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Let's say I have some type of a surface. Let's say that this right over here is the top of your desk. And I were to draw a triangle on that surface. So maybe the triangle looks like this. Something like this. It doesn't have to be a right triangle. And so I'm not implying that this is necessarily a right triangle, although it looks a little bit like one.
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Dilating in 3D Solid geometry High school geometry Khan Academy.mp3
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So maybe the triangle looks like this. Something like this. It doesn't have to be a right triangle. And so I'm not implying that this is necessarily a right triangle, although it looks a little bit like one. And let's call it triangle A, B, and then C. Now what I'm going to do is something interesting. I'm gonna take a fourth point P that's not on the surface of this desk, and it's going to be right above point B. So let me just take that point, go straight up, and I'm going to get to point P right over here.
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Dilating in 3D Solid geometry High school geometry Khan Academy.mp3
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And so I'm not implying that this is necessarily a right triangle, although it looks a little bit like one. And let's call it triangle A, B, and then C. Now what I'm going to do is something interesting. I'm gonna take a fourth point P that's not on the surface of this desk, and it's going to be right above point B. So let me just take that point, go straight up, and I'm going to get to point P right over here. Now what I can do is construct a pyramid using point P as the peak of that pyramid. Now what we're going to start thinking about is what happens if I take cross sections of this pyramid? So in this case, the length of segment PB is the height of this pyramid.
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Dilating in 3D Solid geometry High school geometry Khan Academy.mp3
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So let me just take that point, go straight up, and I'm going to get to point P right over here. Now what I can do is construct a pyramid using point P as the peak of that pyramid. Now what we're going to start thinking about is what happens if I take cross sections of this pyramid? So in this case, the length of segment PB is the height of this pyramid. Now if we were to go halfway along that height, and if we were to take a cross section of this pyramid that is parallel to the surface of our original desk, what would that look like? Well, it would look something, it would look something like this. Now you might be noticing something really interesting.
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Dilating in 3D Solid geometry High school geometry Khan Academy.mp3
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So in this case, the length of segment PB is the height of this pyramid. Now if we were to go halfway along that height, and if we were to take a cross section of this pyramid that is parallel to the surface of our original desk, what would that look like? Well, it would look something, it would look something like this. Now you might be noticing something really interesting. If you were to translate that blue triangle straight down onto the surface of the table, it would look like this. And when you see it that way, it looks like it is a dilation of our original triangle centered at point B. And in fact, it is a dilation centered at point B with a scale factor of 0.5.
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Dilating in 3D Solid geometry High school geometry Khan Academy.mp3
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Now you might be noticing something really interesting. If you were to translate that blue triangle straight down onto the surface of the table, it would look like this. And when you see it that way, it looks like it is a dilation of our original triangle centered at point B. And in fact, it is a dilation centered at point B with a scale factor of 0.5. And you can see it right over here. This length right over here, what BC was dilated down to is half the length of the original BC. This is half the length of the original AB.
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Dilating in 3D Solid geometry High school geometry Khan Academy.mp3
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And in fact, it is a dilation centered at point B with a scale factor of 0.5. And you can see it right over here. This length right over here, what BC was dilated down to is half the length of the original BC. This is half the length of the original AB. And then this is half the length of the original AC. But you could do it at other heights along this pyramid. What if we were to go 0.75 of the way between P and B?
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Dilating in 3D Solid geometry High school geometry Khan Academy.mp3
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This is half the length of the original AB. And then this is half the length of the original AC. But you could do it at other heights along this pyramid. What if we were to go 0.75 of the way between P and B? So if we were to go right over here. So it's closer to our original triangle, closer to our surface. So then the cross section would look like, would look like this.
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Dilating in 3D Solid geometry High school geometry Khan Academy.mp3
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What if we were to go 0.75 of the way between P and B? So if we were to go right over here. So it's closer to our original triangle, closer to our surface. So then the cross section would look like, would look like this. Now if we were to translate that down onto our original surface, what would that look like? Well, it would look like this. It would look like a dilation of our original triangle centered at point B, but this time with a scale factor of 0.75.
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Dilating in 3D Solid geometry High school geometry Khan Academy.mp3
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So then the cross section would look like, would look like this. Now if we were to translate that down onto our original surface, what would that look like? Well, it would look like this. It would look like a dilation of our original triangle centered at point B, but this time with a scale factor of 0.75. And then what if you were to go only a quarter of the way between point P and point B? Well then, you would see something like this. A quarter of the way.
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Dilating in 3D Solid geometry High school geometry Khan Academy.mp3
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It would look like a dilation of our original triangle centered at point B, but this time with a scale factor of 0.75. And then what if you were to go only a quarter of the way between point P and point B? Well then, you would see something like this. A quarter of the way. If you were to take the cross section parallel to our original surface, it would look like this. If you were to translate that straight down onto our table, it would look something like this. And it looks like a dilation centered at point B with a scale factor of 0.25.
