WEBVTT 1 00:00:00.512 --> 00:00:02.816 Welcome to Brainfuse! 2 00:00:05.888 --> 00:00:12.032 A triangle is an example of the simplest polygon. So what makes the triangles on screen similar? 3 00:00:12.288 --> 00:00:14.080 We'll call them T1 4 00:00:14.848 --> 00:00:16.128 for triangle one 5 00:00:16.896 --> 00:00:18.944 And T2 6 00:00:21.760 --> 00:00:27.392 You'll notice that they are different sizes, but that doesn't affect whether or not they're similar 7 00:00:27.904 --> 00:00:34.048 You also see that they have the same angles at each vertex and that's key. We have 60 degrees 8 00:00:34.304 --> 00:00:34.816 here 9 00:00:35.072 --> 00:00:40.192 We have 100 degrees here 10 00:00:40.448 --> 00:00:43.520 And 20 degrees at the vertex 11 00:00:43.776 --> 00:00:51.273 As a result the ratio between any side on T1 and the corresponding side on T2 is always the same 12 00:00:51.273 --> 00:01:01.297 so all sides of T1 are multipled by the same number to give us their corresponding sides on T2 and that number is called 13 00:01:02.438 --> 00:01:03.744 the Scale Factor 14 00:01:04.768 --> 00:01:10.029 So let's find the scale factor of T1 to T2 15 00:01:13.216 --> 00:01:23.674 So on T1 we are told that side a equals 4 and we need to multiply that by some number 16 00:01:27.040 --> 00:01:30.112 and on T2 the corresponding side is equal to 3 17 00:01:30.368 --> 00:01:35.744 So let's solve for n to find the scale factor divide both sides by 4 18 00:01:36.000 --> 00:01:39.072 And n equals 3/4 19 00:01:39.840 --> 00:01:45.984 That's our scale factor so we can multiply the length of any side on T1 by 3/4 20 00:01:46.240 --> 00:01:46.752 21 00:01:47.264 --> 00:01:51.360 To find the the length of its corresponding side on T2 22 00:01:51.616 --> 00:01:54.688 so c times 3/4 23 00:01:55.200 --> 00:01:58.784 would give us the length of z on T2 24 00:01:59.040 --> 00:02:00.064 And b 25 00:02:00.320 --> 00:02:01.856 times 3/4 26 00:02:02.112 --> 00:02:05.440 Would give us the length of y on T2 27 00:02:05.952 --> 00:02:11.149 So in the next few problems we'll use the scale factor to find the length of missing sides 28 00:02:11.149 --> 00:02:18.212 I'll clear the polygons on this screen and we'll move on to some more complex polygons 29 00:02:18.212 --> 00:02:26.944 On screen we have a set of convex polygons, and I'm gonna label this P1 30 00:02:28.224 --> 00:02:29.504 And P2 31 00:02:29.922 --> 00:02:32.832 How can you tell if these polygons are similar? 32 00:02:33.088 --> 00:02:36.160 I see that the shapes are different sizes 33 00:02:36.672 --> 00:02:42.816 But that doesn't tell me whether or not they're similar. I need to see if they have corresponding sides 34 00:02:43.584 --> 00:02:49.728 I need to see if P2 is a reflection of P1 or if the shape has been rotated 35 00:02:49.984 --> 00:02:53.056 and still has sides that correspond with P1. 36 00:02:54.080 --> 00:02:59.200 So make sure to check to see if that shape has been reflected or rotated 37 00:03:01.504 --> 00:03:06.624 Starting with the reflection check if P1 were reflected 38 00:03:06.880 --> 00:03:07.904 Vertically 39 00:03:10.208 --> 00:03:12.000 Just flip over 40 00:03:12.256 --> 00:03:13.280 This line 41 00:03:27.616 --> 00:03:33.760 Ok, and as you can see that is an extremely rough sketch, but the position of the shape doesn't seem to match the 42 00:03:34.016 --> 00:03:37.856 position of P2, so let's try flipping