WEBVTT 1 00:00:01.020 --> 00:00:03.040 One of the places where algebra and geometry 2 00:00:03.040 --> 00:00:05.860 really overlap is in the Cartesian coordinate plane. 3 00:00:05.860 --> 00:00:08.790 You've studied a lot about lines in your algebra one class, 4 00:00:08.790 --> 00:00:11.330 but those same ideas connect into geometry as well. 5 00:00:11.330 --> 00:00:15.480 For example, we know that if two lines have the same slope, 6 00:00:15.480 --> 00:00:16.640 then those lines are parallel. 7 00:00:16.640 --> 00:00:17.810 They'll never cross. 8 00:00:17.810 --> 00:00:18.680 In your slide here, 9 00:00:18.680 --> 00:00:21.070 you see two parallel lines that are white. 10 00:00:21.070 --> 00:00:24.140 We have 2x plus two and 2x minus one. 11 00:00:24.140 --> 00:00:28.150 Both those lines have slope two so they never intersect. 12 00:00:28.150 --> 00:00:30.340 Now perpendicular lines have what we call 13 00:00:30.340 --> 00:00:32.500 opposite sign reciprocal slopes. 14 00:00:32.500 --> 00:00:34.680 So if I wanted to start with the slope two 15 00:00:34.680 --> 00:00:37.040 and find out something that's perpendicular to that, 16 00:00:37.040 --> 00:00:40.510 I would do the opposite sign like negative and reciprocal 17 00:00:40.510 --> 00:00:42.680 so instead of two, it would be 1/2. 18 00:00:42.680 --> 00:00:44.890 So you see on your slide here a yellow line 19 00:00:44.890 --> 00:00:48.630 where we have y equals negative 1/2x plus four 20 00:00:48.630 --> 00:00:50.560 and the negative 1/2 bit, the slope part, 21 00:00:50.560 --> 00:00:52.460 not the four but the negative 1/2 22 00:00:52.460 --> 00:00:54.270 is what tells us that that yellow line 23 00:00:54.270 --> 00:00:56.930 is gonna be perpendicular to both of the white lines. 24 00:00:56.930 --> 00:00:59.460 Again, parallel lines have the exact same slope. 25 00:00:59.460 --> 00:01:02.830 Perpendicular lines have opposite sign reciprocal slopes. 26 00:01:02.830 --> 00:01:03.810 You can also think about 27 00:01:03.810 --> 00:01:05.760 how the slopes of perpendicular lines 28 00:01:05.760 --> 00:01:07.810 will multiply to negative one. 29 00:01:07.810 --> 00:01:09.800 Like for example if I say 2/3, 30 00:01:09.800 --> 00:01:13.510 the perpendicular slope would be negative 3/2. 31 00:01:13.510 --> 00:01:16.900 Change the positive to negative and flip the fraction. 32 00:01:16.900 --> 00:01:18.300 We call that the reciprocal. 33 00:01:19.430 --> 00:01:20.610 Let's talk also about 34 00:01:20.610 --> 00:01:23.360 how we can connect coordinate geometry to polygons. 35 00:01:23.360 --> 00:01:25.610 We can use what we know about the coordinate plane 36 00:01:25.610 --> 00:01:27.980 to verify our conjectures about polygons, 37 00:01:27.980 --> 00:01:30.753 similarity, congruence, and right triangles. 38 00:01:31.860 --> 00:01:33.180 The distance formula will tell us 39 00:01:33.180 --> 00:01:35.470 that congruent segments are side lengths 40 00:01:35.470 --> 00:01:37.320 and the slopes of lines can tell us 41 00:01:37.320 --> 00:01:40.550 if sides are perpendicular or create congruent triangles. 42 00:01:40.550 --> 00:01:42.620 So let's talk a little bit more about that. 43 00:01:42.620 --> 00:01:45.860 If we wanna know if two shapes are similar, 44 00:01:45.860 --> 00:01:47.590 two figures are similar, 45 00:01:47.590 --> 00:01:49.820 that means that their corresponding 46 00:01:49.820 --> 00:01:51.430 side lengths are proportional. 