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## Basic geometry and measurement

# Dilations: scale factor

CCSS.Math: , , , ,

Find the scale factor of a dilation that maps a given figure to another one.

## Want to join the conversation?

- How do you find the scale factor without the origin at zero?(27 votes)
- Dont need to, as for example two and example three, they didnt give any origin so Khan said that we dont even need to draw it since they didnt give enough information; that is the origin.

The video explains how to find the scale factor of a dilation, and how to find the afterimage of a figure that went into a dilation (without the figure being drawn), and how to find the pre-image of a figure that went into a dilation (without the figure being drawn).

Hope this made sense.(22 votes)

- At1:04, why can't the scale factor be 3, not 1/3? Is there a specific rule where it is either a smaller scale or larger? Thanks so much for the help!(20 votes)
- Figures are defined by pre-image and image, so the image would be the dilation while the pre-image is the original figure. So with the preimage being larger, the dilation has to be a fraction less than 1. IF you considered the small pentagon to be the preimage which would be a different question, then the image gets bigger and the scale factor would be 3.(21 votes)

- whats the opposite of a dilation(5 votes)
- A dilation can be an image larger or smaller than the first.(12 votes)

- At0:40how do I know what to find the scale factor of? If it's big shape to small shape or small shape to big shape?(6 votes)
- Great question!

The shape that is labeled prime (with an apostrophe`'`

near each letter) is the image (changed form) of the original shape. Therefore, if the big shape is labeled, say,`A'B'C'D'`

, you'll know that the dilation transformed the small shape to the big shape, and vice-versa.

Get it? Got it? Good.

- littlemissdeena(6 votes)

- any other easy way to slove it?(5 votes)
- Just find the difference between the segments and its just simple division or multiplication after that.(5 votes)

- How do you make this more easy?(4 votes)
- To simplify this process, you can easily multiply the scale factor with the original sides to create the prime sides. Does this help?(3 votes)

- i feel like no one uses this anymore because of how old the comments are(2 votes)
- Well, I Am Here To Prove You Wrong!(4 votes)

- How come in the 3rd example he multiplies 2 by 5/2. but it the last example he divides by 2??(2 votes)
- There are two different questions,

first starts with the pre-image (non-prime figure) in the diagram, and asks about the image (prime figure)

second starts with the image (prime figure) and asks to go backwards to the pre-image (non-prime), going backwards requires to do opposite of multiply by 2(5 votes)

- Why did you have to multiply by 1/3(4 votes)
- He is measuring how much the factor increased so for example 1/3 being divided by 3(0 votes)

- When you dilated 2 by 5/2 how did you get the answer 2? It wasn't clear.(2 votes)
- at3:12, he appears to dilate 2 by 5/2 and get 5, not 2.(3 votes)

## Video transcript

- [Instructor] We are told that pentagon A'B'C'D'E', which is in red right over here, is the image of pentagon ABCDE under a dilation. So that's ABCDE. What is the scale factor of the dilation? So they don't even tell us
the center of the dilation, but in order to figure
out the scale factor you just have to realize
when you do a dilation, the distance between corresponding points will change according to the scale factor. So for example we could
look at the distance between point A and
point B right over here. What is our change in y? Our change in, or even
what is our distance? Our change in y is our distance because we don't have a change in x. Well this is one, two, three, four, five, six. So this length right over here is equal to six. Now what about the corresponding side from A' to B'? Well this length right
over here is equal to two, and so you can see we went from having a length of six to a length of two, so you would have to multiply by 1/3. So our scale factor right over here is 1/3. Now you might be saying okay
that was pretty straightforward because we had a very clear, you could just see the
distance between A and B. How would you do it if
you didn't have a vertical or a horizontal line? Well one way to think about it is, the changes in y and the changes in x would scale accordingly. So if you looked at the distance between point A and point E, our change in y is negative
three right over here, and our change in x is
positive three right over here. And you can see over
here between A' and E', our change in y is negative one, which is 1/3 of negative three, and our change in x is one, which is 1/3 of three. So once again you see our scale factor being 1/3. Let's do another example. So we are told that pentagon A'B'C'D'E' is the image, and they don't, they haven't drawn that here, is the image of pentagon ABCDE under a dilation with
a scale factor of 5/2. So they're giving us our scale factor. What is the length of segment A'E'? So as I was mentioning while I read it, they didn't actually draw this one out. So how do we figure out
the length of a segment? Well I encourage you to pause the video and try to think about it. Well they give us the scale factor, and so what it tells us, the scale factor is 5/2. That means that the
corresponding lengths will change by a factor of 5/2. So to figure out the
length of segment A'E', this is going to be, you could think of it as the image of segment AE. And so you can see that the length of AE is equal to two. And so the length of A'E' is going to be equal to AE which is two times the scale factor, times 5/2, this is our scale
factor right over here. And of course what's two times 5/2? Well it is going to be equal to five, five of these units right over here. So in this case we
didn't even have to draw A'B'C'D'E'. In fact they haven't even
given us enough information. I could draw the scale of that, but I actually don't know where to put it because they didn't even give us our center of dilation. But we know that corresponding sides, or the lengths between
corresponding points, are going to be scaled
by the scale factor. Now with that in mind, let's do another example. So we are told that triangle A'B'C', which they depicted right over here, is the image of triangle ABC, which they did not depict, under a dilation with
a scale factor of two. What is the length of segment AB? Once again they haven't drawn AB here, how do we figure it out? Well it's gonna be a similar
way as the last example, but here they've given us the image and they didn't give us the original. So how do we do it? Well the key, and pause the video again and try to do it on your own. Well the key realization here is that if you take the length of segment AB and you were to multiply
by the scale factor, so you multiply it by two, then you're going to get the length of segment A'B'. The image's length is
equal to the scale factor times the corresponding length
on our original triangle. So what is the length of A'B'? Well this is straightforward
to figure out. It is one, two, three, four, five, six, seven, eight. So this right over here is eight, so we have two times
the length of segment AB is equal to eight. And then you get the length of segment AB, just divide both sides by two, is equal to four. And we're done.