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Dilating in 3D Solid geometry High school geometry Khan Academy.mp3
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A quarter of the way. If you were to take the cross section parallel to our original surface, it would look like this. If you were to translate that straight down onto our table, it would look something like this. And it looks like a dilation centered at point B with a scale factor of 0.25. And the reason why all of these dilations look like dilations centered at point B is because point P is directly above point B. But this is a way to conceptualize dilations or see the relationship between cross sections of a three-dimensional shape, in this case like a pyramid, and how those cross sections relate to the base of the pyramid. Now let me ask you an interesting question.
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Dilating in 3D Solid geometry High school geometry Khan Academy.mp3
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And it looks like a dilation centered at point B with a scale factor of 0.25. And the reason why all of these dilations look like dilations centered at point B is because point P is directly above point B. But this is a way to conceptualize dilations or see the relationship between cross sections of a three-dimensional shape, in this case like a pyramid, and how those cross sections relate to the base of the pyramid. Now let me ask you an interesting question. What if I were to try to take a cross section right at point P? Well then, I would just get a point. I would not get an actual triangle, but you could view that as a dilation with a scale factor of 0.
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Dilating in 3D Solid geometry High school geometry Khan Academy.mp3
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Now let me ask you an interesting question. What if I were to try to take a cross section right at point P? Well then, I would just get a point. I would not get an actual triangle, but you could view that as a dilation with a scale factor of 0. And what if I were to take a cross section at the base? Well then, that would be my original triangle, triangle ABC, and then you could view that as a dilation with a scale factor of one because you've gone all the way down to the base. So hopefully this connects some dots for you between cross sections of a three-dimensional shape that is parallel to the base and notions of dilation.
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Dilating in 3D Solid geometry High school geometry Khan Academy.mp3
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Actually, you probably recognize a few of them already, and I'll write them out. They're the circle, the ellipse, the parabola, and the hyperbola. Hyperbola. And you know what these are already. When I first learned conic sections, I was like, oh, I know what a circle is, I know what a parabola is, and I even know a little bit about ellipses and hyperbolas. Why on earth are they called conic sections? To put things simply, because they're the intersection of a plane and a cone.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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And you know what these are already. When I first learned conic sections, I was like, oh, I know what a circle is, I know what a parabola is, and I even know a little bit about ellipses and hyperbolas. Why on earth are they called conic sections? To put things simply, because they're the intersection of a plane and a cone. I'll draw you down in a second, but just before I do that, it probably makes sense to just draw them by themselves, and I'll switch colors. Circle, we all know what that is. Let me pick a thicker line for my circles.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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To put things simply, because they're the intersection of a plane and a cone. I'll draw you down in a second, but just before I do that, it probably makes sense to just draw them by themselves, and I'll switch colors. Circle, we all know what that is. Let me pick a thicker line for my circles. So a circle looks something like that. It's all the points that are equidistant from some center, and that distance that they all are, that's the radius. So this is r, and this is the center.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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Let me pick a thicker line for my circles. So a circle looks something like that. It's all the points that are equidistant from some center, and that distance that they all are, that's the radius. So this is r, and this is the center. The circle is all of the points that are exactly r away from this center. We learned that early in our education, what a circle is. It makes the world go round, literally.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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So this is r, and this is the center. The circle is all of the points that are exactly r away from this center. We learned that early in our education, what a circle is. It makes the world go round, literally. Ellipse, in layman's terms, is kind of a squished circle. It could look something like this. It could look like, let me do an ellipse in another color.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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It makes the world go round, literally. Ellipse, in layman's terms, is kind of a squished circle. It could look something like this. It could look like, let me do an ellipse in another color. So an ellipse could be like that, could be like that. It's harder to draw using the tool I'm drawing, but it could also be tilted and rotated around, but this is the general sense. Actually, circles are a special case of an ellipse.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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It could look like, let me do an ellipse in another color. So an ellipse could be like that, could be like that. It's harder to draw using the tool I'm drawing, but it could also be tilted and rotated around, but this is the general sense. Actually, circles are a special case of an ellipse. It's an ellipse where it's not stretched in one dimension more than the other. It's kind of perfectly symmetric in every way. Parabola, you've learned that if you've taken Algebra 2, and you probably have if you care about conic sections.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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Actually, circles are a special case of an ellipse. It's an ellipse where it's not stretched in one dimension more than the other. It's kind of perfectly symmetric in every way. Parabola, you've learned that if you've taken Algebra 2, and you probably have if you care about conic sections. But a parabola, let me draw lines here so you know where to separate things. A parabola looks something like this, kind of a U shape. I won't go into the equations right now, but the classic, well I will because you're probably familiar with it.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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Parabola, you've learned that if you've taken Algebra 2, and you probably have if you care about conic sections. But a parabola, let me draw lines here so you know where to separate things. A parabola looks something like this, kind of a U shape. I won't go into the equations right now, but the classic, well I will because you're probably familiar with it. It's y is equal to x squared. And then it could be, you could shift it around, and you could even have a parabola that goes like this. That would be x is equal to y squared.