47 00:01:51.430 --> 00:01:54.180 There's some growth factor or maybe a shrinking factor 48 00:01:54.180 --> 00:01:55.980 that are multiplying each side length by 49 00:01:55.980 --> 00:01:57.570 to get to the other side length. 50 00:01:57.570 --> 00:01:59.350 And if it was in the Cartesian coordinate plane, 51 00:01:59.350 --> 00:02:00.780 I could use the distance formula 52 00:02:00.780 --> 00:02:02.420 to calculate those distances 53 00:02:02.420 --> 00:02:05.190 and see am I multiplying everything by the same number, 54 00:02:05.190 --> 00:02:08.950 like multiplying the smaller distances by two say 55 00:02:08.950 --> 00:02:10.560 to get to the larger ones 56 00:02:10.560 --> 00:02:12.040 and so we could check similarity. 57 00:02:12.040 --> 00:02:13.780 Congruence, of course, is when the side lengths 58 00:02:13.780 --> 00:02:14.613 are the same. 59 00:02:14.613 --> 00:02:15.990 That's important as well. 60 00:02:15.990 --> 00:02:18.090 And then right angles, oftentimes we're asked, 61 00:02:18.090 --> 00:02:20.780 do these three points create a right triangle? 62 00:02:20.780 --> 00:02:22.180 And one of the ways we could tell 63 00:02:22.180 --> 00:02:23.840 is by either finding the distances 64 00:02:23.840 --> 00:02:25.500 and doing the Pythagorean Theorem 65 00:02:25.500 --> 00:02:27.580 or finding the slopes and looking to see 66 00:02:27.580 --> 00:02:30.520 if any of those slopes create perpendicular lines. 67 00:02:30.520 --> 00:02:31.760 Now you could also look to see 68 00:02:31.760 --> 00:02:32.970 if you have perpendicular lines 69 00:02:32.970 --> 00:02:36.390 like it says in your slide using matrices and dot products, 70 00:02:36.390 --> 00:02:38.240 but many times just calculating the slope 71 00:02:38.240 --> 00:02:40.710 using the slope formula is a little bit easier. 72 00:02:40.710 --> 00:02:43.600 We can use coordinate geometry to do proofs, 73 00:02:43.600 --> 00:02:45.610 in particular you combine your algebra skills 74 00:02:45.610 --> 00:02:48.360 with your geometry skills to verify a conjecture. 75 00:02:48.360 --> 00:02:52.050 So here we are asked to classify the quadrilateral ABCD 76 00:02:52.050 --> 00:02:54.670 formed by the vertices and we have four points. 77 00:02:54.670 --> 00:02:56.480 Now this graph is already provided for you. 78 00:02:56.480 --> 00:02:57.520 If you are doing this problem, 79 00:02:57.520 --> 00:02:58.910 you probably wanna draw a sketch 80 00:02:58.910 --> 00:03:00.780 and it's okay if your sketch isn't perfect. 81 00:03:00.780 --> 00:03:03.840 But what you're looking for is to classify a quadrilateral, 82 00:03:03.840 --> 00:03:05.190 some things you might wanna consider 83 00:03:05.190 --> 00:03:07.310 are the distances of the side lengths, 84 00:03:07.310 --> 00:03:08.740 are any of the distances the same 85 00:03:08.740 --> 00:03:10.480 and you would use the distance formula, 86 00:03:10.480 --> 00:03:12.970 or the slopes, are any of the slopes the same, 87 00:03:12.970 --> 00:03:14.920 that would show that you have opposite sides 88 00:03:14.920 --> 00:03:16.510 or some sides that are parallel, 89 00:03:16.510 --> 00:03:19.820 that's a key idea of parallelograms and trapezoids, 90 00:03:19.820 --> 00:03:20.760 those kinds of things. 