WEBVTT 1 00:00:00.256 --> 00:00:02.560 Welcome to Brainfuse! 2 00:00:03.072 --> 00:00:09.216 We are about to work on the Pythagorean Theorem and our sample problem asks us to 3 00:00:09.472 --> 00:00:15.616 Find the length of the hypotenuse in the right triangle onscreen. A right triangle is a triangle with 4 00:00:15.872 --> 00:00:22.016 One 90 degree angle. It's usually, but not always, represented by 5 00:00:22.528 --> 00:00:24.320 A small square like this. 6 00:00:24.576 --> 00:00:29.440 In a right triangle the sides that form the right angle are called legs. 7 00:00:29.696 --> 00:00:30.976 8 00:00:31.138 --> 00:00:32.768 This would be a leg 9 00:00:33.024 --> 00:00:35.840 And it's often called side "a" as well, 10 00:00:36.096 --> 00:00:39.424 This is a leg and often called side "b," 11 00:00:41.216 --> 00:00:45.568 And the side opposite the right angle is the hypotenuse 12 00:00:47.121 --> 00:00:59.136 And that is often called side "c." The lengths of the legs and the hypotenuse are related by the Pythagorean Theorem. 13 00:00:59.136 --> 00:01:05.280 Despite the technology you are currently using we are about to use a formula that dates at least as far 14 00:01:05.536 --> 00:01:07.840 Back as 4000 BCE. 15 00:01:08.096 --> 00:01:09.376 Are you ready? 16 00:01:09.888 --> 00:01:10.656 Here it is: 17 00:01:12.448 --> 00:01:13.728 a squared 18 00:01:13.984 --> 00:01:15.520 Plus b squared 19 00:01:15.776 --> 00:01:16.544 Equals 20 00:01:17.056 --> 00:01:18.336 c squared. 21 00:01:19.616 --> 00:01:21.408 Okay great, so 22 00:01:22.176 --> 00:01:27.040 Apparently this formula has stood the test of time, but how do we know this is true? 23 00:01:27.296 --> 00:01:32.160 We're going to look at a more popular right triangle 24 00:01:32.672 --> 00:01:38.816 Than the one on screen, just just to test it out and we'll go back to our problem. So this is the 25 00:01:39.072 --> 00:01:43.680 Ever-popular 3-4-5 triangle you'll see it on many standardized tests. 26 00:01:43.936 --> 00:01:44.448 Okay, 27 00:01:44.704 --> 00:01:47.520 So we'll make side a, 3 28 00:01:48.288 --> 00:01:50.848 Side b, 4 29 00:01:50.848 --> 00:01:50.848 30 00:01:51.360 --> 00:01:56.222 And I promise you that side c, the hypotenuse, will be 5. 31 00:01:56.736 --> 00:01:58.272 Let's see why. 32 00:01:58.528 --> 00:02:03.392 So right now we're going to make each side of the triangle one side of a square. 33 00:02:11.072 --> 00:02:13.888 Obviously not a precise square. 34 00:02:15.936 --> 00:02:19.008 So if this is a square that is 3 units long 35 00:02:19.520 --> 00:02:22.080 Then it would have to also be 3 units 36 00:02:22.336 --> 00:02:29.847 In width, because each side of the square is the same length. Okay, so we see 9 units there. 37 00:02:30.784 --> 00:02:36.152 Okay, we will do the same for 4. We are making 38 00:02:36.416 --> 00:02:39.232 That leg one side of the square. 39 00:02:43.072 --> 00:02:44.352 Oops 40 00:02:50.496 --> 00:02:58.088 So all sides need to be equal, so if it's 4 units in length its width needs to be 4 units as well. 41 00:03:02.272 --> 00:03:07.904 Okay and last but not least, 42 00:03:11.744 --> 00:03:14.816 I think it's going to crash into the Pythagorean formula 43 00:03:17.632 --> 00:03:19.168 It's understood to be there. 