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### Course: Integrated math 1>Unit 15

Lesson 1: Introduction to rigid transformations

# Dilations intro

Dilations are a type of non-rigid transformation. They change the size of a shape by scaling it up or down, making it bigger or smaller. Unlike rigid transformations, dilations do not keep the shape's size the same.

## Want to join the conversation?

• What is the formula for dilations?
(19 votes)
• If the point (x, y) is dilated by a factor of c about the origin, its image is the point (cx, cy).
More generally, if the point (x, y) is dilated by a factor of c about the point (a,b), its image is the point (c(x - a) + a, c(y - b) + b).
(30 votes)
• What does a translation have to do with dilations? from yours truly Aleiah
(16 votes)
• they are different kinds of transformations, dialations are non-rigid transformations because they do not keep the length of the shape, and translation does. In pre-algebra, you have to do an assignment where you create a reflection, translation, dilation, and rotations on a shape of your own choosing. Hope this helps! -Jay
(8 votes)
• what the difference between rotation and dilation?
(8 votes)
• Rotation means you turn/rotate the shape around a point.

Dilation means enlarging or shrinking the shape.
(17 votes)
• can you have a reflection, dilation and a rigid transformation all at the same time?
(8 votes)
• Yes, you can apply any amount of these at once, think of having a equal triangle on a grid, you can flip it, move it up a bit then stretch out one corner and rotate it so it points in a different direction then move it again.
Whilst it will get harder to keep track the more you go on, it's possible to do as many as you want
(3 votes)
• So dilation is changing the length but preserving the measure of the angles, right?
(6 votes)
• This is almost correct - angles are congruent. The exception to your statement is a scale factor of 1 which is a dilation, but it is the same length so it would also be congruent. Less than 1 is a reduction and greater than 1 it is an enlargement.
(6 votes)
• What is the difference between rigid and nonrigid transformation?
(5 votes)
• A rigid transformation is a transformation that preserves the side lengths.

The more technical way of saying this is that a rigid transformation is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points.

Rigid transformations include translations, rotations, and reflections.
Non-rigid transformations include scaling/dilating.

Hope this helps!
- Convenient Colleague
(7 votes)
• What about angles in a non-rigid transformation? I thought that the angles wouldn't be preserved either, but when Sal was dilating the triangle, I noticed that even though the sides of the figure was getting bigger and smaller, the angles didn't.
(5 votes)
• yeah, i think that angles are preserved in dilation of a shape because Sal was just saying that in dilation which just scale up or down.
(6 votes)
• Are there other kinds of non-rigid transformations besides dilations?
(4 votes)
• Yep! There's a transformation called "shear". Simply put, imagine taking a square and fixing the base of it. Then, push it in one direction to make it a rhombus. This also transforms the image, along with modifying its shape like dilations.
(7 votes)
• what the difference between rotation and dilation?
(4 votes)
• Rotation means to turn the object/shape/line around:
Example:
> to <
Both of these are the exact same size, and have the same ratios. The only difference is that one was rotated, turned around, to face a different direction.

Dilation means to make the object/shape/line larger or smaller, but have the same ratios.
Example:
S to s or s to S
Both have the same ratio, but one is smaller/larger than the other.
(6 votes)
• Whats the difference between rotation and dilation?
(2 votes)
• rotation moves around one point while dilation scales thing bigger or smaller
hope that helps😎
(7 votes)

## Video transcript

- [Instructor] In previous videos, we started talking about the idea of transformations. In particular, we talked about rigid transformations. So for example, you can shift something. This would be a translation. So the thing that I'm moving around is a translation of our original triangle. You could have a rotation. So that thing that I translated, I am now rotating it as you see right over there. And you can also have a reflection. The tool that I'm using doesn't make reflection too easy. But that's essentially flipping it over a line. But what we're going to talk about in this video is a non-rigid transformation. And what makes something a rigid transformation is that lengths between points are reserved. But in a non-rigid transformation, those lengths do not need to be preserved. So for example, this rotated and translated triangle that I'm moving around right here, in fact I'm continuing to translate it as I talk. I can dilate it. And one way to think about dilation is that we're just scaling it down or scaling it up. So for example, here, I am scaling it down. That is a dilation. Or I can scale it up. This is also a dilation or even going off of the graph paper. So the whole point here's just to appreciate that we don't just have the rigid transformations, we can have other types of transformations, and a dilation is one of them in your toolkit that you will often see, especially when you get introduced to the idea of transformation.