Use the Heron's formula for the area of the triangle

    area = .


Here s = 54/2 = 27 is the semi-perimeter, a = 17, b and c are two other sides.

Since the perimeter is 54 and side "a" is 17, we have  b + c = 54 - a = 54 - 17 = 37.

Let "b" be the longest side of the triangle.

Then  b >= 37/2 = 18.5  and since b is integer, we can write  b >= 19.

Also,  b is less than semi-perimeter b < 54/2 = 27;  c = 37-b.


Then the formula takes the form

    area =  = .   (*)


So, we seek for the integer value of "b" in the interval  19 <= b <= 26, which makes 
the right side of expression  (*)  integer number.


   +----------------------------------------------------------+
   |   Then one of the factors (27-b) or (b-10) should be 5,  |
   |                which gives  b = 25.                      |
   +----------------------------------------------------------+


Indeed, then b = 25 is the sought side length, and the area (*) is

    area =  =  =  = 3*2*3*5 = 90 square units.



    Thus the triangle sides are  a= 17,  b= 25  and  c= 37-25 = 12 units; 

                       the longest side is 25 units.

    The triangle inequalities are held, so such triangle does exist.

                 All requirements of the problem are held.



ANSWER.  Such a triangle does exist, and its longest side is 25 units long.

         This solution is a unique :  there is no other solution.

Perimeter = 54.
One of the sides = 17. With another side being a, the 3rd side is: 54 - 17 - a = 37 - a 

           Heron's formula: , with: 

In addition, .

               We now get: 
                                  

In order for the area to be an INTEGER, the RADICAND,  needs to be a PERFECT SQUARE. As such, the 2(3)3(5), or 270
needs to be MULTIPLIED by one MORE 2, one MORE 3, and one MORE 5, or 2(3)(5), in order to make it 22(34)(52), or 8,100 (PERFECT SQUARE). 
This means that 
                 (a - 25)(a - 12) = 0
   a - 25 = 0      or      a - 12 = 0
        a = 25     or           a = 12

So, if a = 25, then b = 37 - 25 = 12, and c = 17 (given)
    if a = 12, then b = 37 - 12 = 25, and c = 17 (given)

Either way, with the 3 sides being 25, 12, and 17, the longest side of the triangle is 25.