44 00:03:25.312 --> 00:03:27.360 Okay, and we make that five by five 45 00:03:37.600 --> 00:03:41.440 Ok, so it's now five units in length 46 00:03:41.440 --> 00:03:42.720 And let's make 47 00:03:42.976 --> 00:03:46.304 That square five units and width so it becomes 48 00:03:47.840 --> 00:03:49.632 At least something like a square 49 00:03:55.264 --> 00:04:01.408 Okay so definitely distorted, but you get the idea. So the area 50 00:04:01.664 --> 00:04:06.016 of this square, as you can see, is 9 units: 3 squared is 9. 51 00:04:07.296 --> 00:04:16.553 The area of the square for side b is 16 units. You can count them up, and 4 squared is 16. 52 00:04:17.279 --> 00:04:23.424 The area for our really distorted square for the hypotenuse 53 00:04:23.680 --> 00:04:26.752 Is 5 squared which is 25 units 54 00:04:30.080 --> 00:04:38.325 Notice that the area of the square on side a and the area of the square on side b add up to 25. 55 00:04:38.899 --> 00:04:43.392 9 plus 16 is equal to 25. 56 00:04:45.696 --> 00:04:49.792 And this is not a coincidence, it happens in all right triangles. 57 00:04:50.560 --> 00:04:56.704 So let's use this theorem to solve our problem. 58 00:04:56.960 --> 00:05:01.056 So we have a squared plus b squared equals c squared 59 00:05:03.616 --> 00:05:09.760 And we need to solve for c, we need to find the length of the hypotenuse. 60 00:05:10.016 --> 00:05:12.320 So I know 61 00:05:12.576 --> 00:05:14.880 Leg a 62 00:05:15.136 --> 00:05:17.184 Is equal to 9 63 00:05:17.696 --> 00:05:20.768 so we'll take 9 squared, 64 00:05:21.280 --> 00:05:23.584 Plus 65 00:05:23.840 --> 00:05:27.168 Leg b is equal to 3 radical 7 66 00:05:29.216 --> 00:05:31.242 so we'll square that. 67 00:05:31.876 --> 00:05:38.688 And c is our mystery length but remember to make, 68 00:05:38.944 --> 00:05:42.016 to square it. Keep it squared 69 00:05:44.320 --> 00:05:47.136 Okay so 9 squared is 81 70 00:05:47.392 --> 00:05:51.744 And 3 radical 7 (squared) 71 00:05:52.000 --> 00:05:56.096 Is equal to 3 squared which is 9 72 00:05:56.608 --> 00:06:01.438 Times radical 7 squared which is 7, because 73 00:06:01.438 --> 00:06:09.152 Radical 7 times radical 7 is 7. The square root of 7 times the square root of 7 is just 7. 74 00:06:09.408 --> 00:06:11.200 Then finally we have 75 00:06:11.712 --> 00:06:12.830 c squared. 76 00:06:12.830 --> 00:06:20.160 So 81 plus, 9 times 7 is 63, 77 00:06:21.440 --> 00:06:24.000 Will give us c squared. 78 00:06:24.768 --> 00:06:30.912 C squared. Okay, so 81 plus 63 is 144, 79 00:06:31.168 --> 00:06:37.312 So we know that c squared is equal to 144 80 00:06:38.336 --> 00:06:49.344 And remember it's squared, so in order to find c you take the square root of both sides and you find that c equals 12. 81 00:06:49.344 --> 00:06:55.488 And that is the length of one side I guess you can put 12 units, because we don't have 82 00:06:55.744 --> 00:06:59.584 A definite unit listed, so 12 units. 83 00:06:59.840 --> 00:07:05.984 So though this theorem had already been widely used in many cultures, a Greek mathematician 84 00:07:06.240 --> 00:07:11.047 Named Pythagoras was given credit for proving how the sides of right triangles are related 85 00:07:11.047 --> 00:07:25.334 In 530 BCE, so that's why this theorem has part of his name. And now it's your turn to use this formula to solve the practice problems. Thank you for watching this lesson.