She should have used this formula
0.5(base)(height)(scale)^2
where the 0.5(base)(height) portion is the area of the original triangle. Beth forgot about the 0.5 out front, aka 1/2. She also forgot to square the "scale" part.
Why do we square the "scale" part? Consider a triangle with base b and height h. Its area is 0.5bh
Now let's scale using the linear factor k. The base b turns into bk. The height turns into hk.
Then,
new area = 0.5*(new base)*(new height)
new area = 0.5*(bk)*(hk)
new area = 0.5bh*k^2
We have the k^2 at the end to signal we square the linear scale factor to get the area scale factor. This applies to any dilated shape. It doesn't have to be a triangle.
If 0 < k < 1, then the shape shrinks. If k > 1, then the shape enlarges.