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It cut into the top right over there. It would get all the way to the bottom right over there. And along this side, it would cut right there. And along that side, it would cut right over there. So what's the resulting shape in two dimensions of essentially the intersection between the slicer and the jello? Well, it would be this thing. It would look like a trapezoid.
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Slice a rectangular pyramid Perimeter, area, and volume Geometry Khan Academy.mp3
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And along that side, it would cut right over there. So what's the resulting shape in two dimensions of essentially the intersection between the slicer and the jello? Well, it would be this thing. It would look like a trapezoid. Let me do that in a new color. I'm overusing that one color. It would look like a trapezoid.
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Slice a rectangular pyramid Perimeter, area, and volume Geometry Khan Academy.mp3
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It would look like a trapezoid. Let me do that in a new color. I'm overusing that one color. It would look like a trapezoid. So this would be the resulting shape. So the resulting shape would look like this. It would look like this, just like that if you made the cut right over there.
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Slice a rectangular pyramid Perimeter, area, and volume Geometry Khan Academy.mp3
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I have this little dilation tool. So the first question is, are the coordinates of the vertices going to be preserved? Well, pause the video and try to think about that. Well, let's just try it out experimentally. We can see under an arbitrary dilation here, the coordinates are not preserved. The point that corresponds to D now has a different coordinate. The vertices, the vertex that corresponds to A now has different coordinates.
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Dilations and shape properties.mp3
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Well, let's just try it out experimentally. We can see under an arbitrary dilation here, the coordinates are not preserved. The point that corresponds to D now has a different coordinate. The vertices, the vertex that corresponds to A now has different coordinates. Same thing for B and C. The corresponding points after the dilation now sit on a different part of the coordinate plane. So in this case, the coordinates of the vertices are not preserved. Now the next question, let me go back to where we were.
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Dilations and shape properties.mp3
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The vertices, the vertex that corresponds to A now has different coordinates. Same thing for B and C. The corresponding points after the dilation now sit on a different part of the coordinate plane. So in this case, the coordinates of the vertices are not preserved. Now the next question, let me go back to where we were. So the next question, the corresponding line segments after dilation, are they sitting on the same line? And so let me dilate again. And so you can see, if you consider this point B prime, because it corresponds to point B, the segment B prime, C prime, this does not sit on the same line as BC.
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Dilations and shape properties.mp3
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Now the next question, let me go back to where we were. So the next question, the corresponding line segments after dilation, are they sitting on the same line? And so let me dilate again. And so you can see, if you consider this point B prime, because it corresponds to point B, the segment B prime, C prime, this does not sit on the same line as BC. But the segment A prime, D prime, the corresponding line segment to line segment AD, that does sit on the same line. And if you think about why that is, well, if we originally draw a line that if we look at the line that contains segment AD, it also goes through point P. And so as we expand out, this segment right over here is going to expand and shift outward along the same lines. But that's not going to be true of these other segments because they don't, because the point P does not sit on the line that those segments sit on.
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Dilations and shape properties.mp3
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And so you can see, if you consider this point B prime, because it corresponds to point B, the segment B prime, C prime, this does not sit on the same line as BC. But the segment A prime, D prime, the corresponding line segment to line segment AD, that does sit on the same line. And if you think about why that is, well, if we originally draw a line that if we look at the line that contains segment AD, it also goes through point P. And so as we expand out, this segment right over here is going to expand and shift outward along the same lines. But that's not going to be true of these other segments because they don't, because the point P does not sit on the line that those segments sit on. And so let's just expand it again. So you see that right over there. Now the next question, are angle measures preserved?
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Dilations and shape properties.mp3
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But that's not going to be true of these other segments because they don't, because the point P does not sit on the line that those segments sit on. And so let's just expand it again. So you see that right over there. Now the next question, are angle measures preserved? Well, it looks like they are. And this is one of the things that is true about a dilation, is that you're going to preserve angle measures. This angle is still a right angle.
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Dilations and shape properties.mp3
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Now the next question, are angle measures preserved? Well, it looks like they are. And this is one of the things that is true about a dilation, is that you're going to preserve angle measures. This angle is still a right angle. This angle here, I guess you could call it angle, the measure of angle B, is the same as the measure of angle B prime. And you can see it with all of these points right over there. And then the last question, are side lengths, perimeter, and area preserved?
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Dilations and shape properties.mp3
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This angle is still a right angle. This angle here, I guess you could call it angle, the measure of angle B, is the same as the measure of angle B prime. And you can see it with all of these points right over there. And then the last question, are side lengths, perimeter, and area preserved? Well, we can immediately see as we dilate outwards, for example, the segment corresponding to AD has gotten longer. In fact, if we dilate outwards, all of the segments, the corresponding segments are getting larger. And if they're all getting larger, then the perimeter is getting larger and the area is getting larger.
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Dilations and shape properties.mp3
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And then the last question, are side lengths, perimeter, and area preserved? Well, we can immediately see as we dilate outwards, for example, the segment corresponding to AD has gotten longer. In fact, if we dilate outwards, all of the segments, the corresponding segments are getting larger. And if they're all getting larger, then the perimeter is getting larger and the area is getting larger. Likewise, if we dilate in like this, they're all getting smaller. So side lengths, perimeter, and area are not preserved. Now let's ask the same questions with another dilation.
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Dilations and shape properties.mp3
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And if they're all getting larger, then the perimeter is getting larger and the area is getting larger. Likewise, if we dilate in like this, they're all getting smaller. So side lengths, perimeter, and area are not preserved. Now let's ask the same questions with another dilation. And this is going to be interesting because we're going to look at a dilation that is centered at one of the vertices of our shape. So let me scroll down here. And so I have the same tool again.
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Dilations and shape properties.mp3
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Now let's ask the same questions with another dilation. And this is going to be interesting because we're going to look at a dilation that is centered at one of the vertices of our shape. So let me scroll down here. And so I have the same tool again. And now here we have a triangle, triangle ABC, and we're gonna dilate about point C. So first of all, do we think the vertices, the coordinates of the vertices are going to be preserved? Let's dilate out. Well, you can see point C is preserved.
