Question 1199461
The lengths of the sides of a triangle are positive integers. One side has length 17 and the perimeter of the triangle is 54. If the area is also an integer, find the length of the longest side.
<pre>Perimeter = 54.
One of the sides = 17. With another side being a, the 3rd side is: 54 - 17 - a = 37 - a 

           Heron's formula: {{{matrix(1,3, Area, "=", sqrt(s(s - a)(s - b)(s - c)))}}}, with: {{{matrix(3,3, a, "=", a, b, "=", 37 - a, c, "=", 17)}}}

In addition, {{{matrix(1,7, "s(semi-perimeter)", "=", perimeter/2, "=", 54/2, "=", 27)}}}.

               We now get: {{{matrix(1,3, Area, "=", sqrt(s(s - a)(s - b)(s - c)))}}}
                                 {{{matrix(5,2, "=", sqrt(27(27 - a)(27 - (37 - a))(27 - 17)), "=", sqrt(27(27 - a)(27 - 37 + a)(10)), "=", sqrt(270(27 - a)(- 10 + a)), "=", sqrt(270(- 270 + 37a - a^2)), "=", sqrt(2*3^3*5(- 270 + 37a - a^2)))}}} 

In order for the area to be an INTEGER, the RADICAND, {{{(2*3^3*5)(- 270 + 37a - a^2)}}} needs to be a PERFECT SQUARE. As such, the 2(3)<sup>3</sup>(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 2<sup>2</sup>(3<sup>4</sup>)(5<sup>2</sup>), or 8,100 (PERFECT SQUARE). 
This means that {{{matrix(4,3, 2(3)(5), "=", - 270 + 37a - a^2, 30, "=", - 270 + 37a - a^2, a^2 - 37a + 270 + 30, 
"=", 0, a^2 - 37a + 300, "=", 0)}}}
                 (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 <font size  = 4><font color = red><b>longest side of the triangle is 25</font></font></b>.</pre>