1 00:00:00,000 --> 00:00:00,670 2 00:00:00,670 --> 00:00:04,045 This triangle that we have right over here is a right triangle. 3 00:00:04,045 --> 00:00:06,600 And it's a right triangle because it has a 90 degree 4 00:00:06,600 --> 00:00:09,240 angle, or has a right angle in it. 5 00:00:09,240 --> 00:00:12,520 Now, we call the longest side of a right triangle, 6 00:00:12,520 --> 00:00:14,599 we call that side, and you could either 7 00:00:14,599 --> 00:00:17,140 view it as the longest side of the right triangle or the side 8 00:00:17,140 --> 00:00:20,980 opposite the 90 degree angle, it is called a hypotenuse. 9 00:00:20,980 --> 00:00:23,740 It's a very fancy word for a fairly simple idea, 10 00:00:23,740 --> 00:00:26,140 just the longest side of a right triangle or the side 11 00:00:26,140 --> 00:00:27,542 opposite the 90 degree angle. 12 00:00:27,542 --> 00:00:29,500 And it's just good to know that because someone 13 00:00:29,500 --> 00:00:30,090 might say hypotenuse. 14 00:00:30,090 --> 00:00:32,548 You're like, oh, they're just talking about this side right 15 00:00:32,548 --> 00:00:36,580 here, the side longest, the side opposite the 90 degree angle. 16 00:00:36,580 --> 00:00:38,860 Now, what I want to do in this video 17 00:00:38,860 --> 00:00:42,167 is prove a relationship, a very famous relationship. 18 00:00:42,167 --> 00:00:43,750 And you might see where this is going. 19 00:00:43,750 --> 00:00:46,370 A very famous relationship between the lengths 20 00:00:46,370 --> 00:00:48,840 of the sides of a right triangle. 21 00:00:48,840 --> 00:00:53,210 So let's say that the length of AC, so uppercase A, uppercase 22 00:00:53,210 --> 00:00:55,930 C, let's call that length lowercase a. 23 00:00:55,930 --> 00:01:00,040 Let's call the length of BC lowercase b right over here. 24 00:01:00,040 --> 00:01:03,420 I'll use uppercases for points, lowercases for lengths. 25 00:01:03,420 --> 00:01:06,630 And let's call the length of the hypotenuse, the length of AB, 26 00:01:06,630 --> 00:01:07,822 let's call that c. 27 00:01:07,822 --> 00:01:10,030 And let's see if we can come up with the relationship 28 00:01:10,030 --> 00:01:12,790 between a, b, and c. 29 00:01:12,790 --> 00:01:14,780 And to do that I'm first going to construct 30 00:01:14,780 --> 00:01:16,410 another line or another segment, I 31 00:01:16,410 --> 00:01:19,520 should say, between c and the hypotenuse. 32 00:01:19,520 --> 00:01:21,600 And I'm going to construct it so that they 33 00:01:21,600 --> 00:01:23,880 intersect at a right angle. 34 00:01:23,880 --> 00:01:25,006 And you can always do that. 35 00:01:25,006 --> 00:01:27,005 And we'll call this point right over here we'll. 36 00:01:27,005 --> 00:01:28,120 Call this point capital D. 37 00:01:28,120 --> 00:01:31,010 And if you're wondering, how can you always do that? 38 00:01:31,010 --> 00:01:33,634 You could imagine rotating this entire triangle like this. 39 00:01:33,634 --> 00:01:36,050 This isn't a rigorous proof, but it just kind of gives you 40 00:01:36,050 --> 00:01:38,100 the general idea of how you can always 41 00:01:38,100 --> 00:01:39,810 construct a point like this. 42 00:01:39,810 --> 00:01:41,260 So if I've rotated it around. 43 00:01:41,260 --> 00:01:44,750 So now our hypotenuse, we're now sitting on our hypotenuse. 44 00:01:44,750 --> 00:01:48,414 This is now point B, this is point A. 45 00:01:48,414 --> 00:01:50,580 So we've rotated the whole thing all the way around. 46 00:01:50,580 --> 00:01:52,710 This is point C. You could imagine just 47 00:01:52,710 --> 00:01:55,820 dropping a rock from point C, maybe with a string attached, 48 00:01:55,820 --> 00:01:59,460 and it would hit the hypotenuse at a right angle. 49 00:01:59,460 --> 00:02:02,980 So that's all we did here to establish segment CD into where 50 00:02:02,980 --> 00:02:05,570 we put our point D right over there. 51 00:02:05,570 --> 00:02:07,220 And the reason why did that is now we 52 00:02:07,220 --> 00:02:09,289 can do all sorts of interesting relationships 53 00:02:09,289 --> 00:02:10,490 between similar triangles. 