Putnam 1982 B6

Problem B-6 Let $K(x, y, z)$ denote the area of a triangle whose sides have lengths $x, y$, and $z$. For any two triangles with sides $a, b, c$ and $a^{\prime}, b^{\prime}, c^{\prime}$, respectively, prove that $$\sqrt{K(a, b, c)}+\sqrt{K\left(a^{\prime}, b^{\prime}, c^{\prime}\right)} \leqslant \sqrt{K\left(a+a^{\prime}, b+b^{\prime}, c+c^{\prime}\right)}$$ and determine the cases of equality.