How many triangles are there with whole number side?

Question

If $a=29$, and $b=21$, how many triangles are there such that side $c$ is a whole number?
My approach: Tried using certain equations to establish relationship between sides to maybe point to right answer, but to no avail.

Answer 1

Since by the triangle inequality $$29+21>c$$ and $$c+21>29,$$ we obtain: $$9\leq c\leq49.$$ Can you end it now?

Answer 2

No equations needed. These two sides must be connected. How long or short could the third side be?

The longest is clearly when $a$ and $b$ are in a line without overlap. Their ends will be $29 + 21 = 50$ apart.

The shortest is again when they are in line but overlap. The unjoined ends will be $29 - 21 = 8$ apart.

Are these degenerate extremes triangles? You decide.

However, all of the lengths between these extremes are certainly triangles.

Answer 3

A bit of geometry:

Draw $a=\overline{BC}=29;$

Draw a circle about $C$ with radius $r=21$.

$2$ Intersection points of circle with $BC$ :

1) $D_1$ where $\overline{BD_1}= 8$;

2) $D_2$ where $\overline{BD_2}=8 +2\cdot 21= 50$.

The locus of $A$ are the points on this circle.

Hence $8< c <50$, or $9 \le c \le 49$.

How many integers c with above constraints?