54 00:02:10,490 --> 00:02:12,180 Because we have three triangles here. 55 00:02:12,180 --> 00:02:15,604 We have triangle ADC, we have triangle DBC, 56 00:02:15,604 --> 00:02:17,520 and then we have the larger original triangle. 57 00:02:17,520 --> 00:02:19,890 And we can hopefully establish similarity 58 00:02:19,890 --> 00:02:21,980 between those triangles. 59 00:02:21,980 --> 00:02:27,590 And first I'll show you that ADC is similar to the larger one. 60 00:02:27,590 --> 00:02:29,710 Because both of them have a right angle. 61 00:02:29,710 --> 00:02:32,070 ADC has a right angle right over here. 62 00:02:32,070 --> 00:02:33,571 Clearly if this angle is 90 degrees, 63 00:02:33,571 --> 00:02:35,653 then this angle is going to be 90 degrees as well. 64 00:02:35,653 --> 00:02:36,660 They are supplementary. 65 00:02:36,660 --> 00:02:38,510 They have to add up to 180. 66 00:02:38,510 --> 00:02:40,440 And so they both have a right angle in them. 67 00:02:40,440 --> 00:02:42,060 So the smaller one has a right angle. 68 00:02:42,060 --> 00:02:43,590 The larger one clearly has a right angle. 69 00:02:43,590 --> 00:02:44,840 That's where we started from. 70 00:02:44,840 --> 00:02:48,690 And they also both share this angle right 71 00:02:48,690 --> 00:02:52,150 over here, angle DAC or BAC, however 72 00:02:52,150 --> 00:02:53,580 you want to refer to it. 73 00:02:53,580 --> 00:02:56,720 So we can actually write down that triangle. 74 00:02:56,720 --> 00:03:00,290 I'm going to start with the smaller one, ADC. 75 00:03:00,290 --> 00:03:02,190 And maybe I'll shade it in right over here. 76 00:03:02,190 --> 00:03:04,023 So this is the triangle we're talking about. 77 00:03:04,023 --> 00:03:05,429 Triangle ADC. 78 00:03:05,429 --> 00:03:07,470 And I went from the blue angle to the right angle 79 00:03:07,470 --> 00:03:10,620 to the unlabeled angle from the point of view of triangle ADC. 80 00:03:10,620 --> 00:03:13,860 This right angle isn't applying to that right over there. 81 00:03:13,860 --> 00:03:15,820 It's applying to the larger triangle. 82 00:03:15,820 --> 00:03:24,820 So we could say triangle ADC is similar to triangle-- 83 00:03:24,820 --> 00:03:27,130 once again, you want to start at the blue angle. 84 00:03:27,130 --> 00:03:29,500 A. Then we went to the right angle. 85 00:03:29,500 --> 00:03:32,220 So we have to go to the right angle again. 86 00:03:32,220 --> 00:03:32,830 So it's ACB. 87 00:03:32,830 --> 00:03:37,190 88 00:03:37,190 --> 00:03:39,270 And because they're similar, we can set up 89 00:03:39,270 --> 00:03:42,220 a relationship between the ratios of their sides. 90 00:03:42,220 --> 00:03:44,705 For example, we know the ratio of corresponding sides 91 00:03:44,705 --> 00:03:47,080 are going to do, well, in general for a similar triangle, 92 00:03:47,080 --> 00:03:48,640 we know the ratio of the corresponding sides 93 00:03:48,640 --> 00:03:49,890 are going to be a constant. 94 00:03:49,890 --> 00:03:54,100 So we could take the ratio of the hypotenuse of the smaller 95 00:03:54,100 --> 00:03:54,960 triangle. 96 00:03:54,960 --> 00:03:57,350 So the hypotenuse is AC. 97 00:03:57,350 --> 00:04:00,710 So AC over the hypotenuse over the larger one, which 98 00:04:00,710 --> 00:04:10,480 is a AB, AC over AB is going to be the same thing as AD 99 00:04:10,480 --> 00:04:14,180 as one of the legs, AD. 100 00:04:14,180 --> 00:04:16,959 And just to show that, I'm just taking corresponding points 101 00:04:16,959 --> 00:04:23,794 on both similar triangles, this is AD over AC. 102 00:04:23,794 --> 00:04:25,960 You could look at these triangles yourself and show, 103 00:04:25,960 --> 00:04:29,930 look, AD, point AD, is between the blue angle 104 00:04:29,930 --> 00:04:31,410 and the right angle. 105 00:04:31,410 --> 00:04:34,760 Sorry, side AD is between the blue angle and the right angle. 106 00:04:34,760 --> 00:04:38,025 Side AC is between the blue angle and the right angle 107 00:04:38,025 --> 00:04:39,010 on the larger triangle. 