If triangle abc is dilated by a scale factor of 2.5 with a center of dilation at vertex b, how does the area of a'b'c' compare with the area of abc?

If you apply a dilation by a scale factor of 2.5 with a center of dilation at vertex b, the point b' will be the same point b (it is the center of the dilation so it does not change).

Then:

b' = b

The length of segment a'b' = 2.5 * the length of segment ab

The length of segment c'b' = 2.5 * the length of segment cb

Given the similarity properties, the height of triangle a'b'c' is 2.5 times the height of  the triangle abc.

You can take ab as the base of the original triangle and a'b' as the base of the new triangle.

In that case, the area of the original triangle is: ab*height

And the area of the new triangle is (2.5ab) * (2.5 height) = (2.5)^2 * ab * height = 6.25*(ab * height)

=> new area = 6.25 * original area.

area of a'b'c' is 6.25 times the area of abc.