If a right triangle has

We've got:

a + b + c = 19

c > 9

a² + b² = c²

and we want ab/2.

Let's start with a little rejiggering:

a² + b² = c²

(a + b)² - 2ab = c²

(a + b + c - c)² - 2ab = c²

(19 - c)² - 2ab = c²

361 - 38c = 2ab

We want ab/2, so we'll divide both sides by 4:

90.25 - 9.5c = ab/2

Since c > 9, we know the left hand side is less than 90.25 - 9.5*9, or less than 4.75.

So any value between 0 and 4.75 should be possible, giving us FOUR integer areas (1, 2, 3, and 4).

For anyone reading this in search of questions, this problem was a lot of fun, but not at all a question you'd see on the GMAT - it's too hard.

Another approach just occurred to me too.

Let's say that c = 9. Since c > 9, whatever we get with c = 9 should give us the max.

If we have a + b + c = 19 and a² + b² = c² and c = 9, then we've got

a + b = 10

a² + b² = 81

Then we can say

a² + b² is (a + b)² - 2ab

so

81 = 10² - 2ab

19 = 2ab

4.75 = ab/2

So as before, we can have any area less than 4.75.