91 00:03:20.760 --> 00:03:22.260 Another thing you might wanna look at 92 00:03:22.260 --> 00:03:24.600 do the slopes show that any of these angles 93 00:03:24.600 --> 00:03:25.850 are right angles? 94 00:03:25.850 --> 00:03:28.700 Again, if you have slopes that are opposite sign reciprocals 95 00:03:28.700 --> 00:03:31.560 like say a slope of two and negative 1/2, 96 00:03:31.560 --> 00:03:34.170 that would show you that you had some right angles. 97 00:03:34.170 --> 00:03:37.800 So let's say you are sketching these four vertices by hand, 98 00:03:37.800 --> 00:03:39.420 not necessarily on graph paper, 99 00:03:39.420 --> 00:03:42.000 it might look like it could be maybe a rectangle. 100 00:03:42.000 --> 00:03:44.270 It might look like you might have right angles in there. 101 00:03:44.270 --> 00:03:45.990 So what you wanna do is find the slopes 102 00:03:45.990 --> 00:03:48.900 of all four of those sides using the slope formula 103 00:03:48.900 --> 00:03:51.430 and you'll find that the two opposite sets of slopes 104 00:03:51.430 --> 00:03:52.500 are perpendicular. 105 00:03:52.500 --> 00:03:56.530 There's two red lines that both have slopes of negative 1/4, 106 00:03:56.530 --> 00:03:59.380 the two yellow lines both have slopes of positive three, 107 00:03:59.380 --> 00:04:01.010 so the opposite sides are parallel. 108 00:04:01.010 --> 00:04:03.450 That way we know it's some type of parallelogram, 109 00:04:03.450 --> 00:04:05.530 but those slopes are not giving us right angles. 110 00:04:05.530 --> 00:04:07.540 They're not opposite sign reciprocals. 111 00:04:07.540 --> 00:04:10.270 Therefore, I know it's not a rectangle or a square. 112 00:04:10.270 --> 00:04:11.750 It's just gonna be a parallelogram. 113 00:04:11.750 --> 00:04:14.700 In isometry or combination of isometries 114 00:04:14.700 --> 00:04:17.790 An often used to transform a shape in the coordinate plane 115 00:04:17.790 --> 00:04:20.090 and by transform we mean rotate, reflect, 116 00:04:20.090 --> 00:04:21.550 or just shift or move. 117 00:04:21.550 --> 00:04:22.970 So here's an example. 118 00:04:22.970 --> 00:04:24.970 A coordinate rule shows a transformation. 119 00:04:24.970 --> 00:04:25.900 So here's an example. 120 00:04:25.900 --> 00:04:28.920 Every x, y point is getting mapped 121 00:04:28.920 --> 00:04:32.600 to the x value plus two and the y values minus three. 122 00:04:32.600 --> 00:04:33.750 Now imagine in your head 123 00:04:33.750 --> 00:04:36.410 if you had a point or a shape in the coordinate plane, 124 00:04:36.410 --> 00:04:40.400 if all of your x numbers become those x numbers plus two, 125 00:04:40.400 --> 00:04:43.260 that's gonna take your shape and move it two to the right 126 00:04:43.260 --> 00:04:45.710 because adding two is a side-to-side shift 127 00:04:45.710 --> 00:04:46.570 to the x coordinate. 128 00:04:46.570 --> 00:04:48.050 We're moving it two to the right. 