1 00:00:00,530 --> 00:00:03,220 In this video we're going to get introduced to the 2 00:00:03,220 --> 00:00:14,190 Pythagorean theorem, which is fun on its own. 3 00:00:14,190 --> 00:00:16,930 But you'll see as you learn more and more mathematics it's 4 00:00:16,930 --> 00:00:21,570 one of those cornerstone theorems of really all of math. 5 00:00:21,570 --> 00:00:24,920 It's useful in geometry, it's kind of the backbone 6 00:00:24,920 --> 00:00:26,750 of trigonometry. 7 00:00:26,750 --> 00:00:29,200 You're also going to use it to calculate distances 8 00:00:29,200 --> 00:00:30,510 between points. 9 00:00:30,510 --> 00:00:33,810 So it's a good thing to really make sure we know well. 10 00:00:33,810 --> 00:00:35,570 So enough talk on my end. 11 00:00:35,570 --> 00:00:38,320 Let me tell you what the Pythagorean theorem is. 12 00:00:38,320 --> 00:00:43,290 So if we have a triangle, and the triangle has to be a right 13 00:00:43,290 --> 00:00:49,110 triangle, which means that one of the three angles in the 14 00:00:49,110 --> 00:00:51,520 triangle have to be 90 degrees. 15 00:00:51,520 --> 00:00:54,580 And you specify that it's 90 degrees by drawing that 16 00:00:54,580 --> 00:00:55,930 little box right there. 17 00:00:55,930 --> 00:00:58,830 So that right there is-- let me do this in a different 18 00:00:58,830 --> 00:01:05,550 color-- a 90 degree angle. 19 00:01:05,550 --> 00:01:09,930 Or, we could call it a right angle. 20 00:01:09,930 --> 00:01:13,390 And a triangle that has a right angle in it is 21 00:01:13,390 --> 00:01:15,850 called a right triangle. 22 00:01:15,850 --> 00:01:21,700 So this is called a right triangle. 23 00:01:21,700 --> 00:01:25,440 Now, with the Pythagorean theorem, if we know two sides 24 00:01:25,440 --> 00:01:28,980 of a right triangle we can always figure out 25 00:01:28,980 --> 00:01:30,920 the third side. 26 00:01:30,920 --> 00:01:34,310 And before I show you how to do that, let me give you one 27 00:01:34,310 --> 00:01:36,560 more piece of terminology. 28 00:01:36,560 --> 00:01:43,230 The longest side of a right triangle is the side opposite 29 00:01:43,230 --> 00:01:46,690 the 90 degree angle-- or opposite the right angle. 30 00:01:46,690 --> 00:01:49,650 So in this case it is this side right here. 31 00:01:49,650 --> 00:01:51,285 This is the longest side. 32 00:01:51,285 --> 00:01:55,020 And the way to figure out where that right triangle is, and 33 00:01:55,020 --> 00:01:58,060 kind of it opens into that longest side. 34 00:01:58,060 --> 00:02:00,150 That longest side is called the hypotenuse. 35 00:02:03,130 --> 00:02:05,330 And it's good to know, because we'll keep referring to it. 36 00:02:05,330 --> 00:02:09,000 And just so we always are good at identifying the hypotenuse, 37 00:02:09,000 --> 00:02:12,560 let me draw a couple of more right triangles. 38 00:02:12,560 --> 00:02:17,090 So let's say I have a triangle that looks like that. 39 00:02:17,090 --> 00:02:19,390 Let me draw it a little bit nicer. 40 00:02:19,390 --> 00:02:22,130 So let's say I have a triangle that looks like that. 41 00:02:22,130 --> 00:02:24,010 And I were to tell you that this angle right 42 00:02:24,010 --> 00:02:25,390 here is 90 degrees. 43 00:02:25,390 --> 00:02:29,860 In this situation this is the hypotenuse, because it is 44 00:02:29,860 --> 00:02:33,410 opposite the 90 degree angle. 45 00:02:33,410 --> 00:02:34,880 It is the longest side. 46 00:02:34,880 --> 00:02:36,670 Let me do one more, just so that we're good at 47 00:02:36,670 --> 00:02:39,420 recognizing the hypotenuse. 48 00:02:39,420 --> 00:02:44,050 So let's say that that is my triangle, and this is the 90 49 00:02:44,050 --> 00:02:45,790 degree angle right there. 