There is some difficulty interpreting this question.
The best interpretation I can offer is that, if the perimeter of two triangles is 144cm, what is the maximum difference in area between the two triangles, if the dimensions of the triangles are whole numbers?
The maximum area of a triangle is when the triangle is equilateral so its area is a/2 * a√3/2=a^2√3/4, where a is the side length, which is 144/3=48cm. The area is 576√3=997.66 sq cm approx.
The cosine formula can be used to relate the lengths of the sides:
a^2=b^2+c^2-2bccosA. If all the sides are integers, angle A must have a rational cosine. The only angles to do so are 0, 60, 90, 180 degrees. 0 and 180 would not make a triangle, so we are left with 60 and 90.
If A=60 we have a^2=b^2+c^2-bc and a+b+c=144cm. So (144-(b+c))^2=b^2+c^2-bc, from which we get:
144^2-288(b+c)+b^2+c^2+2bc=b^2+c^2-bc; and c=(6912-96b)/(96-b)=96(72-b)/(96-b).
When 96-b is a factor of 96 we will get an integer result for c: so b=64, 48 are obvious solutions, giving us c=24, 48. That gives us a=56, 48. When b=60, c=96*12/36=32 and a=52.
If A=90, 144^2-288(b+c)+2bc=0, c=144(72-b)/(144-b), and we get (a,b,c)=(65,63,16) and (60,48,36).
So now we have all the possible triangles: (56,64,24), (48,48,48), (52,60,32), (65,63,16) and (60,48,36). We identified the equilateral triangle area, so we are left with 4. The one with the shortest side is right triangle (65,63,16) and this one has the smallest area: 8*63 (half base of 16 times height of 63)=504 sq cm. the difference between the greatest and smallest area is 576√3-504=72(8√3-7)=493.66 sq cm.
FURTHER INTERPRETATION
After presenting this possible solution, I pondered over the use of the word "rectangle". Perhaps it was a loose translation of "right angle" implying that the only triangles to be considered were right triangles. In my solution, two such triangles were identified: (60,48,36) and (65,63,16), with areas respectively 864 sq cm and 504 sq cm. the greater area is (60,48,36). The areas of the associated rectangles (the triangles are formed by splitting the rectangles into two using the diagonals) are 1728 sq cm and 1008 sq cm, so the difference in the areas of the two triangles is 360 sq cm.