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Dilations and shape properties.mp3
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And so I have the same tool again. And now here we have a triangle, triangle ABC, and we're gonna dilate about point C. So first of all, do we think the vertices, the coordinates of the vertices are going to be preserved? Let's dilate out. Well, you can see point C is preserved. When it gets mapped after the dilation, it sits in the exact same place, but the things that correspond to A and B are not preserved. You could call this A prime, and this definitely has different coordinates than A, and B prime definitely has different coordinates than B. Now what about corresponding line segments?
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Dilations and shape properties.mp3
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Well, you can see point C is preserved. When it gets mapped after the dilation, it sits in the exact same place, but the things that correspond to A and B are not preserved. You could call this A prime, and this definitely has different coordinates than A, and B prime definitely has different coordinates than B. Now what about corresponding line segments? Are they on the same line? Well, some of them are and some of them aren't. So for example, when we dilate, so let's look at the segment AC and the segment BC.
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Dilations and shape properties.mp3
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Now what about corresponding line segments? Are they on the same line? Well, some of them are and some of them aren't. So for example, when we dilate, so let's look at the segment AC and the segment BC. When we dilate, we can see, whoops, when we dilate, we can see the corresponding segments, you could call this A prime, C prime, or B prime, C prime, do still sit on that same line. And that's because the point that we are dilating about, point C, sat on those original segments. So we're essentially just lengthening out on the point that is not the center of dilation.
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Dilations and shape properties.mp3
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So for example, when we dilate, so let's look at the segment AC and the segment BC. When we dilate, we can see, whoops, when we dilate, we can see the corresponding segments, you could call this A prime, C prime, or B prime, C prime, do still sit on that same line. And that's because the point that we are dilating about, point C, sat on those original segments. So we're essentially just lengthening out on the point that is not the center of dilation. We're lengthening out away from it, or if the dilation is going in, we would be shortening along that same line. But some of the segments are not overlapping on the same line. So for example, A prime, B prime does not sit along the same line as A, B.
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Dilations and shape properties.mp3
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So we're essentially just lengthening out on the point that is not the center of dilation. We're lengthening out away from it, or if the dilation is going in, we would be shortening along that same line. But some of the segments are not overlapping on the same line. So for example, A prime, B prime does not sit along the same line as A, B. Now what about the angle measures? Well, we already talked about it. Angle measures are preserved under dilations.
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Dilations and shape properties.mp3
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So for example, A prime, B prime does not sit along the same line as A, B. Now what about the angle measures? Well, we already talked about it. Angle measures are preserved under dilations. The measure of angle C here, this is the exact same angle, and so is the measure of angle, you could call this A prime and B prime right over here. And then finally, what about side lengths? Well, you can clearly see that when I dilate out, my side lengths increase, or if I dilate in, my side lengths decrease.
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Dilations and shape properties.mp3
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Angle measures are preserved under dilations. The measure of angle C here, this is the exact same angle, and so is the measure of angle, you could call this A prime and B prime right over here. And then finally, what about side lengths? Well, you can clearly see that when I dilate out, my side lengths increase, or if I dilate in, my side lengths decrease. And so side lengths are not preserved. And if side lengths are not preserved, then the perimeter is not preserved, and also the area is not preserved. You could view area as a function of the side lengths.
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Dilations and shape properties.mp3
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What we're going to do in this video is see that if we have two different triangles and we have two sets of corresponding sides that have the same length, for example, this blue side has the same length as this blue side here, and this orange side has the same length as this orange side here, and the angle that is formed between those sides, so we have two corresponding angles right over here, that they also have the equal measure. So we could think about, we have a side, an angle, a side, a side, an angle, and a side. If those have the same lengths or measures, then we can deduce that these two triangles must be congruent by the rigid motion definition of congruency, or the shorthand is, if we have side, angle, side in common and the angle is between the two sides, then the two triangles will be congruent. So to be able to prove this, in order to make this deduction, we just have to say that there's always a rigid transformation if we have a side, angle, side in common that will allow us to map one triangle onto the other, because if there is a series of rigid transformations that allow us to do it, then by the rigid transformation definition, the true triangles are congruent. So the first thing that we could do is we could reference back to where we saw that if we have two segments that have the same length, like segment AB and segment DE, if we have two segments with the same length, that they are congruent. You can always map one segment onto the other with a series of rigid transformations. The way that we could do that in this case is we could map point B onto point E, so this would be now, I'll put B prime right over here, and if we just did a transformation to do that, so if we just translated like that, then side, whoops, then side BA would, that orange side would be something like that, but then we could do another rigid transformation that rotates about point E, or B prime, that rotates that orange side and the whole triangle with it onto DE, in which case, once we do that second rigid transformation, point A will now coincide with D, or we could say A prime is equal to D. But the question is, where would C now sit?
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Proving the SAS triangle congruence criterion using transformations Geometry Khan Academy.mp3
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So to be able to prove this, in order to make this deduction, we just have to say that there's always a rigid transformation if we have a side, angle, side in common that will allow us to map one triangle onto the other, because if there is a series of rigid transformations that allow us to do it, then by the rigid transformation definition, the true triangles are congruent. So the first thing that we could do is we could reference back to where we saw that if we have two segments that have the same length, like segment AB and segment DE, if we have two segments with the same length, that they are congruent. You can always map one segment onto the other with a series of rigid transformations. The way that we could do that in this case is we could map point B onto point E, so this would be now, I'll put B prime right over here, and if we just did a transformation to do that, so if we just translated like that, then side, whoops, then side BA would, that orange side would be something like that, but then we could do another rigid transformation that rotates about point E, or B prime, that rotates that orange side and the whole triangle with it onto DE, in which case, once we do that second rigid transformation, point A will now coincide with D, or we could say A prime is equal to D. But the question is, where would C now sit? Well, we can see the distance between A and C, in fact, we can use our compass for it. The distance between A and C is, right, is just like that. And so, since all of these rigid transformations preserve distance, we know that C prime, the point that C gets mapped to after those first two transformations, C prime, its distance is going to stay the same from A prime, so C prime is going to be someplace, someplace along this curve right over here.