it 43 00:03:38.112 --> 00:03:39.392 Horizontally 44 00:03:42.464 --> 00:03:43.488 That's flipping it 45 00:03:44.000 --> 00:03:45.280 Across this line 46 00:03:50.656 --> 00:03:53.984 Again a rough sketch but 47 00:03:54.240 --> 00:04:00.384 This shape does look like it's in the same position as P2, so to make life easier let's 48 00:04:00.640 --> 00:04:02.944 Work with it in this position 49 00:04:04.224 --> 00:04:07.040 six would be there, four 50 00:04:07.350 --> 00:04:09.380 this would be five 51 00:04:11.054 --> 00:04:14.720 2 and 3 52 00:04:15.232 --> 00:04:21.375 By reflecting or flipping the shape horizontally, each side of P1 looks like it has a 53 00:04:21.632 --> 00:04:27.776 Corresponding side on P2. If that's true, we should be able to find the scale factor. 54 00:04:28.032 --> 00:04:30.592 So let's see if we can. 55 00:04:31.104 --> 00:04:36.992 The side with the length of 4 should correspond to the side with the length of 12 56 00:04:39.808 --> 00:04:43.392 And their ratio should equal 57 00:04:43.648 --> 00:04:45.440 6 / 18 58 00:04:45.696 --> 00:04:50.048 6 on P1 corresponds with 18 on P2 59 00:04:54.400 --> 00:04:57.728 Then we have the side with the length of three on P1 60 00:04:57.984 --> 00:05:00.544 Corresponding with the side with a length of 9 61 00:05:03.360 --> 00:05:09.504 and the side with the length of 2 on P1 corresponding with the side that has a length of 6 62 00:05:11.808 --> 00:05:13.856 Then we have 63 00:05:14.112 --> 00:05:15.904 Our side with a length of 5 64 00:05:16.160 --> 00:05:19.232 Which corresponds with x 65 00:05:20.000 --> 00:05:22.304 Value we need to find 66 00:05:23.840 --> 00:05:27.424 So as you can see, all of these ratios 67 00:05:27.680 --> 00:05:29.472 Are fractions 68 00:05:29.728 --> 00:05:32.544 Are equal to 1 / 3 69 00:05:34.592 --> 00:05:36.896 The ratio of the sides 70 00:05:37.664 --> 00:05:38.688 P1 71 00:05:38.944 --> 00:05:40.224 to the sides of 72 00:05:40.480 --> 00:05:41.504 P2 73 00:05:41.760 --> 00:05:43.808 Is one to three 74 00:05:44.320 --> 00:05:47.136 So if we multiply 75 00:05:47.392 --> 00:05:53.536 Each side by 3 we're given the length of the side in P2. 4 times 3 is 12 76 00:05:53.792 --> 00:05:59.936 6 times 3 is 18, 3 times 3 is 9, so that's our scale factor. And we can either just 77 00:06:00.192 --> 00:06:04.288 multiply 5 times 3 or set up a proportion 78 00:06:07.104 --> 00:06:09.664 And cross multiply 79 00:06:10.432 --> 00:06:12.480 That is 80 00:06:13.504 --> 00:06:15.296 X 81 00:06:15.808 --> 00:06:18.368 Equals 3 times 5 is 15 82 00:06:20.416 --> 00:06:26.560 Let's double-check to make sure that it makes sense 5 times our scale factor 3 equals 83 00:06:26.816 --> 00:06:27.840 15 84 00:06:28.608 --> 00:06:34.752 By using the scale factor, we found the length of the missing side. We are going to try 85 00:06:35.008 --> 00:06:40.384 One more set of polygons that are going to look a little bit dented 86 00:06:44.992 --> 00:06:51.673 On-screen we now have some concave polygons notice they look a little bit caved in 87 00:06:51.673 --> 00:06:55.488 at least one vertex. We need to figure out 88 00:06:55.744 --> 00:07:01.888 Whether or not these are similar. Again,they are different sizes, but that doesn't affect whether or not 89 00:07:02.144 --> 00:07:05.216 they are similar. Instead, I'll label them 90 00:07:05.472 --> 00:07:09.824 And then see if one is a reflection of the other 91 00:07:10.080 --> 00:07:15.712 Or if one shape has just been rotated into a different position 92 00:07:25.440 --> 00:07:29.792 Okay so checking the reflection first let's see 93 00:07:30.304 --> 00:07:37.183 If we flip it over this line and at least see what it would look like in this position 94 00:07:37.183 --> 00:07:38.240 Certainly the sides 95 00:07:38.496 --> 00:07:40.288 Won't be as 96 00:07:40.544 --> 00:07:43.244 Long because we would run out of space. 