129 00:04:48.050 --> 00:04:50.340 And in the y values minus three, 130 00:04:50.340 --> 00:04:51.310 if you imagine a shape 131 00:04:51.310 --> 00:04:53.920 and you take all those y values and subtract three, 132 00:04:53.920 --> 00:04:55.380 that shape will move down. 133 00:04:55.380 --> 00:04:56.460 It's called an isometry 134 00:04:56.460 --> 00:04:58.990 'cause we're not changing how big that shape is. 135 00:04:58.990 --> 00:05:00.470 We're not blowing it up or shrinking it. 136 00:05:00.470 --> 00:05:02.910 All we're doing is taking that exact same shape 137 00:05:02.910 --> 00:05:04.480 and shift to get over and down. 138 00:05:04.480 --> 00:05:05.743 It's called an isometry. 139 00:05:07.000 --> 00:05:09.440 The second example says all of the coordinates x, y 140 00:05:09.440 --> 00:05:12.780 become opposite signs, negative x and negative y. 141 00:05:12.780 --> 00:05:15.540 And what happens is you end up reflecting the shape 142 00:05:15.540 --> 00:05:17.160 across both axes. 143 00:05:17.160 --> 00:05:19.100 First you're reflecting across the y-axis 144 00:05:19.100 --> 00:05:20.400 and then the x-axis 145 00:05:20.400 --> 00:05:22.210 and those two put together could be called 146 00:05:22.210 --> 00:05:25.720 either a double reflection or just rotation 180 degrees 147 00:05:25.720 --> 00:05:27.290 around the origin. 148 00:05:27.290 --> 00:05:28.610 The last example we have here 149 00:05:28.610 --> 00:05:30.030 is if you're just changing the sign 150 00:05:30.030 --> 00:05:31.110 on one of your coordinates 151 00:05:31.110 --> 00:05:34.490 like if x, y gets mapped to negative x, y. 152 00:05:34.490 --> 00:05:35.690 Well, if you imagine a shape 153 00:05:35.690 --> 00:05:37.380 with say like in the first quadrant 154 00:05:37.380 --> 00:05:40.590 and now all those x values become negative x values, 155 00:05:40.590 --> 00:05:42.580 it's gonna reflect across the y-axis. 156 00:05:42.580 --> 00:05:45.470 It's gonna go to coordinate two or quadrant two. 157 00:05:45.470 --> 00:05:47.040 It's still gonna be the same shape. 158 00:05:47.040 --> 00:05:48.210 It's still an isometry. 159 00:05:48.210 --> 00:05:49.920 We're not dilating or shrinking it. 160 00:05:49.920 --> 00:05:51.550 They're still congruent, 161 00:05:51.550 --> 00:05:54.550 but it's just we've moved it from one quadrant to the other. 162 00:05:55.470 --> 00:05:58.010 We can also graph shapes in a 3D coordinate plane 163 00:05:58.010 --> 00:06:00.820 if we add a third axis called the z-axis. 164 00:06:00.820 --> 00:06:02.520 And so the way we draw these usually 165 00:06:02.520 --> 00:06:04.700 is by starting with x and y 166 00:06:04.700 --> 00:06:06.100 with a right angle as we usually do 167 00:06:06.100 --> 00:06:09.440 and then we have a diagonal going down to the left z-axis 168 00:06:09.440 --> 00:06:11.010 like you see on your slide. 169 00:06:11.010 --> 00:06:14.010 And so each point now doesn't just have x, y coordinates, 170 00:06:14.010 --> 00:06:16.810 it has x, y, and z. 171 00:06:16.810 --> 00:06:20.020 And we oftentimes if we're plotting a 3D point like that, 172 00:06:20.020 --> 00:06:22.360 we'll draw the whole rectangular prism, 173 00:06:22.360 --> 00:06:24.440 the whole box that creates that 174 00:06:24.440 --> 00:06:28.230 so you can see in the box how it's moving over side to side, 175 00:06:28.230 --> 00:06:30.210 it looks like maybe three units. 176 00:06:30.210 --> 00:06:31.620 It's moving up about two units 177 00:06:31.620 --> 00:06:33.610 and it's coming towards us, three units. 178 00:06:33.610 --> 00:06:37.113 So you could see how you would turn a point into a 3D space.