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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I won't go into the equations right now, but the classic, well I will because you're probably familiar with it. It's y is equal to x squared. And then it could be, you could shift it around, and you could even have a parabola that goes like this. That would be x is equal to y squared. You could also rotate these things around. But I think you know the general shape of a parabola. We'll talk more about how do you graph it, or how do you know what the interesting points on a parabola actually are.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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That would be x is equal to y squared. You could also rotate these things around. But I think you know the general shape of a parabola. We'll talk more about how do you graph it, or how do you know what the interesting points on a parabola actually are. And then the last one, you might have seen this before, is a hyperbola. It almost looks like two parabolas, but not quite, because the curves, they look a little less U-ish, and a little more open, but I'll explain what I mean by that. So a hyperbola usually looks something like this.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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We'll talk more about how do you graph it, or how do you know what the interesting points on a parabola actually are. And then the last one, you might have seen this before, is a hyperbola. It almost looks like two parabolas, but not quite, because the curves, they look a little less U-ish, and a little more open, but I'll explain what I mean by that. So a hyperbola usually looks something like this. So if these are the axes, and if I were to draw, let me draw some asymptotes. I want to go right through. That's pretty good.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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So a hyperbola usually looks something like this. So if these are the axes, and if I were to draw, let me draw some asymptotes. I want to go right through. That's pretty good. These are asymptotes. Those aren't the actual hyperbola. But a hyperbola would look something like this.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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That's pretty good. These are asymptotes. Those aren't the actual hyperbola. But a hyperbola would look something like this. It could either be on, it would be like, they could be like right here, and they get really close. The asymptote, they get closer and closer to those blue lines, like that. And it would happen on this side, too.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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But a hyperbola would look something like this. It could either be on, it would be like, they could be like right here, and they get really close. The asymptote, they get closer and closer to those blue lines, like that. And it would happen on this side, too. So it kind of, the graphs show up here, and then they pop over, and they show up there. So maybe this magenta could be one hyperbola. I haven't done true justice to it.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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And it would happen on this side, too. So it kind of, the graphs show up here, and then they pop over, and they show up there. So maybe this magenta could be one hyperbola. I haven't done true justice to it. Or another hyperbola could be on kind of a, you could kind of call it a vertical hyperbola, and that's not the exact word, but it would look something like that, where it's below the asymptote here, and it's above the asymptote there. So this blue one would be one hyperbola, and then the magenta one would be a different hyperbola. So those are the different graphs.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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I haven't done true justice to it. Or another hyperbola could be on kind of a, you could kind of call it a vertical hyperbola, and that's not the exact word, but it would look something like that, where it's below the asymptote here, and it's above the asymptote there. So this blue one would be one hyperbola, and then the magenta one would be a different hyperbola. So those are the different graphs. So then, you know, the one thing that I'm sure you're asking is, why are they called conic sections? Why are they not called bolas, or variations of circles, or whatever? And in fact, what's even the relationship?
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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So those are the different graphs. So then, you know, the one thing that I'm sure you're asking is, why are they called conic sections? Why are they not called bolas, or variations of circles, or whatever? And in fact, what's even the relationship? It's pretty clear that circles and ellipses are somehow related, that an ellipse is just a squished circle. And maybe it even seems that parabolas and hyperbolas are somewhat related. This is a P, once again.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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And in fact, what's even the relationship? It's pretty clear that circles and ellipses are somehow related, that an ellipse is just a squished circle. And maybe it even seems that parabolas and hyperbolas are somewhat related. This is a P, once again. They both have bola in their name, and they both kind of look like opened U's, although a hyperbola has two of these, kind of opening in different directions, but they look related. But what is the connection behind all these? And that's, frankly, where the word conic comes from.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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This is a P, once again. They both have bola in their name, and they both kind of look like opened U's, although a hyperbola has two of these, kind of opening in different directions, but they look related. But what is the connection behind all these? And that's, frankly, where the word conic comes from. So let me see if I can draw a three-dimensional cone. So if this is a cone, let me see. So that's the top.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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And that's, frankly, where the word conic comes from. So let me see if I can draw a three-dimensional cone. So if this is a cone, let me see. So that's the top. Let me draw... I could have used an ellipse for the top. Looks like that.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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So that's the top. Let me draw... I could have used an ellipse for the top. Looks like that. And it has no top. It would actually keep going on forever in that direction. I'm just kind of slicing it so you see that it's a cone.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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Looks like that. And it has no top. It would actually keep going on forever in that direction. I'm just kind of slicing it so you see that it's a cone. This could be the bottom part of it. So let's take different intersections of a plane with this cone and see if we can at least generate the different shapes that we talked about just now. So if we have a plane that goes directly...
