Maximizing the Length of a Triangle Side

Hard Problem Solving Geometry

In a triangle, the lengths of the sides are represented by the variables $a$, $b$, and $c$. Let the triangle be such that the lengths of the sides meet the triangle inequality, which states that the sum of the lengths of any two sides must be greater than the length of the remaining side. We want to determine the maximum possible length of the longest side, $c$, given the conditions below.

What is the maximum possible value of the length of side $c$ of triangle $ABC$?

Statement (1) alone is sufficient, but statement (2) alone is not sufficient.

Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

Both statements (1) and (2) together are sufficient, but neither statement alone is sufficient.

Each statement alone is sufficient.

Statements (1) and (2) together are not sufficient.

Hint

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