Given only the lengths of the three sides of a triangle, can we find if the triangle is impossible?

Question

I was recently trying to implement Hero's Formula and I was surprised to find it declare that the perimeter of a triangle with sides of length 3, 4, and 9 is a complex number. This suggests that given the lengths of all three sides of a triangle, it is possible to determine if the triangle is possible or not.

To me, this was a surprising result that I do not recall hearing elsewhere. In fact, without restrictions on angels, I was unaware that triangles could be impossible. Is this correct? Given the lengths of the three sides of a triangle, can we find if the triangle is impossible?

Answer 1

Very simply, the longest side cannot be longer than the sum of the shorter sides. If it is the same length as the sum of the shorter sites, it will degenerate to a straight line.

Answer 2

If we have $a\le c$ and $b\le c$ , a triangle with lengths $a,b,c$ is possible if and only if $a+b>c$. This is not the case for the triple $(3/4/9)$

Answer 3

One way to visualize this is to draw a segment length $a$, and then center two circles with radii $b$ and $c$ at the ends of this segment. If the circles do not intersect, then lengths $a,b,c$ cannot form a triangle.

The triangle inequality $$|a+b|\leq|a|+|b|$$ is based on exactly this fact.