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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I'm just kind of slicing it so you see that it's a cone. This could be the bottom part of it. So let's take different intersections of a plane with this cone and see if we can at least generate the different shapes that we talked about just now. So if we have a plane that goes directly... I guess if you call this the axis of this three-dimensional cone. So this is the axis. So if we have a plane that's exactly perpendicular to that axis... Let's see if I can draw it in three dimensions.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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So if we have a plane that goes directly... I guess if you call this the axis of this three-dimensional cone. So this is the axis. So if we have a plane that's exactly perpendicular to that axis... Let's see if I can draw it in three dimensions. So this would look something like this. So it would have a line. This is the front line that's closer to you.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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So if we have a plane that's exactly perpendicular to that axis... Let's see if I can draw it in three dimensions. So this would look something like this. So it would have a line. This is the front line that's closer to you. And then they'll have another line back here. That's close enough. And then, of course, you know these are infinite planes.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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This is the front line that's closer to you. And then they'll have another line back here. That's close enough. And then, of course, you know these are infinite planes. It goes off in every direction. So if this plane is directly perpendicular to the axis of these... So this is where the plane goes behind it.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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And then, of course, you know these are infinite planes. It goes off in every direction. So if this plane is directly perpendicular to the axis of these... So this is where the plane goes behind it. The intersection of this plane and this cone is going to look like this. And we're looking at it from an angle, but if you were to look at it straight down, if you were sitting here and you were to look at this plane, if you were to look at it right above, if I were to just flip this over like this, so we're looking straight down at this plane, that intersection would be a circle. Now, if we take the plane and we tilt it down a little bit, so if instead of that, we have a situation like this, let me see if I can do it justice.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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So this is where the plane goes behind it. The intersection of this plane and this cone is going to look like this. And we're looking at it from an angle, but if you were to look at it straight down, if you were sitting here and you were to look at this plane, if you were to look at it right above, if I were to just flip this over like this, so we're looking straight down at this plane, that intersection would be a circle. Now, if we take the plane and we tilt it down a little bit, so if instead of that, we have a situation like this, let me see if I can do it justice. We have a situation where it's... Let me undo that. Edit, undo. Where it's like this, and it has another side like this, and then I connect them.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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Now, if we take the plane and we tilt it down a little bit, so if instead of that, we have a situation like this, let me see if I can do it justice. We have a situation where it's... Let me undo that. Edit, undo. Where it's like this, and it has another side like this, and then I connect them. So that's the plane. Now the intersection of this plane, which is now not orthogonal, or it's not perpendicular to the axis of this 3-dimensional cone, if you take the intersection of that plane and that cone, and in future videos, and you don't do this in your Algebra 2 class, but eventually we'll kind of do the 3-dimensional intersection and prove that this is definitely the case, and you definitely do get the equations, which I'll show you in the not-too-far-off future, this intersection would look something like this. I think you can visualize it right now.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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Where it's like this, and it has another side like this, and then I connect them. So that's the plane. Now the intersection of this plane, which is now not orthogonal, or it's not perpendicular to the axis of this 3-dimensional cone, if you take the intersection of that plane and that cone, and in future videos, and you don't do this in your Algebra 2 class, but eventually we'll kind of do the 3-dimensional intersection and prove that this is definitely the case, and you definitely do get the equations, which I'll show you in the not-too-far-off future, this intersection would look something like this. I think you can visualize it right now. It would look something like this. And if you were to look straight down on this plane, if you were to look right above the plane, this would look something... This figure I just drew in purple would look something like this.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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I think you can visualize it right now. It would look something like this. And if you were to look straight down on this plane, if you were to look right above the plane, this would look something... This figure I just drew in purple would look something like this. Oh, I didn't draw it that well. It would be an ellipse. You know what an ellipse looks like.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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This figure I just drew in purple would look something like this. Oh, I didn't draw it that well. It would be an ellipse. You know what an ellipse looks like. And if I tilted it the other way, the ellipse would be... would kind of... would squeeze the other way. But that just gives you a general sense of why both of these are conic sections. Now something very interesting, if we keep tilting this plane...
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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You know what an ellipse looks like. And if I tilted it the other way, the ellipse would be... would kind of... would squeeze the other way. But that just gives you a general sense of why both of these are conic sections. Now something very interesting, if we keep tilting this plane... So if we tilt the plane so it's... so let's say we're pivoting around that point. So now my plane... Let me see if I can do this. This is a good exercise in 3-dimensional drawing.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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Now something very interesting, if we keep tilting this plane... So if we tilt the plane so it's... so let's say we're pivoting around that point. So now my plane... Let me see if I can do this. This is a good exercise in 3-dimensional drawing. Let's say it looks something like this. I want to go through that point. Oh.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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This is a good exercise in 3-dimensional drawing. Let's say it looks something like this. I want to go through that point. Oh. So this is my 3-dimensional plane. And I'm drawing it in such a way that it only intersects this bottom cone. And the surface of the plane is parallel to the side of this top cone.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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Oh. So this is my 3-dimensional plane. And I'm drawing it in such a way that it only intersects this bottom cone. And the surface of the plane is parallel to the side of this top cone. In this case, the intersection of the plane and the cone is going to intersect right at that point. I'm kind of... You can almost view that I'm pivoting around this point, the intersection of this point and the plane and the cone. Well, this now would look... the intersection would look something like this.