If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

### Course: Integrated math 1>Unit 15

Lesson 1: Introduction to rigid transformations

# Identifying transformations

Let's look at four types of transformations: rotations (spinning a shape around a point), translations (shifting a shape), reflections (flipping a shape over a line), and dilations (shrinking or expanding a shape). We practice identifying these transformations in different pairs of figures.

## Want to join the conversation?

• What are all the transformations?
(18 votes)
• There are four different types of transformations.

Translation:
the object moves up/down/left/right, but the shape of the object stays exactly the same. Every point of the object moves the same direction and distance.

Rotation:
the object is rotated a certain number of degrees about a fixed point (the point of rotation). A positive rotation moves counterclockwise; a negative rotation moves clockwise.

Reflection:
the object is reflected (or "flipped") across a line of reflection, which might be the x-axis, y-axis, or some other line.

Dilation:
the object stays the same shape, but is either stretched to become larger (an "enlargement") or shrunk to become smaller (a "reduction").

It is possible for an object to undergo more than one transformation at the same time.

Hope this helps!
(93 votes)
• In the 3rd example, I understand that it is reflection, but couldn't it also be rotation.
(7 votes)
• A rotation always preserves clockwise/counterclockwise orientation around a figure, while a reflection always reverses clockwise/counterclockwise orientation.

It can be verified by the distance formula or Pythagorean Theorem that each quadrilateral has four unequal sides (of lengths sqrt(2), 3, sqrt(10), and sqrt(13)). This means there's only one way that the sides of quadrilateral A can correspond to the sides of quadriateral B.

If one travels counterclockwise around the sides of quadrilateral A, then the corresponding sides of quadrilateral B would be in clockwise order. So the transformation reverses clockwise/counterclockwise orientation and therefore cannot be a rotation.

Have a blessed, wonderful day!
(18 votes)
• what is dilation
(4 votes)
• Dilation makes a triangle bigger or smaller while maintaining the same ratio of side lengths.
(16 votes)
• Pause at , it looks like a sting ray!
(10 votes)
• at , sal says the figure has been rotated but I was wondering why it can't be a reflection? both reflection and rotation seem possible, the way I am understanding this.
(2 votes)
• If you put an imaginary line in between the two shapes and tried to flip one onto the other, you would not be able to do it without rotating one shape.
(8 votes)
• Isn't reflection just a rotation?
(3 votes)
• Not quite. A reflection is a flip, while a rotation is a turn. Reflections reverse the direction of orientation, while rotations preserve the direction of orientation.

For example, if we list the vertices of a polygon in counterclockwise order, then the corresponding vertices of the image of a reflection are in clockwise order, while the corresponding vertices of the image of a rotation (of the original polygon) are in counterclockwise order.

Have a blessed, wonderful day!
(5 votes)
• Can a Dilation be a translation and dilation?
(3 votes)
• Yes, a dilation about a point can be expressed as a translation followed by a dilation by the same factor but about a different point.
(4 votes)
• I have no questions at all!
(4 votes)
• the worst part of khan academy is how his voice changes volume every video
(4 votes)
• What do you expect, the site is free. Show a little gratitude.
(1 vote)
• So Dilation is when the figure is smaller
(1 vote)
• Dilation is when the figure retains its shape but its size changes. This can either be from big to small or from small to big. To dilate a figure, all we have to do is multiply every point's coordinates by a scale factor (>1 for an increase in size, <1 for a decrease). Hope this helps!
(6 votes)

## Video transcript

- [Instructor] What we're going to do in this video is get some practice identifying some transformations. And the transformations we're gonna look at are things like rotations where you are spinning something around a point. We're gonna look at translations, where you're shifting all the points of a figure. We're gonna look at reflection, where you flip a figure over some type of a line. And we'll look at dilations, where you're essentially going to either shrink or expand some type of a figure. So with that out of the way, let's think about this question. What single transformation was applied to triangle A to get triangle B? So it looks like triangle A and triangle B, they're the same size, and what's really happened is that every one of these points has been shifted. Or another way I could say it, they have all been translated a little bit to the right and up. And so, right like this, they have all been translated. So this right over here is clearly a translation. Let's do another example. What single transformation was applied to triangle A to get to triangle B? So if I look at these diagrams, this point seems to correspond with that one. This one corresponds with that one. So it doesn't look like a straight translation because they would have been translated in different ways, so it's definitely not a straight translation. Let's think about it. Looks like there might be a rotation here. So maybe it looks like that point went over there. That point went over there. This point went over here, and so we could be rotating around some point right about here. And if you rotate around that point, you could get to a situation that looks like a triangle B. And I don't know the exact point that we're rotating around, but this looks pretty clear, like a rotation. Let's do another example. What single transformation was applied to quadrilateral A to get to quadrilateral B? So let's see, it looks like this point corresponds to that point. And then this point corresponds to that point, and that point corresponds to that point, so they actually look like reflections of each other. If you were to imagine some type of a mirror right over here, they're actually mirror images. This got flipped over the line, that got flipped over the line, and that got flipped over the line. So it's pretty clear that this right over here is a reflection. All right, let's do one more of these. What single transformation was applied to quadrilateral A to get to quadrilateral B? All right, so this looks like, so quadrilateral B is clearly bigger. So this is a non-rigid transformation. The distance between corresponding points looks like it has increased. Now you might be saying, well, wouldn't that be, it looks like if you're making something bigger or smaller, that looks like a dilation. But it looks like this has been moved as well. Has it been translated? And the key here to realize is around, what is your center of dilation? So for example, if your center of dilation is, let's say, right over here, then all of these things are gonna be stretched that way. And so this point might go to there, that point might go over there, this point might go over here, and then that point might go over here. So this is definitely a dilation, where you are, your center where everything is expanding from, is just outside of our trapezoid A.