108 00:04:39,010 --> 00:04:40,950 So both of these are from the larger triangle. 109 00:04:40,950 --> 00:04:43,660 These are the corresponding sides on the smaller triangle. 110 00:04:43,660 --> 00:04:46,990 And if that is confusing looking at them visually, 111 00:04:46,990 --> 00:04:50,199 as long as we wrote our similarity statement correctly, 112 00:04:50,199 --> 00:04:51,990 you can just find the corresponding points. 113 00:04:51,990 --> 00:04:56,590 AC corresponds to AB on the larger triangle, 114 00:04:56,590 --> 00:04:58,840 AD on the smaller triangle corresponds 115 00:04:58,840 --> 00:05:02,330 to AC on the larger triangle. 116 00:05:02,330 --> 00:05:06,920 And we know that AC, we can rewrite that as lowercase a. 117 00:05:06,920 --> 00:05:10,860 AC is lowercase a. 118 00:05:10,860 --> 00:05:16,810 We don't have any label for AD or for AB. 119 00:05:16,810 --> 00:05:18,900 Sorry, we do have a label for AB. 120 00:05:18,900 --> 00:05:20,590 That is c right over here. 121 00:05:20,590 --> 00:05:23,790 We don't have a label for AD. 122 00:05:23,790 --> 00:05:26,840 So AD, let's just call that lowercase d. 123 00:05:26,840 --> 00:05:30,400 So lowercase d applies to that part right over there. 124 00:05:30,400 --> 00:05:33,560 c applies to that entire part right over there. 125 00:05:33,560 --> 00:05:35,905 And then we'll call DB, let's call that length e. 126 00:05:35,905 --> 00:05:38,700 That'll just make things a little bit simpler for us. 127 00:05:38,700 --> 00:05:41,760 So AD we'll just call d. 128 00:05:41,760 --> 00:05:43,850 And so we have a over c is equal to d over a. 129 00:05:43,850 --> 00:05:47,830 If we cross multiply, you have a times a, which is a squared, 130 00:05:47,830 --> 00:05:50,791 is equal to c times d, which is cd. 131 00:05:50,791 --> 00:05:52,790 So that's a little bit of an interesting result. 132 00:05:52,790 --> 00:05:54,789 Let's see what we can do with the other triangle 133 00:05:54,789 --> 00:05:55,930 right over here. 134 00:05:55,930 --> 00:05:57,940 So this triangle right over here. 135 00:05:57,940 --> 00:05:59,490 So once again, it has a right angle. 136 00:05:59,490 --> 00:06:00,865 The larger one has a right angle. 137 00:06:00,865 --> 00:06:04,270 And they both share this angle right over here. 138 00:06:04,270 --> 00:06:07,070 So by angle, angle similarity, the two triangles 139 00:06:07,070 --> 00:06:08,210 are going to be similar. 140 00:06:08,210 --> 00:06:11,040 So we could say triangle BDC, we went from pink 141 00:06:11,040 --> 00:06:12,970 to right to not labeled. 142 00:06:12,970 --> 00:06:20,352 So triangle BDC is similar to triangle. 143 00:06:20,352 --> 00:06:22,310 Now we're going to look at the larger triangle, 144 00:06:22,310 --> 00:06:23,430 we're going to start at the pink angle. 145 00:06:23,430 --> 00:06:25,567 B. Now we're going to go to the right angle. 146 00:06:25,567 --> 00:06:26,066 CA. 147 00:06:26,066 --> 00:06:29,190 148 00:06:29,190 --> 00:06:31,680 BCA. 149 00:06:31,680 --> 00:06:34,979 From pink angle to right angle to non-labeled angle, 150 00:06:34,979 --> 00:06:36,520 at least from the point of view here. 151 00:06:36,520 --> 00:06:38,420 We labeled it before with that blue. 152 00:06:38,420 --> 00:06:40,620 So now let's set up some type of relationship here. 153 00:06:40,620 --> 00:06:45,040 We can say that the ratio on the smaller triangle, BC, side 154 00:06:45,040 --> 00:06:50,130 BC over BA, BC over BA, once again, 155 00:06:50,130 --> 00:06:53,230 we're taking the hypotenuses of both of them. 156 00:06:53,230 --> 00:07:00,593 So BC over BA is going to be equal to BD. 157 00:07:00,593 --> 00:07:02,590 Let me do this in another color. 158 00:07:02,590 --> 00:07:03,450 BD. 159 00:07:03,450 --> 00:07:04,890 So this is one of the legs. 160 00:07:04,890 --> 00:07:05,570 BD. 161 00:07:05,570 --> 00:07:07,430 The way I drew it is the shorter legs. 162 00:07:07,430 --> 00:07:10,370 BD over BC. 163 00:07:10,370 --> 00:07:12,770 I'm just taking the corresponding vertices. 