This problem can be generalized. This is most readily seen by reposing the geometry like so:

-- draw a line segment from (0,a+b) to (a+b,0)

-- pick any point (b,a) on that line

-- the line segment from the origin to that point has length SQRT(a^2+b^2) and its triangle has area ab/2

-- now add a line orthogonal to the new line which is the half its length; using that as the height of a second triangle, its area is (a^2+b^2)/4

The combined areas ab/2+(a^2+b^2)/4=[(a+b)^2]/4 are half of Bella's triangular area and a+b is half her parimeter, so their ratios will be the same. Therefore the ratio is: (a+b)/4. In Bella's case, a+b=10, so the ratio is 5:2, i.e. the area is 50.

However, a+b can just as well be set to any value, say 2. In that case, the ratio is 1 and the area is also 1. Or 40, yielding a ratio of 10:1 with a perimeter of 80 and an area of 800!