WEBVTT 00:00:00.607 --> 00:00:02.229 - [Tutor] Pause this video and see if you can find 00:00:02.229 --> 00:00:04.633 the area of this triangle, 00:00:04.633 --> 00:00:06.583 and I'll give you two hints. 00:00:06.583 --> 00:00:09.375 Recognize, this is an isosceles triangle, 00:00:09.375 --> 00:00:12.033 and another hint is that the Pythagorean Theorem 00:00:12.033 --> 00:00:13.366 might be useful. 00:00:14.254 --> 00:00:16.762 Alright, now let's work through this together. 00:00:16.762 --> 00:00:20.042 So, we might all remember that the area of a triangle 00:00:20.042 --> 00:00:24.701 is equal to one half times our base times our height. 00:00:24.701 --> 00:00:25.945 They give us our base. 00:00:25.945 --> 00:00:28.430 Our base right over here is, 00:00:28.430 --> 00:00:29.680 our base is 10. 00:00:31.207 --> 00:00:32.859 But what is our height? 00:00:32.859 --> 00:00:34.200 Our height would be, 00:00:34.200 --> 00:00:35.901 let me do this in another color, 00:00:35.901 --> 00:00:40.011 our height would be the length of this line right over here. 00:00:40.011 --> 00:00:41.715 So, if we can figure that out, 00:00:41.715 --> 00:00:44.858 then we can calculate what one half times the base 10 00:00:44.858 --> 00:00:46.498 times the height is. 00:00:46.498 --> 00:00:49.154 But how do we figure out this height? 00:00:49.154 --> 00:00:51.484 Well, this is where it's useful to recognize 00:00:51.484 --> 00:00:53.995 that this is an isosceles triangle. 00:00:53.995 --> 00:00:57.521 An isosceles triangle has two sides that are the same. 00:00:57.521 --> 00:01:01.688 And so, these base angles are also going to be congruent. 00:01:02.529 --> 00:01:06.174 And so, and if we drop an altitude right over here 00:01:06.174 --> 00:01:08.197 which is the whole point, that's the height, 00:01:08.197 --> 00:01:12.090 we know that this is, these are going to be right angles. 00:01:12.090 --> 00:01:14.134 And so, if we have two triangles 00:01:14.134 --> 00:01:15.808 where two of the angles are the same, 00:01:15.808 --> 00:01:18.173 we know that the third angle is going to be the same. 00:01:18.173 --> 00:01:21.196 So, that is going to be congruent to that. 00:01:21.196 --> 00:01:23.660 And so, if you have two triangles, 00:01:23.660 --> 00:01:26.620 and this might be obvious already to you intuitively, 00:01:26.620 --> 00:01:29.006 where look, I have two angles in common 00:01:29.006 --> 00:01:31.566 and the side in between them is common, 00:01:31.566 --> 00:01:33.697 it's the same length, 00:01:33.697 --> 00:01:35.730 well that means that these are going to be 00:01:35.730 --> 00:01:37.829 congruent triangles. 00:01:37.829 --> 00:01:39.672 Now, what's useful about that is if we recognize 00:01:39.672 --> 00:01:41.543 that these are congruent triangles, 00:01:41.543 --> 00:01:43.635 notice that they both have a side 13, 00:01:43.635 --> 00:01:46.397 they both have a side, whatever this length in blue is. 00:01:46.397 --> 00:01:49.234 And then, they're both going to have a side length 00:01:49.234 --> 00:01:51.151 that's half of this 10. 00:01:52.571 --> 00:01:55.383 So, this is going to be five, and this is going to be five. 00:01:55.383 --> 00:01:57.112 How was I able to deduce that? 00:01:57.112 --> 00:01:59.293 You might just say, oh that feels intuitively right. 00:01:59.293 --> 00:02:00.650 I was a little bit more rigorous here, 00:02:00.650 --> 00:02:03.420 where I said these are two congruent triangles, 00:02:03.420 --> 00:02:06.143 then we're going to split this 10 in half 00:02:06.143 --> 00:02:07.736 because this is going to be equal to that 00:02:07.736 --> 00:02:09.515 and they add up to 10. 00:02:09.515 --> 00:02:12.283 Alright, now we can use the Pythagorean Theorem 00:02:12.283 --> 00:02:16.072 to figure out the length of this blue side or the height. 00:02:16.072 --> 00:02:19.658 If we call this h, the Pythagorean Theorem tells us 00:02:19.658 --> 00:02:23.221 that h squared plus five squared is equal to 13 squared. 00:02:23.221 --> 00:02:25.554 H squared plus five squared, 00:02:27.131 --> 00:02:31.589 plus five squared is going to be equal to 13 squared, 00:02:31.589 --> 00:02:33.191 is going to be equal to our longest side, 00:02:33.191 --> 00:02:35.472 our hypotenuse squared. 00:02:35.472 --> 00:02:36.387 And so, let's see. 00:02:36.387 --> 00:02:37.970 Five squared is 25. 00:02:40.444 --> 00:02:41.944 13 squared is 169. 00:02:44.491 --> 00:02:47.808 We can subtract 25 from both sides 00:02:47.808 --> 00:02:49.949 to isolate the h squared. 00:02:49.949 --> 00:02:51.599 So, let's do that. 00:02:51.599 --> 00:02:53.728 And what are we left with? 00:02:53.728 --> 00:02:57.420 We are left with h squared is equal to 00:02:57.420 --> 00:03:00.753 these canceled out, 169 minus 25 is 144. 00:03:03.483 --> 00:03:04.843 Now, if you're doing it purely mathematically, 00:03:04.843 --> 00:03:07.017 you say, oh h could be plus or minus 12, 00:03:07.017 --> 00:03:08.322 but we're dealing with the distance, 00:03:08.322 --> 00:03:10.598 so we'll focus on the positive. 00:03:10.598 --> 00:03:15.298 So, h is going to be equal to the principal root of 144. 00:03:15.298 --> 00:03:17.084 So, h is equal to 12. 00:03:17.084 --> 00:03:18.043 Now, we aren't done. 00:03:18.043 --> 00:03:19.149 Remember, they don't want us to just 00:03:19.149 --> 00:03:20.307 figure out the height here, 00:03:20.307 --> 00:03:22.472 they want us to figure out the area. 00:03:22.472 --> 00:03:25.389 Area is one half base times height. 00:03:26.289 --> 00:03:27.430 Well, we already figured out 00:03:27.430 --> 00:03:31.052 that our base is this 10 right over here, 00:03:31.052 --> 00:03:32.809 let me do this in another color. 00:03:32.809 --> 00:03:36.309 So, our base is that distance which is 10, 00:03:37.443 --> 00:03:39.533 and now we know our height. 00:03:39.533 --> 00:03:40.950 Our height is 12. 00:03:42.328 --> 00:03:45.925 So, now we just have to compute one half times 10 times 12. 00:03:45.925 --> 00:03:47.875 Well, that's just going to be equal to 00:03:47.875 --> 00:03:50.096 one half times 10 is five, 00:03:50.096 --> 00:03:52.072 times 12 is 60, 00:03:52.072 --> 00:03:55.947 60 square units, whatever our units happen to be. 00:03:55.947 --> 00:03:57.364 That is our area.