,0000,0000,0000,,So, h is equal to 12.
Dialogue: 0,0:03:17.08,0:03:18.04,Default,,0000,0000,0000,,Now, we aren't done.
Dialogue: 0,0:03:18.04,0:03:19.15,Default,,0000,0000,0000,,Remember, they don't want us to just
Dialogue: 0,0:03:19.15,0:03:20.31,Default,,0000,0000,0000,,figure out the height here,
Dialogue: 0,0:03:20.31,0:03:22.47,Default,,0000,0000,0000,,they want us to figure out the area.
Dialogue: 0,0:03:22.47,0:03:25.39,Default,,0000,0000,0000,,Area is one half base times height.
Dialogue: 0,0:03:26.29,0:03:27.43,Default,,0000,0000,0000,,Well, we already figured out
Dialogue: 0,0:03:27.43,0:03:31.05,Default,,0000,0000,0000,,that our base is this 10 right over here,
Dialogue: 0,0:03:31.05,0:03:32.81,Default,,0000,0000,0000,,let me do this in another color.
Dialogue: 0,0:03:32.81,0:03:36.31,Default,,0000,0000,0000,,So, our base is that distance which is 10,
Dialogue: 0,0:03:37.44,0:03:39.53,Default,,0000,0000,0000,,and now we know our height.
Dialogue: 0,0:03:39.53,0:03:40.95,Default,,0000,0000,0000,,Our height is 12.
Dialogue: 0,0:03:42.33,0:03:45.92,Default,,0000,0000,0000,,So, now we just have to compute\None half times 10 times 12.
Dialogue: 0,0:03:45.92,0:03:47.88,Default,,0000,0000,0000,,Well, that's just going to be equal to
Dialogue: 0,0:03:47.88,0:03:50.10,Default,,0000,0000,0000,,one half times 10 is five,
Dialogue: 0,0:03:50.10,0:03:52.07,Default,,0000,0000,0000,,times 12 is 60,
Dialogue: 0,0:03:52.07,0:03:55.95,Default,,0000,0000,0000,,60 square units, whatever\Nour units happen to be.
Dialogue: 0,0:03:55.95,0:03:57.36,Default,,0000,0000,0000,,That is our area.