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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And the surface of the plane is parallel to the side of this top cone. In this case, the intersection of the plane and the cone is going to intersect right at that point. I'm kind of... You can almost view that I'm pivoting around this point, the intersection of this point and the plane and the cone. Well, this now would look... the intersection would look something like this. It would look like that. And it would keep going down. So if I were to draw it, it would look like this.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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Well, this now would look... the intersection would look something like this. It would look like that. And it would keep going down. So if I were to draw it, it would look like this. If I was right above the plane, if I were to just draw the plane. And there you get your parabola. So that's interesting.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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So if I were to draw it, it would look like this. If I was right above the plane, if I were to just draw the plane. And there you get your parabola. So that's interesting. If you keep kind of tilting... If you start with a circle, tilt a little bit, you get an ellipse, you get an ellipse, you get kind of a more and more skewed ellipse, more and more skewed ellipse. And at some point, the ellipse kind of... You know, the ellipse keeps getting more and more skewed like that.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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So that's interesting. If you keep kind of tilting... If you start with a circle, tilt a little bit, you get an ellipse, you get an ellipse, you get kind of a more and more skewed ellipse, more and more skewed ellipse. And at some point, the ellipse kind of... You know, the ellipse keeps getting more and more skewed like that. At some point, it kind of pops. Right when you... Right when you become exactly parallel to the side of this top cone.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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And at some point, the ellipse kind of... You know, the ellipse keeps getting more and more skewed like that. At some point, it kind of pops. Right when you... Right when you become exactly parallel to the side of this top cone. And I'm doing it all very inexact right now, but I think I want to give you the intuition. It pops, and it turns into a parabola. So you can kind of view a parabola.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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Right when you become exactly parallel to the side of this top cone. And I'm doing it all very inexact right now, but I think I want to give you the intuition. It pops, and it turns into a parabola. So you can kind of view a parabola. There is this relationship. It's kind of... The parabola is what happens when one side of an ellipse pops open, and you get this parabola.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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So you can kind of view a parabola. There is this relationship. It's kind of... The parabola is what happens when one side of an ellipse pops open, and you get this parabola. And then if you keep tilting this plane, if you keep tilting the plane... I'll do it in another color. So it intersects both sides of the cone.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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The parabola is what happens when one side of an ellipse pops open, and you get this parabola. And then if you keep tilting this plane, if you keep tilting the plane... I'll do it in another color. So it intersects both sides of the cone. So let me see if I can draw that. So if this is my new plane... Whoops. That's good enough.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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So it intersects both sides of the cone. So let me see if I can draw that. So if this is my new plane... Whoops. That's good enough. So if my plane looks like this... I know it's very hard to read now. And you wanted the intersection of this plane, this green plane in the cone.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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That's good enough. So if my plane looks like this... I know it's very hard to read now. And you wanted the intersection of this plane, this green plane in the cone. I should probably redraw it all, but hopefully you're not getting overwhelmingly confused. The intersection would look like this. It would intersect the bottom cone there, and it would intersect the top cone over there.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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And you wanted the intersection of this plane, this green plane in the cone. I should probably redraw it all, but hopefully you're not getting overwhelmingly confused. The intersection would look like this. It would intersect the bottom cone there, and it would intersect the top cone over there. And then you would have something like this. You would have... This would be the intersection of the plane in the bottom cone, and then up here would be the intersection of the plane in the top cone.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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It would intersect the bottom cone there, and it would intersect the top cone over there. And then you would have something like this. You would have... This would be the intersection of the plane in the bottom cone, and then up here would be the intersection of the plane in the top cone. Remember, this plane goes off in every direction infinitely. So that's just a general sense of what the conic sections are and why, frankly, they're called conic sections. And let me know if this got confusing, because maybe I'll do another video where I'll redraw it a little bit cleaner, or maybe I can find some kind of neat 3-D application that can do it better than I can do it.
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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This would be the intersection of the plane in the bottom cone, and then up here would be the intersection of the plane in the top cone. Remember, this plane goes off in every direction infinitely. So that's just a general sense of what the conic sections are and why, frankly, they're called conic sections. And let me know if this got confusing, because maybe I'll do another video where I'll redraw it a little bit cleaner, or maybe I can find some kind of neat 3-D application that can do it better than I can do it. But this is kind of just the reason why they all are conic sections, and why they really are related to each other. And we'll do that in a little more depth mathematically in a few videos. But in the next video, now that you know what they are and why they're all called conic sections, I'll actually talk about the formulas about these and how do you recognize the formulas, and given a formula, how do you actually plot the graphs of these conic sections?
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Introduction to conic sections Conic sections Algebra II Khan Academy.mp3
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So pause this video and see if you can do that, and you might wanna leverage this diagram. All right, so let's work through this together. So we can start with this diagram, and what we know is that segment ED is parallel to segment CB. So we can write that down. Segment ED is parallel to segment CB. And so segment ED is what they're talking about. That is a line or a line segment that is parallel to one side of the triangle.
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Proof Parallel lines divide triangle sides proportionally Similarity Geometry Khan Academy.mp3
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So we can write that down. Segment ED is parallel to segment CB. And so segment ED is what they're talking about. That is a line or a line segment that is parallel to one side of the triangle. So really, given what we know and what's already been written over here on this triangle, we need to prove another way of writing it, another way of saying it divides the other two sides proportionately is that the ratio between the part of the original triangle side that is on one side of the dividing line to the length on the other side is going to be the same on both sides that it is intersecting. So another way to say that it divides the other two sides proportionately, if we look at this triangle over here, it would mean that the length of segment AE over the length of segment EC is going to be equal to the length of segment AD over the length of segment DB. This statement right over here and what I underlined up here are equivalent given this triangle.
