if the sides of the triangle are given by 20 cm, 30 cm, and 60 cm find the area of the triangle.
I tried a long time. Apparently, Heron's formula does not seem to work
$\sqrt{s(s-a)(s-b)(s-c)}$
where $s = (a+b+c)/2$
In the above problem $s=55$ and thus we end up with a negative number inside square root. I am not sure if there is any other formula to be applied to this problem .
Heron's formula works if the triangle exists. This triangle does not exist. By the triangle inequality applied to this triangle, we should have $60\le20+30$, which is false.
If the triangle inequality holds, then each of $s-a=\frac{b+c-a}{2}$, $s-b=\frac{a+c-b}{2}$, and $s-c=\frac{a+b-c}{2}$ is non-negative and so $s(s-a)(s-b)(s-c)\ge0$.