Dilations and properties

- [Instructor] What we're going to do in this video is think about how shapes' properties might be preserved or not preserved from dilations and so here we have this quadrilateral and we're going to dilate it about point P here and 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. 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. 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 'cause it corresponds to point B, the segment B prime C prime, this does not sit on the same line as BC but the segment 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? 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 can 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's getting larger and the area's 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 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? 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. 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 AB. 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 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. As we dilate out like this, the perimeter grows and so does the area. If we dilate in like this, the perimeter shrinks and so does the area.