50 00:02:45,790 --> 00:02:47,710 And I think you know how to do this already. 51 00:02:47,710 --> 00:02:49,620 You go right what it opens into. 52 00:02:49,620 --> 00:02:51,530 That is the hypotenuse. 53 00:02:51,530 --> 00:02:53,200 That is the longest side. 54 00:02:57,940 --> 00:03:00,400 So once you have identified the hypotenuse-- and let's say 55 00:03:00,400 --> 00:03:02,050 that that has length C. 56 00:03:02,050 --> 00:03:03,980 And now we're going to learn what the Pythagorean 57 00:03:03,980 --> 00:03:05,210 theorem tells us. 58 00:03:05,210 --> 00:03:08,680 So let's say that C is equal to the length of the hypotenuse. 59 00:03:08,680 --> 00:03:11,630 So let's call this C-- that side is C. 60 00:03:11,630 --> 00:03:17,910 Let's call this side right over here A. 61 00:03:17,910 --> 00:03:21,890 And let's call this side over here B. 62 00:03:21,890 --> 00:03:28,620 So the Pythagorean theorem tells us that A squared-- so 63 00:03:28,620 --> 00:03:32,880 the length of one of the shorter sides squared-- plus 64 00:03:32,880 --> 00:03:36,890 the length of the other shorter side squared is going to 65 00:03:36,890 --> 00:03:41,370 be equal to the length of the hypotenuse squared. 66 00:03:41,370 --> 00:03:43,740 Now let's do that with an actual problem, and you'll see 67 00:03:43,740 --> 00:03:45,820 that it's actually not so bad. 68 00:03:45,820 --> 00:03:49,820 So let's say that I have a triangle that looks like this. 69 00:03:49,820 --> 00:03:51,050 Let me draw it. 70 00:03:51,050 --> 00:03:54,210 Let's say this is my triangle. 71 00:03:54,210 --> 00:03:57,160 It looks something like this. 72 00:03:57,160 --> 00:04:00,560 And let's say that they tell us that this is the right angle. 73 00:04:00,560 --> 00:04:02,940 That this length right here-- let me do this in different 74 00:04:02,940 --> 00:04:06,830 colors-- this length right here is 3, and that this 75 00:04:06,830 --> 00:04:09,170 length right here is 4. 76 00:04:09,170 --> 00:04:14,490 And they want us to figure out that length right there. 77 00:04:14,490 --> 00:04:17,130 Now the first thing you want to do, before you even apply the 78 00:04:17,130 --> 00:04:19,660 Pythagorean theorem, is to make sure you have your 79 00:04:19,660 --> 00:04:20,710 hypotenuse straight. 80 00:04:20,710 --> 00:04:23,350 You make sure you know what you're solving for. 81 00:04:23,350 --> 00:04:26,120 And in this circumstance we're solving for the hypotenuse. 82 00:04:26,120 --> 00:04:30,440 And we know that because this side over here, it is the side 83 00:04:30,440 --> 00:04:33,310 opposite the right angle. 84 00:04:33,310 --> 00:04:36,540 If we look at the Pythagorean theorem, this is C. 85 00:04:36,540 --> 00:04:38,160 This is the longest side. 86 00:04:38,160 --> 00:04:41,920 So now we're ready to apply the Pythagorean theorem. 87 00:04:41,920 --> 00:04:48,070 It tells us that 4 squared-- one of the shorter sides-- plus 88 00:04:48,070 --> 00:04:53,260 3 squared-- the square of another of the shorter sides-- 89 00:04:53,260 --> 00:04:56,080 is going to be equal to this longer side squared-- the 90 00:04:56,080 --> 00:05:00,590 hypotenuse squared-- is going to be equal to C squared. 91 00:05:00,590 --> 00:05:02,310 And then you just solve for C. 92 00:05:02,310 --> 00:05:06,380 So 4 squared is the same thing as 4 times 4. 93 00:05:06,380 --> 00:05:08,460 That is 16. 94 00:05:08,460 --> 00:05:11,910 And 3 squared is the same thing as 3 times 3. 95 00:05:11,910 --> 00:05:13,810 So that is 9. 96 00:05:13,810 --> 00:05:18,580 And that is going to be equal to C squared. 97 00:05:18,580 --> 00:05:20,610 Now what is 16 plus 9? 98 00:05:20,610 --> 00:05:22,480 It's 25. 99 00:05:22,480 --> 00:05:25,195 So 25 is equal to C squared. 