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## High school geometry

### Course: High school geometry > Unit 3

Lesson 1: Transformations & congruence# Non-congruent shapes & transformations

CCSS.Math: ,

A student reasons about whether a pair of circles are congruent by trying to map one onto the other using rigid transformations. Created by Sal Khan.

## Want to join the conversation?

- When I take the test (at school) on this topic, we're not going to have the tools that you have when you showed us. So my question is, how do you know if the two shapes are congruent without using the tools?

In other words, is there a way to use the graph coordinates to figure out if the two shapes are congruent or not??(16 votes)- i think even though this comment was a while ago you can take the dimensions of one and the dimensions of the other and see if they are the exact same(7 votes)

- Is radi plural for radius?(4 votes)
- Yes, but the correct spelling is radii (ray·dee·ai).(7 votes)

- Would it make a difference if she instead first dilated then translated the circle?(3 votes)
- It doesn't matter. Dilation is not a rigid transformation and will not conserve congruence.(8 votes)

- Is there such thing as a non-rigid transformation?(4 votes)
- Yes, most transformations of the plane are non-rigid. The transformation which maps each point (x, y) to (x²+y², xy) is non-rigid, since it doesn't map straight lines to straight lines.(4 votes)

- is there other basic rigid motions other than reflect,translate, and rotate?

as said in1:07(4 votes)- Those three translations are the three basic geometric translations besides dilation.(3 votes)

- When reflecting a figure in a plane, do the images have to be exactly the same size as their preimages? Cant they be a few centimeters bigger or smaller?(4 votes)
- If the images weren't the same size as their preimages, they would not be part of the same transformation. When you reflect a figure across a axis, the on ly thing changing is the orientation of the figure, not the size or shape.(3 votes)

- how do you find out if 2 are similar or congruent on regular pencil and paper?(3 votes)
- You can use distance formula to prove that the sides are congruent, and that if the sides are congruent, then the shapes are congruent, but that takes a long time and is annoying. Later you will learn a bunch of postulates that prove congruence.(3 votes)

- What is exactly a translation and a dilation?(3 votes)
- EDIT

I messed up with my explanation of dilation here, so look at my next response in comments. Thanks to david severin for pointing it out.

translation is moving up, down, left, right or a combination, then a dilation is more complicated

Dilation is basically stretching or compressing, so making a graph wider, or longer or skinner or something. for example, tae the simple x^2. if you had 4x^2 the graph would be stretched upward by a factor of 4, so it would get skinnier. if instead it was (2x)^2 this is a horizontal compress by a factor of 2. It's worth noting these two are the same, so a vertical stretch is a horizontal compress and vice versa.

What do they mean? well a vertical stretch means the point (2,4) will increase vertically by quadrupling the y value, so it turens to (2,16). so the y vlue is multiplied. a compress divides the coordinates. so horizontal compress would cut the x value in half so (2,4) would become (1,4)

Let me know if this doesn't make sense.(3 votes)

- they say at2:03nothing(3 votes)
- Because that is after the video ended. ඞ(3 votes)

- i did the same question on the next practice(3 votes)
- This link could give you a variety of questions if you keep doing the problems.

https://www.khanacademy.org/math/basic-geo/basic-geo-transformations-congruence/congruent-similar/e/exploring-rigid-transformations-and-congruence

Hope this helps!

- Sam(3 votes)

## Video transcript

- [Instructor] We are told,
Brenda was able to map circle M onto circle N using a translation and a dilation. This is circle M right over here. Here's the center of it. This is circle M, this
circle right over here. It looks like at first, she translates it. The center goes from this
point to this point here. After the translation, we have the circle right over here. Then she dilates it. The center of dilation
looks like it is point N. She dilates it with some
type of a scale factor in order to map it exactly onto N. That all seems right. Brenda concluded, "I
was able to map circle M "onto circle N using a sequence "of rigid transformations, "so the figures are congruent." Is she correct? Pause this video and think about that. Let's work on this together. She was able to map circle M onto circle N using a sequence of transformations. She did a translation and then a dilation. Those are all transformations, but they are not all
rigid transformations. I'll put a question mark right over there. A translation is a rigid transformation. Remember, rigid transformations are ones that preserve distances, preserve angle measures, preserve lengths, while a dilation is not
a rigid transformation. As you can see very clearly, it is not preserving lengths. It is not, for example,
preserving the radius of the circle. In order for two figures to be congruent, the mapping has to be only
with rigid transformations. Because she used a dilation, in fact, you have to use a dilation if you wanna be able to map M onto N because they have different radii, then she's not correct. These are not congruent figures. She cannot make this conclusion.