97 00:08:08.233 --> 00:08:18.944 Flipping it over this line, horizontally, this is a squished version but it would look something like that 98 00:08:19.200 --> 00:08:24.320 Ok, as you can see neither shape is in the same position as P2 99 00:08:24.576 --> 00:08:30.720 P2 is not a reflection of P1, but there's still hope because we need to check to see 100 00:08:30.976 --> 00:08:34.048 If it might be rotated 101 00:08:34.559 --> 00:08:37.376 Okay, so we'll clear this away 102 00:08:40.192 --> 00:08:46.336 We rotate that shape 90 degrees we end up with this one 103 00:08:46.592 --> 00:08:51.712 Looks familiar right? It is in the same position as P1 104 00:08:51.968 --> 00:08:52.992 So, 105 00:08:53.248 --> 00:08:56.320 I'll just put it below our original shape 106 00:08:58.368 --> 00:09:01.593 And let's label the corresponding sides 107 00:09:01.593 --> 00:09:15.520 We have 3 on top, 5, 2, 6, 3, 4 and q. We need to solve for q. 108 00:09:15.776 --> 00:09:21.920 So these shapes now look like they're in the same position. Let's see if there is a scale factor 109 00:09:22.176 --> 00:09:25.760 Then we'll know these shapes are similar and be able to solve for q 110 00:09:27.808 --> 00:09:29.856 P1 111 00:09:30.624 --> 00:09:31.648 to 112 00:09:31.904 --> 00:09:33.184 P2 113 00:09:35.744 --> 00:09:38.048 We have 114 00:09:39.072 --> 00:09:40.096 15 over 3 115 00:09:42.656 --> 00:09:45.984 we have 25 over 5 116 00:09:48.544 --> 00:09:50.592 And to 117 00:09:50.848 --> 00:09:56.992 Prevent you from falling asleep during this lesson we will just do three of these sides 118 00:09:57.248 --> 00:10:00.064 As you can see this concave polygon has many sides 119 00:10:00.320 --> 00:10:02.624 30 over 6 120 00:10:03.904 --> 00:10:06.464 so that gives us a scale factor of 121 00:10:06.720 --> 00:10:08.000 5 to 1 122 00:10:08.512 --> 00:10:14.400 Where the sides on P1 are 5 times larger than the sides on P2 123 00:10:14.912 --> 00:10:16.448 To solve for q 124 00:10:16.704 --> 00:10:21.312 All we need to do is divide by 5 125 00:10:21.568 --> 00:10:24.128 q corresponds with 45 126 00:10:24.640 --> 00:10:27.200 And let's set up a proportion 127 00:10:27.712 --> 00:10:30.016 So that we can solve for q 128 00:10:38.976 --> 00:10:41.536 q times 5 is 129 00:10:42.048 --> 00:10:43.072 5q 130 00:10:43.584 --> 00:10:49.728 And 1 times 45 is 45. We divide both sides by 5 and that shows 131 00:10:49.984 --> 00:10:53.824 us that q is equal to 9 132 00:10:57.408 --> 00:11:03.552 So using the scale factor allowed us to find the missing sides. If shapes 133 00:11:04.064 --> 00:11:10.208 Don't look similar, make sure to check to see if one of them has been reflected or rotated 134 00:11:10.464 --> 00:11:16.608 If so, find the scale factor and you can use that scale factor to solve for missing sides. 135 00:11:16.864 --> 00:11:23.008 Thanks so much for watching this lesson and now it's your turn to search for similar polygons.