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Proving the SAS triangle congruence criterion using transformations Geometry Khan Academy.mp3
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The way that we could do that in this case is we could map point B onto point E, so this would be now, I'll put B prime right over here, and if we just did a transformation to do that, so if we just translated like that, then side, whoops, then side BA would, that orange side would be something like that, but then we could do another rigid transformation that rotates about point E, or B prime, that rotates that orange side and the whole triangle with it onto DE, in which case, once we do that second rigid transformation, point A will now coincide with D, or we could say A prime is equal to D. But the question is, where would C now sit? Well, we can see the distance between A and C, in fact, we can use our compass for it. The distance between A and C is, right, is just like that. And so, since all of these rigid transformations preserve distance, we know that C prime, the point that C gets mapped to after those first two transformations, C prime, its distance is going to stay the same from A prime, so C prime is going to be someplace, someplace along this curve right over here. We also know that the rigid transformations preserve angle measures, and so we also know that as we do the mapping, the angle will be preserved. So either side AC will be mapped to this side right over here, and if that's the case, then F would be equal to C prime, and we would have found our rigid transformation based on SAS, and so therefore, the two triangles would be congruent. But there's another possibility that the angle gets conserved, but side AC is mapped down here.
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Proving the SAS triangle congruence criterion using transformations Geometry Khan Academy.mp3
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And so, since all of these rigid transformations preserve distance, we know that C prime, the point that C gets mapped to after those first two transformations, C prime, its distance is going to stay the same from A prime, so C prime is going to be someplace, someplace along this curve right over here. We also know that the rigid transformations preserve angle measures, and so we also know that as we do the mapping, the angle will be preserved. So either side AC will be mapped to this side right over here, and if that's the case, then F would be equal to C prime, and we would have found our rigid transformation based on SAS, and so therefore, the two triangles would be congruent. But there's another possibility that the angle gets conserved, but side AC is mapped down here. So there's another possibility that side AC, due to our rigid transformations, or after our first set of rigid transformations, looks something like this. So it looks, it looks something like that, in which case C prime would be mapped right over there. And in that case, we can just do one more rigid transformation.
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Proving the SAS triangle congruence criterion using transformations Geometry Khan Academy.mp3
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But there's another possibility that the angle gets conserved, but side AC is mapped down here. So there's another possibility that side AC, due to our rigid transformations, or after our first set of rigid transformations, looks something like this. So it looks, it looks something like that, in which case C prime would be mapped right over there. And in that case, we can just do one more rigid transformation. We can just do a reflection about DE, or A prime, B prime, to reflect point C prime over that to get right over there. How do we know that C prime would then be mapped to F? Well, this angle would be preserved due to the rigid transformation.
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Proving the SAS triangle congruence criterion using transformations Geometry Khan Academy.mp3
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And in that case, we can just do one more rigid transformation. We can just do a reflection about DE, or A prime, B prime, to reflect point C prime over that to get right over there. How do we know that C prime would then be mapped to F? Well, this angle would be preserved due to the rigid transformation. So as we flip it over, as we do the reflection over DE, the angle would be preserved, and A prime, C prime will then map to DF. And then we'd be done. We have just shown that there's always a series of rigid transformations, as long as you meet this SAS criteria, that can map one triangle onto the other, and therefore, they are congruent.
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Proving the SAS triangle congruence criterion using transformations Geometry Khan Academy.mp3
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It's transforming from one thing to another. So what would transformation mean in a mathematical context? Well, it could mean that you're taking something mathematical and you're changing it into something else mathematical, and that's exactly what it is. It's talking about taking a set of coordinates or a set of points and then changing them into a different set of coordinates or a different set of points. For example, this right over here, this is a quadrilateral, we've plotted it on the coordinate plane. This is a set of points, not just the four points that represent the vertices of the quadrilateral, but all the points along the sides too. There's a bunch of points along this.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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It's talking about taking a set of coordinates or a set of points and then changing them into a different set of coordinates or a different set of points. For example, this right over here, this is a quadrilateral, we've plotted it on the coordinate plane. This is a set of points, not just the four points that represent the vertices of the quadrilateral, but all the points along the sides too. There's a bunch of points along this. You could argue there's an infinite, or there are an infinite number of points along this quadrilateral. This right over here, the point x equals zero, y equals negative four, this is a point on the quadrilateral. Now, we can apply a transformation to this.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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There's a bunch of points along this. You could argue there's an infinite, or there are an infinite number of points along this quadrilateral. This right over here, the point x equals zero, y equals negative four, this is a point on the quadrilateral. Now, we can apply a transformation to this. The first one I'm going to show you is a translation, which just means moving all the points in the same direction and the same amount in that same direction. I'm using the Khan Academy translation widget to do it. Let's translate this, and I can do it by grabbing onto one of the vertices.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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Now, we can apply a transformation to this. The first one I'm going to show you is a translation, which just means moving all the points in the same direction and the same amount in that same direction. I'm using the Khan Academy translation widget to do it. Let's translate this, and I can do it by grabbing onto one of the vertices. Notice, I've now shifted it to the right by two. Every point here, not just the orange points, has shifted to the right by two. This one has shifted to the right by two.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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Let's translate this, and I can do it by grabbing onto one of the vertices. Notice, I've now shifted it to the right by two. Every point here, not just the orange points, has shifted to the right by two. This one has shifted to the right by two. This point right over here has shifted to the right by two. Every point has shifted in the same direction by the same amount. That's what a translation is.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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This one has shifted to the right by two. This point right over here has shifted to the right by two. Every point has shifted in the same direction by the same amount. That's what a translation is. Now, I've shifted, let's see, if I put it here, everything, every point has shifted to the right one and up one. They've all shifted by the same amount in the same direction. That is a translation, but you can imagine a translation is not the only kind of transformation.