Lowest possible value for k for triangle with an integer area

Question

There is a triangle with sides length (9 + k), (39 + k), and (48 + k). The triangle has an area that is an integer. What is the smallest possible value for k?

I already tried pythagorean theorem.

Answer 1

The semi-perimeter $s = \frac12(96+3k) = 48+3k/2 $.

Hero's formula states that, if $A$ is the area,

$\begin{array}\\ A^2 &=s(s-a)(s-b)(s-c)\\ &=(48+3k/2)(39+k/2)(9+k/2)(k/2)\\ &=k(96+3k)(78+k)(18+k)/16\\ &=3k(32+k)(78+k)(18+k)/16\\ \end{array} $

Throwing this into Wolfy, when $k=52$, $A = 2730$.

Answer 2

Hint: use Henon's formula. It states that if a triangle has sides $a, b, c$, then its area is given by $$\sqrt{s(s-a)(s-b)(s-c)},$$ where $s=\frac{a+b+c}{2}$ is half the perimeter.