100 00:05:25,195 --> 00:05:29,020 And we could take the positive square root of both sides. 101 00:05:29,020 --> 00:05:30,960 I guess, just if you look at it mathematically, it could 102 00:05:30,960 --> 00:05:33,160 be negative 5 as well. 103 00:05:33,160 --> 00:05:34,870 But we're dealing with distances, so we only care 104 00:05:34,870 --> 00:05:37,050 about the positive roots. 105 00:05:37,050 --> 00:05:41,170 So you take the principal root of both sides and 106 00:05:41,170 --> 00:05:44,280 you get 5 is equal to C. 107 00:05:44,280 --> 00:05:50,260 Or, the length of the longest side is equal to 5. 108 00:05:50,260 --> 00:05:52,640 Now, you can use the Pythagorean theorem, if we give 109 00:05:52,640 --> 00:05:54,620 you two of the sides, to figure out the third side no matter 110 00:05:54,620 --> 00:05:55,690 what the third side is. 111 00:05:55,690 --> 00:05:59,300 So let's do another one right over here. 112 00:05:59,300 --> 00:06:10,670 Let's say that our triangle looks like this. 113 00:06:10,670 --> 00:06:12,610 And that is our right angle. 114 00:06:12,610 --> 00:06:17,820 Let's say this side over here has length 12, and let's say 115 00:06:17,820 --> 00:06:21,080 that this side over here has length 6. 116 00:06:21,080 --> 00:06:27,210 And we want to figure out this length right over there. 117 00:06:27,210 --> 00:06:29,870 Now, like I said, the first thing you want to do is 118 00:06:29,870 --> 00:06:31,350 identify the hypotenuse. 119 00:06:31,350 --> 00:06:34,130 And that's going to be the side opposite the right angle. 120 00:06:34,130 --> 00:06:35,550 We have the right angle here. 121 00:06:35,550 --> 00:06:37,650 You go opposite the right angle. 122 00:06:37,650 --> 00:06:41,460 The longest side, the hypotenuse, is right there. 123 00:06:41,460 --> 00:06:46,100 So if we think about the Pythagorean theorem-- that A 124 00:06:46,100 --> 00:06:50,820 squared plus B squared is equal to C squared-- 12 125 00:06:50,820 --> 00:06:52,220 you could view as C. 126 00:06:52,220 --> 00:06:54,740 This is the hypotenuse. 127 00:06:54,740 --> 00:06:56,670 The C squared is the hypotenuse squared. 128 00:06:56,670 --> 00:06:59,030 So you could say 12 is equal to C. 129 00:06:59,030 --> 00:07:00,880 And then we could say that these sides, it doesn't matter 130 00:07:00,880 --> 00:07:02,580 whether you call one of them A or one of them B. 131 00:07:02,580 --> 00:07:04,970 So let's just call this side right here. 132 00:07:04,970 --> 00:07:06,990 Let's say A is equal to 6. 133 00:07:06,990 --> 00:07:11,780 And then we say B-- this colored B-- is equal 134 00:07:11,780 --> 00:07:12,640 to question mark. 135 00:07:12,640 --> 00:07:15,070 And now we can apply the Pythagorean theorem. 136 00:07:15,070 --> 00:07:25,940 A squared, which is 6 squared, plus the unknown B squared is 137 00:07:25,940 --> 00:07:28,330 equal to the hypotenuse squared-- is equal 138 00:07:28,330 --> 00:07:29,760 to C squared. 139 00:07:29,760 --> 00:07:33,250 Is equal to 12 squared. 140 00:07:33,250 --> 00:07:35,260 And now we can solve for B. 141 00:07:35,260 --> 00:07:36,370 And notice the difference here. 142 00:07:36,370 --> 00:07:38,110 Now we're not solving for the hypotenuse. 143 00:07:38,110 --> 00:07:40,210 We're solving for one of the shorter sides. 144 00:07:40,210 --> 00:07:42,790 In the last example we solved for the hypotenuse. 145 00:07:42,790 --> 00:07:43,790 We solved for C. 146 00:07:43,790 --> 00:07:46,570 So that's why it's always important to recognize that A 147 00:07:46,570 --> 00:07:49,190 squared plus B squared plus C squared, C is the length 148 00:07:49,190 --> 00:07:49,670 of the hypotenuse. 