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### Course: High school geometry>Unit 1

Lesson 2: 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?
• 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).
• What does a translation have to do with dilations? from yours truly Aleiah
• 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
• what the difference between rotation and dilation?
• Rotation means you turn/rotate the shape around a point.

Dilation means enlarging or shrinking the shape.
• can you have a reflection, dilation and a rigid transformation all at the same time?
• 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
• So dilation is changing the length but preserving the measure of the angles, right?
• 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.
• What is the difference between rigid and nonrigid transformation?
• 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
• 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.
• 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.
• Are there other kinds of non-rigid transformations besides dilations?
• 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.
• this was all strange but im glad i'm getting a hang of it now. i really hope i will pass my test
• what the difference between rotation and dilation?
• 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.