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Proof Parallel lines divide triangle sides proportionally Similarity Geometry Khan Academy.mp3
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That is a line or a line segment that is parallel to one side of the triangle. So really, given what we know and what's already been written over here on this triangle, we need to prove another way of writing it, another way of saying it divides the other two sides proportionately is that the ratio between the part of the original triangle side that is on one side of the dividing line to the length on the other side is going to be the same on both sides that it is intersecting. So another way to say that it divides the other two sides proportionately, if we look at this triangle over here, it would mean that the length of segment AE over the length of segment EC is going to be equal to the length of segment AD over the length of segment DB. This statement right over here and what I underlined up here are equivalent given this triangle. So the way that we can try to do it is to establish a similarity between triangle AED and triangle ACB. So how do we do that? Well, because these two lines are parallel, we can view segment AC as a transversal intersecting two parallel lines.
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Proof Parallel lines divide triangle sides proportionally Similarity Geometry Khan Academy.mp3
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This statement right over here and what I underlined up here are equivalent given this triangle. So the way that we can try to do it is to establish a similarity between triangle AED and triangle ACB. So how do we do that? Well, because these two lines are parallel, we can view segment AC as a transversal intersecting two parallel lines. So that tells us that these two corresponding angles are going to be congruent. So we could say that angle one is congruent to angle three. And the reason why is because they are corresponding corresponding angles.
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Proof Parallel lines divide triangle sides proportionally Similarity Geometry Khan Academy.mp3
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Well, because these two lines are parallel, we can view segment AC as a transversal intersecting two parallel lines. So that tells us that these two corresponding angles are going to be congruent. So we could say that angle one is congruent to angle three. And the reason why is because they are corresponding corresponding angles. I'm just trying to write a little bit of shorthand. This is short for corresponding angles. That's the rationale.
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Proof Parallel lines divide triangle sides proportionally Similarity Geometry Khan Academy.mp3
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And the reason why is because they are corresponding corresponding angles. I'm just trying to write a little bit of shorthand. This is short for corresponding angles. That's the rationale. And we also know that angle two is congruent to angle four for the same reason. So angle two is congruent to angle four, once again, because they are corresponding angles. This time we have a different transversal, corresponding angles where a transversal intersects two parallel lines.
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Proof Parallel lines divide triangle sides proportionally Similarity Geometry Khan Academy.mp3
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That's the rationale. And we also know that angle two is congruent to angle four for the same reason. So angle two is congruent to angle four, once again, because they are corresponding angles. This time we have a different transversal, corresponding angles where a transversal intersects two parallel lines. And so now if you look at triangle AED and triangle ACB, you see that they have two sets of corresponding angles that are congruent. And if you have two sets of corresponding angles, that means that all of the angles are congruent. And you actually see that over here if you care about it.
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Proof Parallel lines divide triangle sides proportionally Similarity Geometry Khan Academy.mp3
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This time we have a different transversal, corresponding angles where a transversal intersects two parallel lines. And so now if you look at triangle AED and triangle ACB, you see that they have two sets of corresponding angles that are congruent. And if you have two sets of corresponding angles, that means that all of the angles are congruent. And you actually see that over here if you care about it. But two is enough, but you actually have a third because angle, I guess you call it BAC, is common to both triangles. And so we can say that triangle AED is similar to triangle ACB, ACB, by angle similarity, similarity. And then given that these two are similar, then we can set up a proportion.
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Proof Parallel lines divide triangle sides proportionally Similarity Geometry Khan Academy.mp3
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And you actually see that over here if you care about it. But two is enough, but you actually have a third because angle, I guess you call it BAC, is common to both triangles. And so we can say that triangle AED is similar to triangle ACB, ACB, by angle similarity, similarity. And then given that these two are similar, then we can set up a proportion. That tells us that the ratio of the length of segment AE to this entire side to AC, AC, is equal to the ratio of AD, the length of that segment, to the length of the entire thing, to AB. Now, this implies, and I'm just gonna start writing it to the right here to save space. This is the same thing as the ratio of AE over, AC is AE plus EC's length.
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Proof Parallel lines divide triangle sides proportionally Similarity Geometry Khan Academy.mp3
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And then given that these two are similar, then we can set up a proportion. That tells us that the ratio of the length of segment AE to this entire side to AC, AC, is equal to the ratio of AD, the length of that segment, to the length of the entire thing, to AB. Now, this implies, and I'm just gonna start writing it to the right here to save space. This is the same thing as the ratio of AE over, AC is AE plus EC's length. So AE's length plus EC's length. And then this is going to be equal to the length of segment AD over segment AB. Its length is the length of segment AD, AD plus segment DB, plus DB.
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Proof Parallel lines divide triangle sides proportionally Similarity Geometry Khan Academy.mp3
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This is the same thing as the ratio of AE over, AC is AE plus EC's length. So AE's length plus EC's length. And then this is going to be equal to the length of segment AD over segment AB. Its length is the length of segment AD, AD plus segment DB, plus DB. Now, really what I need to do is figure out how do I algebraically manipulate it so I get what I have up here. Let me scroll down a little bit. So one way I could try to simplify this is to essentially cross multiply.
