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### Course: 8th grade (Illustrative Mathematics) > Unit 2

Lesson 1: Lesson 4: Dilations on a square grid# Dilating points

Dilations are a way to stretch or shrink shapes around a point called the center of dilation. The amount we stretch or shrink is called the scale factor. If the scale factor is greater than 1, the shape stretches. If it's between 0 and 1, the shape shrinks.

## Want to join the conversation?

- What's the correct order? Do you across and then down or the opposite?(27 votes)
- It actually doesn't matter! The key thing is that the dilation value affects the distance between two points. As in the first example (dilation by a factor of 3), A is originally 1 unit down from P and 2 units to the left of P.

1*3 = 3, so A' (the dilated point) should be 3 units down from P.

2*3 = 6, so A' should be 6 units to the left of P.

It doesn't matter if you go left first or down first, because you always determine the location of A' with respect to P based on the location of A (which doesn't move) with respect to P.(52 votes)

- What does that apostrophe at the end of "a"supposed to mean?(7 votes)
- It is called "prime", it's there to say that the point is not the original point, but the image of the original one after transformation.

A = original point

A' = image of A

Hope that helps?(45 votes)

- How would you do it if there is no coordinate plane?(13 votes)
- That's a great question!

While a coordinate plane is helpful in making our measurements more exact and accurate, it is by no means necessary. In fact, in real world you wouldn't be using one.

Let's imagine you are building a wall out of lego bricks that are all the same size. Each brick is 1 unit in length. When you line up 2 bricks end to end, that is dilation by a factor of 2! If you have four and you take one away, dilation by a factor of 3/4!

This analogy can be extended to any number of real world objects, or even a line on a paper. Draw a line on a paper and measure it's length. Lay a ruler at one end and increase the line by two lengths, dilation by a factor of 3!

I hope this answer was clear and understandable :)(21 votes)

- is this enlargement ? and what is a shear(11 votes)
- No, unless you’re dilating shapes by a factor that is greater than 1.

According to Wikipedia, a shear is “the component of stress coplanar with a material cross section.”

Hope that helps.(11 votes)

- SO I'm in 8th grade algerbra and um we're learning this at the begininng of the year and we have a test tomorrow and only talked about this for 3 days. I'm so confused when a problem just gives you the scale factor and doessn't give you an orgin. What do I do if the problem doesn't tell me the orgin but tells me the scale factor? How do I go about that?(8 votes)
- I think the origin is always the coordinate 0,0.(7 votes)

- How do you determine the direction of the dilation?(8 votes)
- If the point that you are dilating is directly above the point of dilation and you are dilating by 3, you take the distance from the point of dilation and the point you are dilating and you multiply it by 3. That is where you put your new point. If you draw a line from the point of dilation to the new point it should pass through the dilated point.

Hope this helps. God bless!(4 votes)

- I'm confused, how does this work? If you dilate points you go how many ever points from the pre-image or the point of origin?

Please help me because 2 hours of not understanding is killing my brain slowly!(6 votes)- Excellent question.

I don't know what you mean but a basic understanding of the thing is if P is 3 units right, and the scale factor is 2, you get 3 nuits right of when you started. Sorry I'm a year late.

Hope it answered your question(0 votes)

- Having some issues with determining whether you go down or across first in order to plot the dilation correctly. In a mastery question, Plot the image of point

D under a dilation about point P with a scale factor of 1/3. since the graph does not have X,Y coordinates, P is in the top left of the graph and D is in the lower right. I went down then across in order to plot the dilation. This was wring because I was supposed to go across first, then down. How do I determine whether I go down or across first when calculating the dilation? thanks.(5 votes) - why would you go 6 down and 3 to the right? (2:35)?(3 votes)
- To get from the point of origin to A. Then divide those by 3 and you have A'. Do you think you can figure out where A' might be?

Hope this helps. God bless!(3 votes)

- why are you going negative?? I'm confused(4 votes)

## Video transcript

- [Instructor] We're asked
to plot the image of point A under a dilation about point P
with a scale factor of three. So what they're saying when
they say under a dilation, they're saying stretching
it or scaling it up or down around the point P. Now so what we're going
to do is just think about, well how far is point A and
then we want to dilate it with a scale factor of three. So however far A is from point P, it's going to be three times
further under the dilation. Three times further in the same direction. So how do we think about that? Well, one way to think about it is to go from P to A
you have to go one down and two to the left, so
minus one and minus two. And so if you dilate it
with a factor of three, then you're going to want to
go three times as far down. So minus three, and three
times as far to the left, so you'll go minus six. So one, let me do this, so
negative one, negative two, negative three, negative four,
negative five, negative six. So you will end up right over there. And you can even see
it, that this is indeed three times as far from
P in the same direction. So we could call the image of point A, maybe we call that A prime,
and so there you have it. It has been dilated with
a scale factor of three. And so you might be saying, wait, I'm used to dilating being
stretching or scaling. How have I stretched or scaled something? Well imagine a bunch of points here that represents some type of picture and if you push them all three
times further from point P, which you could do as
your center of dilation, then you would expand
the size of your picture by a scale factor of three. Let's do another example with a point. So, here we're told,
plot the image of point A under a dilation about the
origin with a scale factor of 1/3 so first of all we don't
even see the point A here, so it's actually below the fold. So let's see, there we
go, that's our point A. We want it to be about the origin, so about the point zero zero. This is what we want to, the
dilation about the origin with a scale factor of 1/3, scale is 1/3. Scale factor, I should say. So how do we do this? Well here, however far
A is from the origin we now want to be in the same
direction, but 1/3 as far. So one way to think about it,
to go from the origin to A you have to go six down
and three to the right. So 1/3 of that would be two
down and one to the right. Two is 1/3 of six and one is 1/3 of three, so you will end up right over here. That would be our A prime. So notice, you are 1/3
away from the origin as we were before because once again, this is point A under a
dilation about the origin with a scale factor of 1/3.