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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That's what a translation is. Now, I've shifted, let's see, if I put it here, everything, every point has shifted to the right one and up one. They've all shifted by the same amount in the same direction. That is a translation, but you can imagine a translation is not the only kind of transformation. In fact, there's an unlimited variation, there's an unlimited number of different transformations. For example, I could do a rotation. I have another set of points here that's represented by quadrilateral, I guess we could call it CD or BCDE, and I could rotate it.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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That is a translation, but you can imagine a translation is not the only kind of transformation. In fact, there's an unlimited variation, there's an unlimited number of different transformations. For example, I could do a rotation. I have another set of points here that's represented by quadrilateral, I guess we could call it CD or BCDE, and I could rotate it. I would rotate it around a point. For example, I could rotate it around the point D. This is what I start with. If I, let me see if I can do this.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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I have another set of points here that's represented by quadrilateral, I guess we could call it CD or BCDE, and I could rotate it. I would rotate it around a point. For example, I could rotate it around the point D. This is what I start with. If I, let me see if I can do this. I could rotate it like, actually let me see. If I start like this, I could rotate it 90 degrees. I could rotate 90 degrees.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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If I, let me see if I can do this. I could rotate it like, actually let me see. If I start like this, I could rotate it 90 degrees. I could rotate 90 degrees. I could rotate it like, that looks pretty close to a 90 degree rotation. Every point that was on the original or in the original set of points, I've now shifted it relative to that point that I'm rotating around, I've now rotated it 90 degrees. This point has now mapped to this point over here.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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I could rotate 90 degrees. I could rotate it like, that looks pretty close to a 90 degree rotation. Every point that was on the original or in the original set of points, I've now shifted it relative to that point that I'm rotating around, I've now rotated it 90 degrees. This point has now mapped to this point over here. This point has now mapped to this point over here. I'm just picking the vertices because those are a little bit easier to think about. This point has mapped to this point.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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This point has now mapped to this point over here. This point has now mapped to this point over here. I'm just picking the vertices because those are a little bit easier to think about. This point has mapped to this point. The point of rotation, actually, since D is actually the point of rotation, that one actually has not shifted. Just so you get some terminology, the set of points after you apply the transformation, this is called the image of the transformation. I had quadrilateral BCDE.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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This point has mapped to this point. The point of rotation, actually, since D is actually the point of rotation, that one actually has not shifted. Just so you get some terminology, the set of points after you apply the transformation, this is called the image of the transformation. I had quadrilateral BCDE. I applied a 90 degree counterclockwise rotation around the point D. This new set of points, this is the image of our original quadrilateral after the transformation. I don't have to just, let me undo this, I don't have to rotate around just one of the points that are on the original set, that are on our quadrilateral. I could rotate around the origin.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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I had quadrilateral BCDE. I applied a 90 degree counterclockwise rotation around the point D. This new set of points, this is the image of our original quadrilateral after the transformation. I don't have to just, let me undo this, I don't have to rotate around just one of the points that are on the original set, that are on our quadrilateral. I could rotate around the origin. I could do something like that. Notice it's a different rotation now. It's a different rotation.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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I could rotate around the origin. I could do something like that. Notice it's a different rotation now. It's a different rotation. I could rotate around any point. Let's look at another transformation. That would be the notion of a reflection.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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It's a different rotation. I could rotate around any point. Let's look at another transformation. That would be the notion of a reflection. You know what reflection means in everyday life. You kind of imagine the reflection of an image in a mirror or on the water. That's exactly what we're going to do over here.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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That would be the notion of a reflection. You know what reflection means in everyday life. You kind of imagine the reflection of an image in a mirror or on the water. That's exactly what we're going to do over here. If we reflect, we reflect across a line. Let me do that. This, what is this, one, two, three, four, five, this not irregular pentagon, let's reflect it.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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That's exactly what we're going to do over here. If we reflect, we reflect across a line. Let me do that. This, what is this, one, two, three, four, five, this not irregular pentagon, let's reflect it. To reflect it, let me actually make a line like this. I could reflect it across a whole series of lines. Whoops, let me see if I can.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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This, what is this, one, two, three, four, five, this not irregular pentagon, let's reflect it. To reflect it, let me actually make a line like this. I could reflect it across a whole series of lines. Whoops, let me see if I can. Let's reflect it across this. What does it mean to reflect across something? One way I imagine is if this was, we're going to get its mirror image and you kind of imagine this as kind of the line of symmetry, that the image and the original set, the original shape, they should be mirror images across this line.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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Whoops, let me see if I can. Let's reflect it across this. What does it mean to reflect across something? One way I imagine is if this was, we're going to get its mirror image and you kind of imagine this as kind of the line of symmetry, that the image and the original set, the original shape, they should be mirror images across this line. We could see that. Let's do the reflection. There you go.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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One way I imagine is if this was, we're going to get its mirror image and you kind of imagine this as kind of the line of symmetry, that the image and the original set, the original shape, they should be mirror images across this line. We could see that. Let's do the reflection. There you go. You see we have a mirror image. This is this far away from the line. This corresponding point in the image is on the other side of the line, but the same distance.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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There you go. You see we have a mirror image. This is this far away from the line. This corresponding point in the image is on the other side of the line, but the same distance. This point over here is this distance from the line and this point over here is the same distance, but on the other side. Now all of the transformations that I've just showed you, the translation, the reflection, the rotation, these are called rigid transformations. Once again, you can just think about what does rigid mean in everyday life?
