2023 AMC 10A Problem 4

This asks for the greatest possible length, so there are two parts: show the length cannot be too big, then show the best value really happens.

Part 1: the longest side is at most 12.

1. Picture the four sides as sticks joined end to end. For the shape to close up, the other three sticks must reach from one end of the longest stick to the other while bending around it. That only works if their total length is more than the longest stick.
2. Call the longest side $L$. The other three sides add up to $26 - L$. So we need
$$L < 26 - L, \quad\text{which means}\quad 2L < 26, \quad L < 13.$$
3. The sides are whole numbers, so $L$ is at most $12$.

Part 2: a side of 12 is possible.

1. One side must be $4$. With a longest side of $12$, the last two sides add to $26 - 12 - 4 = 10$. Take $5$ and $5$.
2. Check the sides $4, 5, 5, 12$: the three short sides add to $14$, which is more than $12$. So this quadrilateral can be built.

The greatest possible side length is $12$.

The answer is $\boxed{\textbf{(D)}\ 12}$.

This is the triangle inequality for any number of sides: one side can never be as long as all the others put together. For a "greatest possible" question, first find the limit, then give an example that reaches it. The side of length $4$ does not change the limit; it only matters when you build the example.