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Proof Parallel lines divide triangle sides proportionally Similarity Geometry Khan Academy.mp3
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Its length is the length of segment AD, AD plus segment DB, plus DB. Now, really what I need to do is figure out how do I algebraically manipulate it so I get what I have up here. Let me scroll down a little bit. So one way I could try to simplify this is to essentially cross multiply. That's equivalent to multiplying both sides by both of these denominators, and we've covered that in other videos. And so this is going to be equal to the length of segment AE times AD plus DB, those segment lengths. That's gotta be equal to length of AD times AE plus the length of segment EC.
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Proof Parallel lines divide triangle sides proportionally Similarity Geometry Khan Academy.mp3
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So one way I could try to simplify this is to essentially cross multiply. That's equivalent to multiplying both sides by both of these denominators, and we've covered that in other videos. And so this is going to be equal to the length of segment AE times AD plus DB, those segment lengths. That's gotta be equal to length of AD times AE plus the length of segment EC. And I can distribute this over here. I have length of segment AE times length of segment AD plus length of segment AE times length of segment DB is equal to length of segment AD times length of segment AE plus length of segment AD times length of segment EC. And let's see, is there anything that I can simplify here?
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Proof Parallel lines divide triangle sides proportionally Similarity Geometry Khan Academy.mp3
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That's gotta be equal to length of AD times AE plus the length of segment EC. And I can distribute this over here. I have length of segment AE times length of segment AD plus length of segment AE times length of segment DB is equal to length of segment AD times length of segment AE plus length of segment AD times length of segment EC. And let's see, is there anything that I can simplify here? Well, I have an AE times AD on both sides, so let me just subtract AE times AD from both sides. And so then I'm just left with that this is equal to that. So scroll down a little bit more, and let me actually just rewrite this cleanly.
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Proof Parallel lines divide triangle sides proportionally Similarity Geometry Khan Academy.mp3
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And let's see, is there anything that I can simplify here? Well, I have an AE times AD on both sides, so let me just subtract AE times AD from both sides. And so then I'm just left with that this is equal to that. So scroll down a little bit more, and let me actually just rewrite this cleanly. So I have AE times DB is equal to AD times EC. These are all the segment lengths right over here. Now, if you divide both sides by EC, you're going to get an EC down here, and then this would cancel out.
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Proof Parallel lines divide triangle sides proportionally Similarity Geometry Khan Academy.mp3
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In previous videos, we started talking about the idea of transformations. In particular, we talked about rigid transformations. So, for example, you can shift something. This would be a translation. So the thing that I'm moving around is a translation of our original triangle. You could have a rotation. So that thing that I translated, I am now rotating it, as you see right over there.
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Transformations - dilation.mp3
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This would be a translation. So the thing that I'm moving around is a translation of our original triangle. You could have a rotation. So that thing that I translated, I am now rotating it, as you see right over there. And you could also have a reflection, and the tool that I'm using doesn't make reflection too easy but that's essentially flipping it over a line. But what we're going to talk about in this video is a non-rigid transformation. And what makes something a rigid transformation is that lengths between points are preserved.
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Transformations - dilation.mp3
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So that thing that I translated, I am now rotating it, as you see right over there. And you could also have a reflection, and the tool that I'm using doesn't make reflection too easy but that's essentially flipping it over a line. But what we're going to talk about in this video is a non-rigid transformation. And what makes something a rigid transformation is that lengths between points are preserved. But in a non-rigid transformation, those lengths do not need to be preserved. So, for example, this rotated and translated triangle that I'm moving around right here, in fact, I'm continuing to translate it as I talk, I can dilate it. And one way to think about dilation is that we're just scaling it down or scaling it up.
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Transformations - dilation.mp3
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And what makes something a rigid transformation is that lengths between points are preserved. But in a non-rigid transformation, those lengths do not need to be preserved. So, for example, this rotated and translated triangle that I'm moving around right here, in fact, I'm continuing to translate it as I talk, I can dilate it. And one way to think about dilation is that we're just scaling it down or scaling it up. So, for example, here, I am scaling it down. That is a dilation. Or I could scale it up.
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Transformations - dilation.mp3
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And one way to think about dilation is that we're just scaling it down or scaling it up. So, for example, here, I am scaling it down. That is a dilation. Or I could scale it up. This is also a dilation. We're even going off of the graph paper. So the whole point here is just to appreciate that we don't just have the rigid transformations, we can have other types of transformations.
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Transformations - dilation.mp3
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So what they're saying here is if you were to take the adjacent leg length over the hypotenuse leg length for 25 degree angle, it would be a ratio of approximately 0.91. For a 35 degree angle, it would be a ratio of 0.82, and then they do this for 45 degrees, and they do the different ratios right over here. So we're gonna use the table to approximate the measure of angle D in the triangle below. So pause this video and see if you can figure that out. All right, now let's work through this together. Now what information do they give us about angle D in this triangle? Well, we are given the opposite length right over here.
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Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3
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So pause this video and see if you can figure that out. All right, now let's work through this together. Now what information do they give us about angle D in this triangle? Well, we are given the opposite length right over here. Let me label that. That is the opposite leg length, which is 3.4. We're also given, what is this right over here?