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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This corresponding point in the image is on the other side of the line, but the same distance. This point over here is this distance from the line and this point over here is the same distance, but on the other side. Now all of the transformations that I've just showed you, the translation, the reflection, the rotation, these are called rigid transformations. Once again, you can just think about what does rigid mean in everyday life? It means something that's not flexible. It means something that you can't kind of stretch or scale up or scale down. It kind of maintains its shape.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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Once again, you can just think about what does rigid mean in everyday life? It means something that's not flexible. It means something that you can't kind of stretch or scale up or scale down. It kind of maintains its shape. That's what rigid transformations are fundamentally about. If you want to think a little bit more mathematically, a rigid transformation is one in which lengths and angles are preserved. You can see in this transformation right over here, the distance between this point and this point, between points T and R, and the distance between their corresponding image points, that distance is the same.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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It kind of maintains its shape. That's what rigid transformations are fundamentally about. If you want to think a little bit more mathematically, a rigid transformation is one in which lengths and angles are preserved. You can see in this transformation right over here, the distance between this point and this point, between points T and R, and the distance between their corresponding image points, that distance is the same. The angle here, angle RTY, the measure of this angle over here, if you look at the corresponding angle in the image, it's going to be the same angle. So you might, and the same thing is true if you're doing a translation. You can imagine these are kind of acting like rigid objects.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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You can see in this transformation right over here, the distance between this point and this point, between points T and R, and the distance between their corresponding image points, that distance is the same. The angle here, angle RTY, the measure of this angle over here, if you look at the corresponding angle in the image, it's going to be the same angle. So you might, and the same thing is true if you're doing a translation. You can imagine these are kind of acting like rigid objects. You can't stretch them. They're not flexible. They're maintaining their shape.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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You can imagine these are kind of acting like rigid objects. You can't stretch them. They're not flexible. They're maintaining their shape. Now what would be examples of transformations that are not rigid transformations? Well, you can imagine scaling things up and down. If I were to zoom, if I were to scale this out where it has maybe the angles are preserved, but the lengths aren't preserved, that would not be a rigid transformation.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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They're maintaining their shape. Now what would be examples of transformations that are not rigid transformations? Well, you can imagine scaling things up and down. If I were to zoom, if I were to scale this out where it has maybe the angles are preserved, but the lengths aren't preserved, that would not be a rigid transformation. If I were to just stretch one side of it, or if I were to just pull this point while the other point stayed where they are, I'd be kind of distorting it or stretching it. That would not be a rigid transformation. So hopefully this gets you, it's actually very, very interesting.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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If I were to zoom, if I were to scale this out where it has maybe the angles are preserved, but the lengths aren't preserved, that would not be a rigid transformation. If I were to just stretch one side of it, or if I were to just pull this point while the other point stayed where they are, I'd be kind of distorting it or stretching it. That would not be a rigid transformation. So hopefully this gets you, it's actually very, very interesting. I mean, when you use an art program or actually use a lot of computer graphics or you play a video game, most of what the video game is doing is actually doing transformations, sometimes in two dimensions, sometimes in three dimensions. And once you get into more advanced math, especially things like linear algebra, there's a whole field that's really focused around transformations. In fact, some of the computers with really good graphics processors, a graphics processor is just a piece of hardware that is really good at performing mathematical transformations so that you can immerse yourself in a 3D reality or whatever else.
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Introduction to transformations Transformations Geometry Khan Academy.mp3
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And what I want to do is look at the midpoints of each of the sides of ABC. So this is the midpoint of one of the sides of side BC, let's call that point D. Let's call this midpoint E, and let's call this midpoint right over here F. And since it's the midpoint, we know that the distance between BD is equal to the distance from D to C. So this distance is equal to this distance. We know that AE is equal to EC, so this distance is equal to that distance. And we know that AF is equal to FB, so this distance is equal to this distance. Instead of drawing medians going from these midpoints to the vertices, what I want to do is I want to connect these midpoints and see what happens. So if I connect them, I clearly have three points. So if you connect three nonlinear points like this, you will get another triangle.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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And we know that AF is equal to FB, so this distance is equal to this distance. Instead of drawing medians going from these midpoints to the vertices, what I want to do is I want to connect these midpoints and see what happens. So if I connect them, I clearly have three points. So if you connect three nonlinear points like this, you will get another triangle. And this triangle that's formed from the midpoints of the sides of this larger triangle, we call this a medial triangle. And that's all nice and cute by itself. But what we're going to see in this video is that the medial triangle actually has some very neat properties.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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So if you connect three nonlinear points like this, you will get another triangle. And this triangle that's formed from the midpoints of the sides of this larger triangle, we call this a medial triangle. And that's all nice and cute by itself. But what we're going to see in this video is that the medial triangle actually has some very neat properties. What we're actually going to show is that it divides any triangle into four smaller triangles that are congruent to each other, that all four of these triangles are identical to each other and they're all similar to the larger triangle. And you could think of them each as having 1 4th of the area of the larger triangle. So let's go about proving it.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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But what we're going to see in this video is that the medial triangle actually has some very neat properties. What we're actually going to show is that it divides any triangle into four smaller triangles that are congruent to each other, that all four of these triangles are identical to each other and they're all similar to the larger triangle. And you could think of them each as having 1 4th of the area of the larger triangle. So let's go about proving it. So first, let's focus on this triangle down here, triangle CDE. And it looks similar to the larger triangle, to triangle CBA. But let's prove it to ourselves.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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So let's go about proving it. So first, let's focus on this triangle down here, triangle CDE. And it looks similar to the larger triangle, to triangle CBA. But let's prove it to ourselves. So one thing we can say is, well, look, both of them share this angle right over here. Both the larger triangle, triangle CBA, shares this angle. And the smaller triangle, CDE, has this angle.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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But let's prove it to ourselves. So one thing we can say is, well, look, both of them share this angle right over here. Both the larger triangle, triangle CBA, shares this angle. And the smaller triangle, CDE, has this angle. So they definitely share that angle. And then let's think about the ratios of the sides. We know that the ratio of CD to CB is equal to 1 over 2.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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And the smaller triangle, CDE, has this angle. So they definitely share that angle. And then let's think about the ratios of the sides. We know that the ratio of CD to CB is equal to 1 over 2. This is half of this entire side, is equal to 1 over 2. And that's the same thing as the ratio of CE to CA. CE is exactly half of CA, because E is the midpoint.