Dilations and shape properties

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. 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 that the coordinates are not preserved. The point that corresponds to D now has different coordinates. The vertex that corresponds to A now has different coordinates; the 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: Are the corresponding line segments after dilation sitting on the same line? Let me dilate again. You can see if you consider this point B prime, because it corresponds to point B, the segment B prime C prime does not sit on the same line as BC. But the segment A prime D prime, the corresponding line segment to line segment AD, does sit on the same line.

If you think about it, why is that? Well, if we originally draw a line that contains segment AD, it also goes through point P. As we expand out, this segment right over here is going to expand and shift outward along the same line. But that's not going to be true of these other segments because the point P does not sit on the line that those segments sit on.

Let's just expand it again. 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. You're going to preserve angle measures. This angle is still a right 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.

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 corresponding segments are getting larger. 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. 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 I have the same tool again. Now here we have a triangle, triangle ABC, and we're going to 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. 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 the corresponding line segments? Are they on the same line? Well, some of them are, and some of them aren't. For example, when we dilate, look at the segment AC and the segment BC. 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. 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 is the exact same angle. So is the measure of angle—you could call this A prime and B prime right over here. Finally, what about side lengths? You can clearly see that when I dilate out, my side lengths increase, or if I dilate in, my side lengths decrease.

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. If 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.