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## High school geometry

### Unit 1: Lesson 6

Dilations# Dilations: center

CCSS.Math:

Determining the center of dilation, given a figure and its image under a dilation.

## Want to join the conversation?

- I don't understand what exactly a center of dilation is. Could someone explain?(24 votes)
- It is a point where a dilation is based off of. For example: if a center of dilation is the center of a circle with radius 5 and is under a dilation with a scale factor 2, then the radius would be 10.(33 votes)

- Could someone explain how to find the center in a short, and sweet way, please?(4 votes)
- If you draw an imaginary line from each of the corresponding points of the two figures. The center would be the point that all the lines converge at.(27 votes)

- why isn't the center of dilations C?(5 votes)
- The triangle isn't shrinking. Instead, it is growing. So the obvious answer is point D because the scale factor is greater
**than**1. Also remember that when the scale factor is greater than 1, the figure runs**away**from the center of dilation, so the center of dilation is point D.(15 votes)

- why isn't the center of dilations C?(2 votes)
- The dilation is from N to N' the only way to get there is to "expand" the triangle not "shrink" it(2 votes)

- what if i don't have the center displayed for me how do i find it?(7 votes)
- What if the scale factor is 1/4 and the center of dilation is (0,0)?(3 votes)
- Multiply the points by 1/4, so if the point is (8,4) it would end up at (2,1), or (-12,-16) it would be (-3,-4). It could even end up as fractions (5,6) would go to (1.25,1.5).(5 votes)

- How do you find the center of dilation with just two coordinates not graphed(2 votes)
- In order to find the center of dilation, (𝑥₀, 𝑦₀),

we need to know the coordinates of the point that is dilated, (𝑥₁, 𝑦₁),

the coordinates of its image, (𝑥₂, 𝑦₂),

and the scale factor, 𝑘.

We know that 𝑥₂ − 𝑥₀ = 𝑘 ∙ (𝑥₁ − 𝑥₀), from which we can solve for 𝑥₀ as

𝑥₀ = (𝑘 ∙ 𝑥₁ − 𝑥₂)∕(𝑘 − 1)

Similarly, 𝑦₀ = (𝑘 ∙ 𝑦₁ − 𝑦₂)∕(𝑘 − 1)

– – –

Example: Find the center of dilation if dilating (3, −7) by a factor 3 results in (9, −3).

Let (𝑥, 𝑦) be the center of dilation.

Then we have

𝑥 = (3 ∙ 3 − 9)∕(3 − 1) = 0

𝑦 = (3 ∙ (−7) − (−3))∕(3 − 1) = −9

So, the center of dilation is (0, −9)(5 votes)

- why is the answer D not C?(2 votes)
- With a dilation, every point on a figure and the corresponding point on the dilated image of the figure should be collinear with the center of dilation, as Sal explains in the video. Point C matches with one of the endpoints of triangle N and its image, but it has to match every single point to be the center of dilation.(5 votes)

- But, wouldn't the answer be C? I'm so confused...(2 votes)
- C is on the line connecting those two points, but it isn't the center. Remember, N is the original triangle and N' is the one being dilated. N' is to the right of N and it increased, so the center would be to the left.(4 votes)

- I am not understanding the center of dilation in my math class or homework and I looked in the comments below and saw a few but did not make any sense at all.(3 votes)

## Video transcript

- [Instructor] We are
told that triangle N' is the image of triangle N under a dilation. So this is N' in this red color, and then N is the original
N is in this blue color. What is the center of dilation? And they give us some
choices here, choice A, B, C, or D as the center of dilation. So pause this video and see
if you can figure it out on your own. So there's a couple of
ways to think about it. One way I like to just first think about, well what is the scale factor here? So in our original N,
we have this side here, it has a length of two, and then once we dilated it by, and used that scale factor, the corresponding side
has a length of four. So we went from two to four. So we can figure out our scale factor, scale factor is equal to two. Two times two is equal to four. Now what about our center of dilation? So one way to think about it is, pick two corresponding points. So let's say we were to pick this point and this point. So the image, the corresponding point on N', is going to be the scale factor as far away from our center of dilation as the original point. So in this example we know
the scale factor is two, so this is going to be twice as far from our center of dilation as the corresponding point. Well you can immediately see, and it's going to be
in the same direction, so actually if you just draw
a line connecting these two, there's actually only one
choice that sits on that line, and that is choice D right over here as being the center of dilation. And you can also verify that notice, this first point on the original triangle, its change in x is two and
its change in y is three, two three, to go from
from point D to point to that point. And then if you wanna go
to point D to its image, well now you gotta go twice as far. Your change in x is four, and your change in y is six. You could use the Pythagorean Theorem to calculate this distance and then the longer distance, but what you see is, is that the corresponding
point is now twice as far from your center of dilation. So there's a couple of
ways to think about it. One, if you connect corresponding points, your center of dilation
is going to be on a line that connects those two points. And that the image should
be the scale factor as far away from the center of dilation, in this case it should be twice as far from the center of dilation as the point that it is the image of.