Shape properties after a sequence of transformations

In past videos, we've thought about whether segment lengths or angle measures are preserved with a transformation. What we're now going to think about is what is preserved with a sequence of transformations, and in particular, we're going to think about angle measure, angle measure, and segment lengths.

So if you're transforming some type of a shape, segment segment lengths. So let's look at this first example. They say a sequence of transformations is described below. So we first do a translation, then we do a reflection over a horizontal line pq, then we do a vertical stretch about pq.

What is this going to do? Is this going to preserve angle measures, and is this going to preserve segment lengths? Well, a translation is a rigid transformation, and so that will preserve both angle measures and segment lengths. So after that, angle measures and segment lengths are still going to be the same.

A reflection over a horizontal line pq, well, a reflection is also a rigid transformation, and so we will continue to preserve angle measure and segment lengths. Then they say a vertical stretch about pq. Well, let's just think about what a vertical stretch does.

So if I have some triangle right over here, if I have some triangle that looks like this, it says triangle ABC. And if you were to do a vertical stretch, what's going to happen? Well, let's just imagine that we take these sides and we stretch them out so that we now have a is over here or a prime I should say is over there. Let's say that b prime is now over here. This isn't going to be exact.

Well, what just happened to my triangle? Well, the measure of angle C is for sure going to be different now, and my segment lengths are for sure going to be different now. a prime c prime is going to be different than ac in terms of segment length. So a vertical stretch, if we're talking about a stretch in general, this is going to preserve neither. So neither preserved, neither preserved.

So in general, if you're doing rigid transformation after rigid transformation, you're going to preserve both angles and segment lengths. But if you throw a stretch in there, then all bets are off; you're not going to preserve either of them.

Let's do another example. A sequence of transformations is described below, and so they give three transformations. So pause this video and think about whether angle measures, segment lengths, or well, either both, or neither, or only one of them be preserved.

All right, so first we have a rotation about a point P. That's a rigid transformation; it would preserve both segment lengths and angle measures. Then you have a translation, which is also a rigid transformation, and so that would preserve both again. Then we have a rotation about point P, so once again another rigid transformation. So in this situation, everything is going to be preserved.

So both angle measure, angle measure, and segment length are going to be preserved in this example. Let's do one more example. So here once again we have a sequence of transformations, so pause this video again and see if you can figure out whether angle measures, segment lengths, both, or neither are going to be preserved.

So the first transformation is a dilation. So dilation is a non-rigid transformation, so segment lengths not preserved; segment lengths not preserved. And we've seen this in multiple videos already, but in a dilation, angles are preserved; angles preserved. So already we've lost our segment lengths, but we still got our angles.

Then we have a rotation about another point Q, so this is a rigid transformation; it would preserve both. But we've already lost our segment lengths, but angles are going to continue to be preserved. And then finally, a reflection, which is still a rigid transformation, and it would preserve both.

But once again, our segment lengths got lost through the dilation, but we will continue to preserve the angles. So in this series of, after these three transformations, the only thing that's going to be preserved are going to be your angles.