164 00:07:12,770 --> 00:07:14,600 Over BC. 165 00:07:14,600 --> 00:07:18,203 And once again, we know BC is the same thing as lowercase b. 166 00:07:18,203 --> 00:07:20,322 BC is lowercase b. 167 00:07:20,322 --> 00:07:22,926 BA is lowercase c. 168 00:07:22,926 --> 00:07:25,570 169 00:07:25,570 --> 00:07:29,740 And then BD we defined as lowercase e. 170 00:07:29,740 --> 00:07:31,260 So this is lowercase e. 171 00:07:31,260 --> 00:07:33,210 We can cross multiply here and we 172 00:07:33,210 --> 00:07:37,830 get b times b, which, and I've mentioned this in many videos, 173 00:07:37,830 --> 00:07:40,310 cross multiplying is really the same thing as multiplying 174 00:07:40,310 --> 00:07:42,680 both sides by both denominators. 175 00:07:42,680 --> 00:07:47,960 b times b is b squared is equal to ce. 176 00:07:47,960 --> 00:07:50,010 And now we can do something kind of interesting. 177 00:07:50,010 --> 00:07:51,406 We can add these two statements. 178 00:07:51,406 --> 00:07:53,030 Let me rewrite the statement down here. 179 00:07:53,030 --> 00:07:56,100 So b squared is equal to ce. 180 00:07:56,100 --> 00:07:58,310 So if we add the left hand sides, 181 00:07:58,310 --> 00:08:02,120 we get a squared plus b squared. 182 00:08:02,120 --> 00:08:09,420 a squared plus b squared is equal to cd plus ce. 183 00:08:09,420 --> 00:08:12,595 184 00:08:12,595 --> 00:08:14,917 And then we have a c both of these terms, 185 00:08:14,917 --> 00:08:16,000 so we could factor it out. 186 00:08:16,000 --> 00:08:19,880 So this is going to be equal to-- we can factor out the c. 187 00:08:19,880 --> 00:08:22,952 It's going to be equal to c times d plus e. 188 00:08:22,952 --> 00:08:29,790 c times d plus e and close the parentheses. 189 00:08:29,790 --> 00:08:31,460 Now what is d plus e? 190 00:08:31,460 --> 00:08:34,159 d is this length, e is this length. 191 00:08:34,159 --> 00:08:37,169 So d plus e is actually going to be c as well. 192 00:08:37,169 --> 00:08:38,496 So this is going to be c. 193 00:08:38,496 --> 00:08:41,039 So you have c times c, which is just the same thing 194 00:08:41,039 --> 00:08:43,030 as c squared. 195 00:08:43,030 --> 00:08:45,700 So now we have an interesting relationship. 196 00:08:45,700 --> 00:08:51,150 We have that a squared plus b squared is equal to c squared. 197 00:08:51,150 --> 00:08:52,580 Let me rewrite that. 198 00:08:52,580 --> 00:08:54,300 a squared. 199 00:08:54,300 --> 00:08:58,623 Well, let me just do an arbitrary new color. 200 00:08:58,623 --> 00:09:02,380 I deleted that by accident, so let me rewrite it. 201 00:09:02,380 --> 00:09:07,390 So we've just established that a squared plus b squared 202 00:09:07,390 --> 00:09:09,400 is equal to c squared. 203 00:09:09,400 --> 00:09:11,320 And this is just an arbitrary right triangle. 204 00:09:11,320 --> 00:09:13,590 This is true for any two right triangles. 205 00:09:13,590 --> 00:09:17,120 We've just established that the sum of the squares of each 206 00:09:17,120 --> 00:09:20,060 of the legs is equal to the square of the hypotenuse. 207 00:09:20,060 --> 00:09:22,550 And this is probably what's easily 208 00:09:22,550 --> 00:09:26,220 one of the most famous theorem in mathematics, named 209 00:09:26,220 --> 00:09:27,360 for Pythagoras. 210 00:09:27,360 --> 00:09:30,370 Not clear if he's the first person to establish this, 211 00:09:30,370 --> 00:09:32,310 but it's called the Pythagorean Theorem. 212 00:09:32,310 --> 00:09:38,290 213 00:09:38,290 --> 00:09:41,469 And it's really the basis of, well, all not all of geometry, 214 00:09:41,469 --> 00:09:43,510 but a lot of the geometry that we're going to do. 215 00:09:43,510 --> 00:09:45,880 And it forms the basis of a lot of the trigonometry we're 216 00:09:45,880 --> 00:09:46,230 going to do. 217 00:09:46,230 --> 00:09:47,550 And it's a really useful way, if you 218 00:09:47,550 --> 00:09:49,299 know two of the sides of a right triangle, 219 00:09:49,299 --> 00:09:51,890 you can always find the third.