Question

If triangle ABC is defined by the coordinates A(-4, -4), B(2, -2), C(0, 4) is dilated by a scale factor of 1 2 , with resulting vertex A' at (-2, -2). What is the center of the dilation?

Answer 1

the origin, (0, 0)

Explanation:

The coordinates of A' are 1/2 those of A, meaning each has been multiplied by the scale factor 1/2. When the dilated points are all the original points multiplied by the scale factor, the center of dilation is the origin.

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For center of dilation Q, the image of a point A after dilation by a factor of k is ...

A' = kA + (k-1)Q

Then for points A, A', and dilation factor k, the center of dilation can be found to be ...

(A' -kA)/(k-1) = Q

Here, that is ...

Q = ((-2, -2) -(1/2)(-4, -4))/(1/2 -1) = (0, 0)/(-1/2)

Q = (0, 0)