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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We know that the ratio of CD to CB is equal to 1 over 2. This is half of this entire side, is equal to 1 over 2. And that's the same thing as the ratio of CE to CA. CE is exactly half of CA, because E is the midpoint. It's equal to CE over CA. So we have an angle, and corresponding angles that are congruent. And then the ratios of two corresponding sides on either side of that angle are the same.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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CE is exactly half of CA, because E is the midpoint. It's equal to CE over CA. So we have an angle, and corresponding angles that are congruent. And then the ratios of two corresponding sides on either side of that angle are the same. CD over CB is 1 half. CE over CA is 1 half. And the angle in between is congruent.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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And then the ratios of two corresponding sides on either side of that angle are the same. CD over CB is 1 half. CE over CA is 1 half. And the angle in between is congruent. So by SAS similarity, we know that triangle CDE is similar to triangle CBA. And just from that, you can get some interesting results. Because then we know that the ratio of this side of the smaller triangle to the longer triangle is also going to be 1 half.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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And the angle in between is congruent. So by SAS similarity, we know that triangle CDE is similar to triangle CBA. And just from that, you can get some interesting results. Because then we know that the ratio of this side of the smaller triangle to the longer triangle is also going to be 1 half. Because the other two sides have a ratio of 1 half, and we're dealing with similar triangles. So this is going to be 1 half of that. And we know 1 half of AB is just going to be the length of FA.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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Because then we know that the ratio of this side of the smaller triangle to the longer triangle is also going to be 1 half. Because the other two sides have a ratio of 1 half, and we're dealing with similar triangles. So this is going to be 1 half of that. And we know 1 half of AB is just going to be the length of FA. So we know that this length right over here is going to be the same as FA or FB. And we get that straight from similar triangles. Because these are similar, we know that DE over BA has got to be equal to these ratios, the other corresponding sides, which is equal to 1 half.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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And we know 1 half of AB is just going to be the length of FA. So we know that this length right over here is going to be the same as FA or FB. And we get that straight from similar triangles. Because these are similar, we know that DE over BA has got to be equal to these ratios, the other corresponding sides, which is equal to 1 half. And so that's how we got that right over there. Now let's think about this triangle up here. Triangle, we could call it BDF.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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Because these are similar, we know that DE over BA has got to be equal to these ratios, the other corresponding sides, which is equal to 1 half. And so that's how we got that right over there. Now let's think about this triangle up here. Triangle, we could call it BDF. So first of all, if you compare triangle BDF to the larger triangle, they both share this angle right over here, angle ABC. They both have that angle in common. And we're going to have the exact same argument.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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Triangle, we could call it BDF. So first of all, if you compare triangle BDF to the larger triangle, they both share this angle right over here, angle ABC. They both have that angle in common. And we're going to have the exact same argument. You can just look at this diagram, and you know that the ratio of BA, the ratio of BF to BA is equal to 1 half, which is also the ratio of BD to BC. The ratio of this to that is the same as the ratio of this to that, which is 1 half. Because BD is half of this whole length, BF is half of that whole length.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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And we're going to have the exact same argument. You can just look at this diagram, and you know that the ratio of BA, the ratio of BF to BA is equal to 1 half, which is also the ratio of BD to BC. The ratio of this to that is the same as the ratio of this to that, which is 1 half. Because BD is half of this whole length, BF is half of that whole length. And so you have corresponding sides have the same ratio on the two triangles. And they share an angle in between. So once again, by SAS similarity, we know that triangle, I'll write it this way, DBF is similar to triangle CBA.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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Because BD is half of this whole length, BF is half of that whole length. And so you have corresponding sides have the same ratio on the two triangles. And they share an angle in between. So once again, by SAS similarity, we know that triangle, I'll write it this way, DBF is similar to triangle CBA. And once again, we use this exact same kind of argument that we did with this triangle. Well, if it's similar, the ratio of all the corresponding sides have to be the same. And that ratio is 1 half.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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So once again, by SAS similarity, we know that triangle, I'll write it this way, DBF is similar to triangle CBA. And once again, we use this exact same kind of argument that we did with this triangle. Well, if it's similar, the ratio of all the corresponding sides have to be the same. And that ratio is 1 half. So the ratio of this side to this side, the ratio of FD to AC has to be 1 half. Or FD has to be 1 half of AC. And 1 half of AC is just the length of AE.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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And that ratio is 1 half. So the ratio of this side to this side, the ratio of FD to AC has to be 1 half. Or FD has to be 1 half of AC. And 1 half of AC is just the length of AE. So that is just going to be that length right over there. I think you see where this is going. And also, because it's similar, all of the corresponding angles have to be the same.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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And 1 half of AC is just the length of AE. So that is just going to be that length right over there. I think you see where this is going. And also, because it's similar, all of the corresponding angles have to be the same. And we know that the larger triangle has a yellow angle right over there. So we'd have that yellow angle right over here. And this triangle right over here was also similar to the larger triangle.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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And also, because it's similar, all of the corresponding angles have to be the same. And we know that the larger triangle has a yellow angle right over there. So we'd have that yellow angle right over here. And this triangle right over here was also similar to the larger triangle. So it will have that same angle measure up here. We already showed that in this first part. So now let's go to this third triangle.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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And this triangle right over here was also similar to the larger triangle. So it will have that same angle measure up here. We already showed that in this first part. So now let's go to this third triangle. I think you see the pattern. I'm sure you might be able to just pause this video and prove it for yourself. But we see that the ratio of AF over AB is going to be the same as the ratio of AE over AC, which is equal to 1 half.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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So now let's go to this third triangle. I think you see the pattern. I'm sure you might be able to just pause this video and prove it for yourself. But we see that the ratio of AF over AB is going to be the same as the ratio of AE over AC, which is equal to 1 half. So we have two corresponding sides where the ratio is 1 half from the smaller to the larger triangle. And they share a common angle. They share this angle in between the two sides.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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But we see that the ratio of AF over AB is going to be the same as the ratio of AE over AC, which is equal to 1 half. So we have two corresponding sides where the ratio is 1 half from the smaller to the larger triangle. And they share a common angle. They share this angle in between the two sides. So by SAS similarity, this is getting repetitive now, we know that triangle EFA is similar to triangle CBA. And so the ratio of all of the corresponding sides need to be 1 half. So the ratio of FE to BC needs to be 1 half, or FE needs to be half of that, which is just the length of BD.