149 00:07:49,670 --> 00:07:51,850 So let's just solve for B here. 150 00:07:51,850 --> 00:07:59,280 So we get 6 squared is 36, plus B squared, is equal 151 00:07:59,280 --> 00:08:04,700 to 12 squared-- this 12 times 12-- is 144. 152 00:08:04,700 --> 00:08:08,550 Now we can subtract 36 from both sides of this equation. 153 00:08:11,420 --> 00:08:13,270 Those cancel out. 154 00:08:13,270 --> 00:08:17,510 On the left-hand side we're left with just a B squared 155 00:08:17,510 --> 00:08:23,410 is equal to-- now 144 minus 36 is what? 156 00:08:23,410 --> 00:08:27,000 144 minus 30 is 114. 157 00:08:27,000 --> 00:08:30,080 And then you subtract 6, is 108. 158 00:08:30,080 --> 00:08:33,910 So this is going to be 108. 159 00:08:33,910 --> 00:08:36,630 So that's what B squared is, and now we want to take the 160 00:08:36,630 --> 00:08:40,600 principal root, or the positive root, of both sides. 161 00:08:40,600 --> 00:08:44,430 And you get B is equal to the square root, the 162 00:08:44,430 --> 00:08:48,650 principal root, of 108. 163 00:08:48,650 --> 00:08:50,550 Now let's see if we can simplify this a little bit. 164 00:08:50,550 --> 00:08:53,550 The square root of 108. 165 00:08:53,550 --> 00:08:54,930 And what we could do is we could take the prime 166 00:08:54,930 --> 00:08:56,670 factorization of 108 and see how we can 167 00:08:56,670 --> 00:08:58,410 simplify this radical. 168 00:08:58,410 --> 00:09:07,590 So 108 is the same thing as 2 times 54, which is the same 169 00:09:07,590 --> 00:09:15,570 thing as 2 times 27, which is the same thing as 3 times 9. 170 00:09:15,570 --> 00:09:19,780 So we have the square root of 108 is the same thing as the 171 00:09:19,780 --> 00:09:24,550 square root of 2 times 2 times-- well actually, 172 00:09:24,550 --> 00:09:25,520 I'm not done. 173 00:09:25,520 --> 00:09:28,760 9 can be factorized into 3 times 3. 174 00:09:28,760 --> 00:09:34,170 So it's 2 times 2 times 3 times 3 times 3. 175 00:09:34,170 --> 00:09:36,820 And so, we have a couple of perfect squares in here. 176 00:09:36,820 --> 00:09:38,680 Let me rewrite it a little bit neater. 177 00:09:38,680 --> 00:09:41,160 And this is all an exercise in simplifying radicals that you 178 00:09:41,160 --> 00:09:44,200 will bump into a lot while doing the Pythagorean theorem, 179 00:09:44,200 --> 00:09:46,460 so it doesn't hurt to do it right here. 180 00:09:46,460 --> 00:09:55,820 So this is the same thing as the square root of 2 times 2 181 00:09:55,820 --> 00:10:00,790 times 3 times 3 times the square root of that last 182 00:10:00,790 --> 00:10:02,510 3 right over there. 183 00:10:02,510 --> 00:10:04,090 And this is the same thing. 184 00:10:04,090 --> 00:10:05,785 And, you know, you wouldn't have to do all of 185 00:10:05,785 --> 00:10:07,960 this on paper. 186 00:10:07,960 --> 00:10:08,970 You could do it in your head. 187 00:10:08,970 --> 00:10:09,530 What is this? 188 00:10:09,530 --> 00:10:11,780 2 times 2 is 4. 189 00:10:11,780 --> 00:10:14,200 4 times 9, this is 36. 190 00:10:14,200 --> 00:10:18,030 So this is the square root of 36 times the square root of 3. 191 00:10:18,030 --> 00:10:20,610 The principal root of 36 is 6. 192 00:10:20,610 --> 00:10:25,380 So this simplifies to 6 square roots of 3. 193 00:10:25,380 --> 00:10:28,730 So the length of B, you could write it as the square root of 194 00:10:28,730 --> 00:10:34,040 108, or you could say it's equal to 6 times the 195 00:10:34,040 --> 00:10:35,040 square root of 3. 196 00:10:35,040 --> 00:10:37,150 This is 12, this is 6. 197 00:10:37,150 --> 00:10:40,580 And the square root of 3, well this is going to be a 1 198 00:10:40,580 --> 00:10:41,600 point something something. 199 00:10:41,600 --> 00:10:45,360 So it's going to be a little bit larger than 6.