The dilation do,2.5 is applied to triangle abc below. use the dilation of triangle abc and its image to prove that the dilation takes a line passing through the center of the dilation to the same line.

A dilation 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} 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



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



As an be seen the line AC and the line A'C' is the same line.

We’re gonna need a picture to refer to in order to figure it out. try reposting your question with the picture included.