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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They share this angle in between the two sides. So by SAS similarity, this is getting repetitive now, we know that triangle EFA is similar to triangle CBA. And so the ratio of all of the corresponding sides need to be 1 half. So the ratio of FE to BC needs to be 1 half, or FE needs to be half of that, which is just the length of BD. So this is just going to be that length right over there. And you can also say that this triangle, this triangle, and this triangle, we haven't talked about this middle one yet, they're all similar to the larger triangle so they're also all going to be similar to each other. So they're all going to have the same corresponding angles.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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So the ratio of FE to BC needs to be 1 half, or FE needs to be half of that, which is just the length of BD. So this is just going to be that length right over there. And you can also say that this triangle, this triangle, and this triangle, we haven't talked about this middle one yet, they're all similar to the larger triangle so they're also all going to be similar to each other. So they're all going to have the same corresponding angles. So if the larger triangle had this yellow angle here, then all of the triangles are going to have this yellow angle right over there. As the larger triangle had this blue angle, right over here, then in the corresponding vertex, all of the triangles are going to have that blue angle. All of the ones that we've shown are similar.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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So they're all going to have the same corresponding angles. So if the larger triangle had this yellow angle here, then all of the triangles are going to have this yellow angle right over there. As the larger triangle had this blue angle, right over here, then in the corresponding vertex, all of the triangles are going to have that blue angle. All of the ones that we've shown are similar. We haven't thought about this middle triangle just yet. And of course, if this is similar to the whole, it'll also have this angle at this vertex right over here, because this corresponds to that vertex based on the similarity. So that's interesting.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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All of the ones that we've shown are similar. We haven't thought about this middle triangle just yet. And of course, if this is similar to the whole, it'll also have this angle at this vertex right over here, because this corresponds to that vertex based on the similarity. So that's interesting. Now let's compare the triangles to each other. We've now shown that all of these triangles have the exact same three sides. It has this blue side, or actually, I don't know, this one marked side, this two marked side, this three marked side, one marked, two marked, three marked, one marked, two marked, three marked.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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So that's interesting. Now let's compare the triangles to each other. We've now shown that all of these triangles have the exact same three sides. It has this blue side, or actually, I don't know, this one marked side, this two marked side, this three marked side, one marked, two marked, three marked, one marked, two marked, three marked. And that even applies to this middle triangle right over here. So by side, side, side congruency, we now know, and we want to be careful to get our corresponding sides right, we now know that triangle CDE is congruent to triangle DBF, I want to get the corresponding sides. I'm looking at the colors.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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It has this blue side, or actually, I don't know, this one marked side, this two marked side, this three marked side, one marked, two marked, three marked, one marked, two marked, three marked. And that even applies to this middle triangle right over here. So by side, side, side congruency, we now know, and we want to be careful to get our corresponding sides right, we now know that triangle CDE is congruent to triangle DBF, I want to get the corresponding sides. I'm looking at the colors. I went from yellow to magenta to blue, yellow, magenta to blue, which is going to be congruent to triangle EFA, which is going to be congruent to this triangle in here, but we want to make sure that we're getting the right corresponding sides here. So to make sure we do that, we just have to think about the angles. So we know, and this is interesting, because the interior angles of a triangle add up to 180 degrees, we know this magenta angle plus this blue angle plus this yellow angle equal 180.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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I'm looking at the colors. I went from yellow to magenta to blue, yellow, magenta to blue, which is going to be congruent to triangle EFA, which is going to be congruent to this triangle in here, but we want to make sure that we're getting the right corresponding sides here. So to make sure we do that, we just have to think about the angles. So we know, and this is interesting, because the interior angles of a triangle add up to 180 degrees, we know this magenta angle plus this blue angle plus this yellow angle equal 180. Here, we have the blue angle and the magenta angle, and clearly they will all add up to 180, so you must have the blue angle must be right over here. Same argument, yellow angle and blue angle, we must have the magenta angle right over here. They add up to 180, so this must be the magenta angle.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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So we know, and this is interesting, because the interior angles of a triangle add up to 180 degrees, we know this magenta angle plus this blue angle plus this yellow angle equal 180. Here, we have the blue angle and the magenta angle, and clearly they will all add up to 180, so you must have the blue angle must be right over here. Same argument, yellow angle and blue angle, we must have the magenta angle right over here. They add up to 180, so this must be the magenta angle. And then finally, magenta and blue, this must be the yellow angle right over there. And so when we wrote the congruency here, we started at CDE. We went yellow, magenta, blue.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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They add up to 180, so this must be the magenta angle. And then finally, magenta and blue, this must be the yellow angle right over there. And so when we wrote the congruency here, we started at CDE. We went yellow, magenta, blue. So over here, we're going to go yellow, magenta, blue. So it's going to be congruent to triangle FED. And so that's pretty cool.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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We went yellow, magenta, blue. So over here, we're going to go yellow, magenta, blue. So it's going to be congruent to triangle FED. And so that's pretty cool. We just showed that all three, that this triangle, this triangle, this triangle, and that triangle are congruent. And also, we can look at the corresponding, and that they all have ratios relative to the large. They're all similar to the larger triangle, the triangle ABC, and that the ratio between the sides is 1 to 2.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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And so that's pretty cool. We just showed that all three, that this triangle, this triangle, this triangle, and that triangle are congruent. And also, we can look at the corresponding, and that they all have ratios relative to the large. They're all similar to the larger triangle, the triangle ABC, and that the ratio between the sides is 1 to 2. And also, because we've looked at corresponding angles, we see, for example, that this angle is the same as that angle. So if you viewed DC or if you viewed BC as a transversal, all of a sudden, it becomes pretty clear that FD is going to be parallel to AC because the corresponding angles are congruent. So this is going to be parallel to that right over there.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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They're all similar to the larger triangle, the triangle ABC, and that the ratio between the sides is 1 to 2. And also, because we've looked at corresponding angles, we see, for example, that this angle is the same as that angle. So if you viewed DC or if you viewed BC as a transversal, all of a sudden, it becomes pretty clear that FD is going to be parallel to AC because the corresponding angles are congruent. So this is going to be parallel to that right over there. And then you could use that same exact argument to say, well, then this side, because once again, corresponding angles here and here, you could say that this is going to be parallel to that right over there. And then finally, you make the same argument over here. You have, I want to make sure I get the right corresponding angles, you have this line and this line and this angle corresponds to that angle.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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So this is going to be parallel to that right over there. And then you could use that same exact argument to say, well, then this side, because once again, corresponding angles here and here, you could say that this is going to be parallel to that right over there. And then finally, you make the same argument over here. You have, I want to make sure I get the right corresponding angles, you have this line and this line and this angle corresponds to that angle. They're the same. So this DE must be parallel to BA. So that's another neat property of this medial triangle.