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Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3
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Well, we are given the opposite length right over here. Let me label that. That is the opposite leg length, which is 3.4. We're also given, what is this right over here? Is this adjacent or is this a hypotenuse? You might be tempted to say, well, this is right next to the angle, or this is one of the lines, or it's on the ray that helps form the angle, so maybe it's adjacent. But remember, adjacent is the adjacent side that is not the hypotenuse.
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Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3
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We're also given, what is this right over here? Is this adjacent or is this a hypotenuse? You might be tempted to say, well, this is right next to the angle, or this is one of the lines, or it's on the ray that helps form the angle, so maybe it's adjacent. But remember, adjacent is the adjacent side that is not the hypotenuse. And this is clearly the hypotenuse. It is the longest side. It is the side opposite the 90 degree angle.
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Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3
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But remember, adjacent is the adjacent side that is not the hypotenuse. And this is clearly the hypotenuse. It is the longest side. It is the side opposite the 90 degree angle. So this right over here is the hypotenuse. Hypotenuse. So we're given the opposite leg length and the hypotenuse length.
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Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3
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It is the side opposite the 90 degree angle. So this right over here is the hypotenuse. Hypotenuse. So we're given the opposite leg length and the hypotenuse length. And so let's see, which of these ratios deal with the opposite and the hypotenuse? And if we, let's see, this first one is adjacent and hypotenuse. The second one here is hypotenuse, sorry, opposite and hypotenuse.
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Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3
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So we're given the opposite leg length and the hypotenuse length. And so let's see, which of these ratios deal with the opposite and the hypotenuse? And if we, let's see, this first one is adjacent and hypotenuse. The second one here is hypotenuse, sorry, opposite and hypotenuse. So that's exactly what we're talking about. We're talking about the opposite leg length over the hypotenuse, over the hypotenuse length. So in this case, what is going to be our opposite leg length over our hypotenuse leg length?
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Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3
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The second one here is hypotenuse, sorry, opposite and hypotenuse. So that's exactly what we're talking about. We're talking about the opposite leg length over the hypotenuse, over the hypotenuse length. So in this case, what is going to be our opposite leg length over our hypotenuse leg length? It's going to be 3.4 over eight. 3.4 over eight, which is approximately going to be equal to, let me do this down here, this is eight goes into 3.4. Eight goes in, doesn't go into three.
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Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3
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So in this case, what is going to be our opposite leg length over our hypotenuse leg length? It's going to be 3.4 over eight. 3.4 over eight, which is approximately going to be equal to, let me do this down here, this is eight goes into 3.4. Eight goes in, doesn't go into three. Eight goes into 34 four times. Four times eight is 32. If I subtract, and I could scroll down a little bit, I get a two.
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Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3
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Eight goes in, doesn't go into three. Eight goes into 34 four times. Four times eight is 32. If I subtract, and I could scroll down a little bit, I get a two. I can bring down a zero. Eight goes into 20 two times. And that's about as much precision as any of these have.
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Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3
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If I subtract, and I could scroll down a little bit, I get a two. I can bring down a zero. Eight goes into 20 two times. And that's about as much precision as any of these have. And so it looks like for this particular triangle and this angle of the triangle, if I were to take a ratio of the opposite length and the hypotenuse length, opposite over hypotenuse, I get 0.42. So that looks like this situation right over here. So that would imply that this is a 25 degree, 25 degree angle approximately.
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Using right triangle ratios to approximate angle measure High school geometry Khan Academy.mp3
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You might recognize the first part of geometry right over here. You have the root word geo, the same word that you see in things like geography and geology. And this comes, this refers to the Earth. This refers, my E looks like a C right over there. This refers to the Earth. And then you see this metri part. And you see metri in things like trigonometry as well.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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This refers, my E looks like a C right over there. This refers to the Earth. And then you see this metri part. And you see metri in things like trigonometry as well. And metri, or the metric system, and this comes from measurement. This comes from measurement or measure. So when someone's talking about geometry, the word itself comes from Earth measurement.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And you see metri in things like trigonometry as well. And metri, or the metric system, and this comes from measurement. This comes from measurement or measure. So when someone's talking about geometry, the word itself comes from Earth measurement. And that's kind of not so bad of a name because it is such a general subject. Geometry really is the study and trying to understand how shapes and space and things that we see relate to each other. So when you start learning about geometry, you learn about lines and triangles and circles.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So when someone's talking about geometry, the word itself comes from Earth measurement. And that's kind of not so bad of a name because it is such a general subject. Geometry really is the study and trying to understand how shapes and space and things that we see relate to each other. So when you start learning about geometry, you learn about lines and triangles and circles. And you learn about angles. And we'll define all of these things more and more precisely as we go further and further on. But also encapsulate things like patterns and three-dimensional shapes.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So when you start learning about geometry, you learn about lines and triangles and circles. And you learn about angles. And we'll define all of these things more and more precisely as we go further and further on. But also encapsulate things like patterns and three-dimensional shapes. So it's almost everything that we see, all of the visually mathematical things that we understand can in some way be categorized in geometry. Now with that out of the way, let's just start from the basics, a basic starting point from geometry. And then we can just grow from there.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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