jmc

Either both given sides are legs, or the 20 cm side is the hypotenuse. If both sides are legs, then the area of the triangle is $(12)(20)/2=120$ square centimeters.

If the 20 cm side is the hypotenuse, then the ratio of the given leg length to the hypotenuse is $12/20=3/5$, so the triangle is a 3-4-5 triangle and the other leg has length 16 cm. The triangle then has area $(12)(16)/2=96$ square centimeters.

The largest possible area then is $\boxed{120}$ square centimeters.