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Exploring medial triangles Special properties and parts of triangles Geometry Khan Academy.mp3
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So if I have a sphere, so this isn't just a circle, this is a sphere, you could view it as a globe of some kind, so I'm going to shade it a little bit so you can tell that it's three dimensional. They're giving us the diameter, so if we go from one side of the sphere straight through the center of it, so we're imagining that we can see through the sphere, and we go straight through the centimeter, that distance right over there is 14 centimeters. Now, to find the volume of a sphere, we prove this, or you will see a proof for this later when you learn calculus, but the formula for the volume of a sphere is volume is equal to 4 thirds pi r cubed, where r is the radius of the sphere. So they've given us the diameter, and just like for circles, the radius of the sphere is half of the diameter. So in this example, our radius is going to be 7 centimeters. In fact, the sphere itself is a set of all points in three dimensions that is exactly the radius away from the center. But with that out of the way, let's just apply this radius being 7 centimeters to this formula right over here.
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Volume of a sphere Perimeter, area, and volume Geometry Khan Academy.mp3
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So they've given us the diameter, and just like for circles, the radius of the sphere is half of the diameter. So in this example, our radius is going to be 7 centimeters. In fact, the sphere itself is a set of all points in three dimensions that is exactly the radius away from the center. But with that out of the way, let's just apply this radius being 7 centimeters to this formula right over here. So we're going to have a volume is equal to 4 thirds pi times 7 centimeters to the third power. So I'll do that in that pink color. So times 7 centimeters to the third power.
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Volume of a sphere Perimeter, area, and volume Geometry Khan Academy.mp3
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But with that out of the way, let's just apply this radius being 7 centimeters to this formula right over here. So we're going to have a volume is equal to 4 thirds pi times 7 centimeters to the third power. So I'll do that in that pink color. So times 7 centimeters to the third power. And since it already involves pi, and you can approximate pi with 3.14, some people even approximate it with 22 over 7, but we'll actually just get the calculator out to get the exact value for this volume. So this is going to be, so my volume is going to be 4 divided by 3, and then I don't want to just put a pi there, because that might interpret it as 4 divided by 3 pi. So 4 divided by 3 times pi times 7 to the third power.
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Volume of a sphere Perimeter, area, and volume Geometry Khan Academy.mp3
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So times 7 centimeters to the third power. And since it already involves pi, and you can approximate pi with 3.14, some people even approximate it with 22 over 7, but we'll actually just get the calculator out to get the exact value for this volume. So this is going to be, so my volume is going to be 4 divided by 3, and then I don't want to just put a pi there, because that might interpret it as 4 divided by 3 pi. So 4 divided by 3 times pi times 7 to the third power. In order of operations, it'll do the exponent before it does the multiplication, so this should work out. And the units are going to be in centimeters cubed, or cubic centimeters. So we get 1436, they don't tell us what to round it to, so I'll just round it to the nearest tenth.
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Volume of a sphere Perimeter, area, and volume Geometry Khan Academy.mp3
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So we have these two parallel lines, line segment A, B, and line segment C, D. They are parallel, I should say, they're parallel line segments. And then we have these transversals that go across them. So you have this transversal B, C right over here, and you have this transversal A, D. And what this diagram tells us is that the distance between A and E, this little hash mark, says that this line segment is the same distance as the distance between E and D. Or another way to think about it is that point E is at the midpoint, or is the midpoint of line segment A, D. And what I want to think about in this video is, is point E also the midpoint of line segment B, C? So this is the question right over here. So is E the midpoint of line segment B, C? And you can imagine, based on a lot of the videos we've been seeing lately, maybe it has something to do with congruent triangles. So let's see if we can set up some congruency relationship between the two obvious triangles in this diagram.
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Congruent triangle proof example Congruence Geometry Khan Academy.mp3
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So this is the question right over here. So is E the midpoint of line segment B, C? And you can imagine, based on a lot of the videos we've been seeing lately, maybe it has something to do with congruent triangles. So let's see if we can set up some congruency relationship between the two obvious triangles in this diagram. We have this triangle up here on the left, and we have this diagram down here. This one kind of looks like it's pointing up, this one looks like it's pointing down. So there's a bunch of things we know about vertical angles and angles of transversals.
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Congruent triangle proof example Congruence Geometry Khan Academy.mp3
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So let's see if we can set up some congruency relationship between the two obvious triangles in this diagram. We have this triangle up here on the left, and we have this diagram down here. This one kind of looks like it's pointing up, this one looks like it's pointing down. So there's a bunch of things we know about vertical angles and angles of transversals. The most obvious one is that we have this vertical. We know that angle AEB is going to be congruent, or its measure is going to be equal to the measure of angle CED. So we know that angle AEB is going to be congruent to angle DEC, which really just means they have the exact same measure.
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Congruent triangle proof example Congruence Geometry Khan Academy.mp3
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