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## Geometry (all content)

### Unit 12: Lesson 1

Definitions of similarity# Similar shapes & transformations

CCSS.Math: ,

Sal is given a pair of polygons, and then he determines whether they are similar by trying to map one onto the other using angle-preserving transformations. Created by Sal Khan.

## Video transcript

- [Instructor] We are
told that Shui concluded the quadrilaterals, these two over here, have four pairs of congruent
corresponding angles. We can see these right over there. And so, based on that she concludes that the figures are similar. What error if any, did Shui
make in her conclusion? Pause this video and try to
figure this out on your own. All right, so let's just remind ourselves one definition of similarity that we often use on geometry class, and that's two figures are similar is if you can through a series of rigid transformations and dilations, if you can map one figure onto the other. Now, when I look at these two figures, you could try to do something. You could say okay, let me shift it so that K gets mapped onto H. And if you did that, it looks like L would get mapped onto G. But these sides KN and LM right over here, they seem a good bit longer. So, and then if you try to dilate it down so that the length of KN is
the same as the length of HI well then the lengths of KL
and GH would be different. So it doesn't seem like you could do this. So it is strange that Shui
concluded that they are similar. So let's find the mistake. I'm already, I'll already rule out C, that it's a correct conclusion 'cause I don't think they are similar. So let's see. Is the error that a rigid
transformation, a translation would map HG onto KL? Yep, we just talked about that. HG can be mapped onto KL so the quadrilaterals are
congruent, not similar. Oh, choice A is making an
even stronger statement because anything that is
congruent is going to be similar. You actually can't have
something that's congruent and not similar. And so, choice A does not make any sense. So our deductive reasoning
tells us it's probably choice B. But let's just read it. It's impossible to map quadrilateral GHIJ onto quadrilateral LKNM using only rigid transformations and dilations so the figures are not similar. Yeah, that's right. You could try, you could map HG onto KL, but then segment IJ would
look something like this, IJ would go right over here. And then, if you tried to dilate it, so that the length of HI
and GJ matched KN or LM, then you're gonna make HG bigger as well. So, you're never gonna be able
to map them onto each other even if you can use dilations. So I like choice B.