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### Course: 8th grade (Illustrative Mathematics)>Unit 2

Lesson 2: Lesson 6: Similarity

# Similar shapes & transformations

Two shapes are similar if we can change one shape into the other using rigid transformations (like moving or rotating) and dilations (making it bigger or smaller). Other kinds of transformations can change the angles or the ratios of lengths in a figure. If we need those types of transformations, the shapes are not similar. Created by Sal Khan.

## Want to join the conversation?

• Is there a video where Sal formally defines "Dilation"?
• I practiced the concept and it is totally different on what you have to do... help?
• This video is fairly old, the exercise has probably been updated since the video was recorded. However, I'm sure it shouldn't be too different.
1 minute later after Ayaka has rewatched the video and looked at the exercise
It is that much different x.x
-Translation: Instead of dragging a shape around, it requires you to enter in how much you want to move it by. Remember, x comes first, y second. So putting in (-5, 6) would move it 5 to the left and 6 up.
-Rotation: Rather than dragging the arrow around, it wants you to put in where you're rotating it about. Best to enter in the coordinates of one of the points of your shape, preferably the one that overlaps a point of the other shape. Then put in how many degrees you want to rotate your shape by. There are 360 degrees in a circle, so entering in 360 will bring it all the way round. 180 will effectively flip it over and 90 will bring it round by a quarter, or half of a half.
-Reflection: Reflection baffled even me at first glance. It is extremely hard to use! Imagine a line segment coming from the first set of coordinates you give, ending at the second set. The shape you have will flip over that line.
-Dilation: The coordinates are the coordinates of where you would have put the center of the original circle dilation tool. The third number you put in is how much to dilate by. If you say 2, the shape will be 2 times as big. If you say 0.5, the shape will be 0.5 times as big.
-Rage quitting: Don't.
• So two figures are congruent if you can map them onto each other using a series of rigid transformations so that corresponding parts are equal to each other, and two figures are similar if you can map them onto each other using rigid transformations and a dilation. Sal mentions that all congruent shapes are similar, but aren't congruent shapes limited to rigid transformations? How is this possible? Please correct me If I'm missing something, thanks!
• Hi RN,
I think that what Sal meant was all congruent shapes are similar but not all similar shapes are congruent if that isn't too confusing.
For instance, you can map 2 congruent triangles onto each other with rigid transformations and technically, that would make it similar too, but you couldn't map 2 similar triangles onto each other without using a non-rigid transformation which would mean the triangles aren't congruent.
I think a shape doesn't have to be dilated in order to be similar but all dilations cause shapes that are similar.
I hope this helps!
(Let me know if it doesn't or if I'm wrong.)
• What is an example of a transformation that is not a similarity transformation?
• Good question!
I'm not sure if there even is one. You can apply all geometric transformation for similarity, so I guess the transformations that are in geometry are also similarity transformations. Hope this helped!
• I am doing a problem that asks me to determine which of two people are correct in solving a problem. The possible solutions that I have to choose from are transformation and glide-reflection. I understand what transformation is and that it is most likely the correct answer, but what is glide-reflection? I have never heard of it, and I need to explain why this would not be the correct way to solve the problem. Is there a video where Sal explains glide-reflections? Thanks!
• Yes, glide reflection is less commonly taught in school. A glide reflection is a sequence of transformations consisting of a reflection across a line and a translation in a direction parallel to that line (in either order).

Have a blessed, wonderful day!
• So i guess a vertical stretch wouldn't preserve anything, right?

Or is that the same as dilation? it seems as though if sal or "shui" could use a vertical stretch in the problem, but does that not keep similarity?

thanks :)
• Vertical stretch occurs when a base graph is multiplied by a certain factor that is greater than 1. This results in the graph being pulled outward but retaining the input values (or x). When a function is vertically stretched, we expect its graph’s y values to be farther from the x-axis.
I don't think this will help, but it might.
Calc-Ya-Later!
• Couldn't find anything on the web about there being a fifth transformation, but the answer to this problem speaks as if there are more transformations past dilation,(-) so does anyone know if there are more non-rigid transformations besides dilation or not? Or do they just mean that a rotation or reflection would have to go either replace and/or go into play?
• The one you're thinking of might be glide reflection, which is a translation followed by a reflection across that line of translation. But there are no other transformations of this type. Every similarity can be written as a single dilation, followed by a single reflection, translation, glide reflection, or rotation.
• Are there formulas to find similarity & congruence instead of using the tools?
• Yes congruence can be determined with postulates such as SAA SSS SAS and sometimes SSA. Similarity can be determined with postulates such as AAA or AA. I hope this helps.
• Okay, so the shapes have to have the exact same side sizes to be congruent, right?

I was doing some of the exercises on congruency and similarity yesterday, but I kept on getting the answer wrong, and then I changed and it still didn't like me. I think you might need to change them or something.
• do what Arnaud Jasperse said but I like how you said that you changed the answer but it still didnt like you lol
• What is congruence
• congruent shapes have the same number of sides, angles between each sides and sides that match up in length.

Essentially a shape is congruent only if you rotate it or reflect it.

## 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.