Dilations and properties

We are told that quadrilateral ABCD is dilated about point P. So, this is our quadrilateral that's going to be dilated around point P, and then they ask us some questions: Are the coordinates of the vertices preserved? Are corresponding line segments on the same line? Are angle measures preserved? Are side lengths, perimeter, and area preserved?

So, pause this video and see if you can answer it.

So now let's work through it together. One thing you might be thinking is they're just saying it's dilated. They didn't say what the scale factor is. We know it's dilated about point P, but what's the scale factor? Well, since they didn't tell us, I guess these questions would apply to any scale factor. But to help us visualize this, let's just apply a scale factor here. Let's just say that our scale factor—scale factor, so it fits on this page—is equal to one-half. So, what would result here? Well, right now point A is 1, 2, 3, 4, 5, 6, 7, 8 away from point B. So, its image, if we do a scale factor of one-half, would be four away: 1, 2, 3, 4. So, right over there, point A prime could be right over there. It would be along the same line; it would just be half as far.

Point D is right now 1, 2, 3, 4, 5, 6 away, so D prime would be three away, half as far away. And so, this would be the segment A prime D prime, which corresponds to the segment AD. Now let's see point B. If we have a scale factor of one-half, it is 1, 2, 3, 4, 5, 6 below A, but if we have a scale factor of one-half, it's going to be 3 below A. So, this would be our B prime. And so, you could also see that if you connect a line going from point P to B, that B prime is also on that line; it's just half as far—scale factor of one-half.

Now, point C—there's a couple of ways we could think about it: to go from point P to point C, we have to go four down and to the left. So, if you apply a scale factor of one-half, you go 2 down and 1 to the left. So, this would be C prime right over here. And so, A prime, B prime, C prime, D prime—this is the image of quadrilateral ABCD after the dilation around point B.

I picked an arbitrary scale factor here that I felt that I could draw; you see it right over here. The reason why I did that is it makes it a little bit easier to answer this question: Are the coordinates of the vertices preserved? Well, every vertex—all the vertices are different coordinates here, so we would say no, none of those are preserved.

And the whole idea here is to appreciate what gets preserved or doesn't get preserved under dilations. Are corresponding line segments on the same line? Well, let's see—no, they're all parallel to each other, so AB is parallel to A prime B prime, AD is parallel to A prime D prime, and so on and so forth, but they're not all—none of them are on the same line, so I would say no.

And then we say, are angle measures preserved? So, this is something that dilations do preserve. So, this angle is still the same as that angle; this was a right angle before—it's still a right angle. This angle is still equal to that angle, or the measure of those angles, likewise the measure of that angle is still equal to the measure of that angle. So, angle measures are preserved. That's a general thing that you could take away about dilations.

Are side lengths, perimeter, and area preserved? Well, dilations are a non-rigid transformation, which tells you that side lengths are not preserved. The distance between corresponding points are not preserved, and you could see it very clearly here. And if side lengths aren't preserved, then perimeter is definitely not preserved, and then area would not be preserved. So, this would be no for all of these. Under a dilation, if we were talking about a rigid transformation like a translation or a rotation, then these three would be preserved.

Let's do another example. So, we're told triangle ABC is dilated about point C. Once again, they don't tell us what the scale factor is, and then they ask us the same four questions. So, like before, pause this video, see if you can figure it out on your own.

Now let's work through it together, and I'm just going to apply a scale factor—in fact, you know, might as well do a scale factor of one-half. So, scale factor—I'll just do SF for short—let's say it's equal to one-half. And I'm just picking one so that we can visualize what its image would look like after the dilation.

So, let's see—right now, remember this is interesting because we're dilating around a point that happens to be one of the vertices of the triangle. So, let's see point A. To go from C to A, we go 3 to the left and we go 2, 4, 6 up—6 up, just like that. So, 3 to the left and then 6 up. If you have a scale factor of one-half—oh, sorry, not two to the left, three to the left—three to the left and 6 up.

And so, if you have a scale factor of one-half, instead you're gonna go 1.5 to the left and then 3 up. And so, you're going to end up right over there—now this is A prime. Notice this is actually sitting on the segment AC, and that happens to be the case because, once again, our center of dilation is one of the vertices of our actual triangle.

Now let's look up—think about where B prime would be. To go from C to B, we go 2 to the right and then we go 4 up—plus 4. So, if we are talking about a scale factor of one-half, we'll go one-half as far to the right and one-half as far up. So, one to the right and two up. So, this would be where B prime is, and so there we have the image. And of course, C would map to C prime because it's actually the point that we are dilating about.

So, this would be our image right over here: A prime, B prime, and C prime. So, are the coordinates of the vertices preserved? Well, only C is preserved; the other ones are not preserved. The only reason why C was preserved is because that is the point we are dilating about.

I apologize for this; this is the third time I said it. Our corresponding line segments on the same line—well, A prime C prime, same—on the same line, I guess you could say, on the same line as AC. Similarly, C prime B prime—or I could say B prime C prime, same—I'm just going to use a little shorthand. Once again, that is because we are dilating about a point that is one of the vertices.

The other segment A prime B prime is not—A prime B prime—it’s not on the same line; it is parallel. This is parallel to that, but it is not on the same line. Are angle measures preserved? Well, we don't even have to look at this; this is going to be a yes. This is one of the things about dilation: angle measures are preserved. That angle is equal to that angle; the measure of this angle is congruent to that angle, or this angle is congruent to that angle, and of course, this is the same angle in both cases.

Are side lengths, perimeter, and area preserved? Once again, no. When we're doing a dilation, side lengths—and the side lengths aren't preserved; perimeter is not going to be preserved, and area is not going to be preserved. And we're done.