A dilation
![D_{O,\,k}](https://tex.z-dn.net/?f=D_%7BO%2C%5C%2Ck%7D)
is a dilation with a scale factor, k centered at the origin.
A scale factor, k means that the distance of the image from the the center of dilation is k times the distance of the pre-image from the center of dilation and the size of the image is k times the size of the pre-image.
Given triangle ABC with vertices A(-2, 2), B(1, 2) and C(1, -1), a dilation
![D_{O,\,k}](https://tex.z-dn.net/?f=D_%7BO%2C%5C%2Ck%7D)
will result in the image with vertices A'2.5(-2, 2) = A'(-5, 5), B'2.5(1, 2) = B'(2.5, 5) and C'2.5(1, -1) = C'(2.5, -2.5)
Now, consider line AC, the equation of the line is given by
![\frac{y-2}{x-(-2)} = \frac{-1-2}{1-(-2)} \\ \\ \Rightarrow \frac{y-2}{x+2} = \frac{-3}{1+2} = \frac{-3}{3} =-1 \\ \\ \Rightarrow y-2=-(x+2)=-x-2 \\ \\ \Rightarrow y=-x](https://tex.z-dn.net/?f=%5Cfrac%7By-2%7D%7Bx-%28-2%29%7D%20%3D%20%5Cfrac%7B-1-2%7D%7B1-%28-2%29%7D%20%5C%5C%20%5C%5C%20%5CRightarrow%20%5Cfrac%7By-2%7D%7Bx%2B2%7D%20%3D%20%5Cfrac%7B-3%7D%7B1%2B2%7D%20%3D%20%5Cfrac%7B-3%7D%7B3%7D%20%3D-1%20%5C%5C%20%5C%5C%20%5CRightarrow%20y-2%3D-%28x%2B2%29%3D-x-2%20%5C%5C%20%5C%5C%20%5CRightarrow%20y%3D-x)
Notice that line y = -x is a line passing through the origin which is the center of dilation.
Consider line A'C', the equation of the line is given by
![\frac{y-5}{x-(-5)} = \frac{-2.5-5}{2.5-(-5)} \\ \\ \frac{y-5}{x+5} = \frac{-7.5}{2.5+5} = \frac{-7.5}{7.5} =-1 \\ \\ \Rightarrow y-5=-(x+5)=-x-5 \\ \\ \Rightarrow y=-x](https://tex.z-dn.net/?f=%5Cfrac%7By-5%7D%7Bx-%28-5%29%7D%20%3D%20%5Cfrac%7B-2.5-5%7D%7B2.5-%28-5%29%7D%20%5C%5C%20%5C%5C%20%5Cfrac%7By-5%7D%7Bx%2B5%7D%20%3D%20%5Cfrac%7B-7.5%7D%7B2.5%2B5%7D%20%3D%20%5Cfrac%7B-7.5%7D%7B7.5%7D%20%3D-1%20%5C%5C%20%5C%5C%20%5CRightarrow%20y-5%3D-%28x%2B5%29%3D-x-5%20%5C%5C%20%5C%5C%20%5CRightarrow%20y%3D-x)
As an be seen the line AC and the line A'C' is the same line.