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### Course: High school geometry>Unit 1

Lesson 6: Dilations

# Dilations: center

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

## Want to join the conversation?

• why isn't the center of dilations C?
• The dilation is from N to N' the only way to get there is to "expand" the triangle not "shrink" it
• I don't understand what exactly a center of dilation is. Could someone explain?
• 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.
• bro ima actually eat my desk
• Well, did you?
• Could someone explain how to find the center in a short, and sweet way, please?
• 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.
• This video does not help at all.
• why isn't the center of dilations C?
• 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.
• what if i don't have the center displayed for me how do i find it?
• How do you use a stove.
• How do you find the center of dilation with just two coordinates not graphed
(1 vote)
• 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)