2023 AMC 10A Problem 4

In any quadrilateral, each side is less than the sum of the other three

If the longest side is $s,$ the other three sum to $26 - s,$ so $s \lt 26 - s$

In any quadrilateral each side is shorter than the sum of the other three. Call the longest side $s.$ The rest sum to $26 - s,$ so $s \lt 26 - s,$ which gives $s \lt 13$ and hence $s \le 12.$ Can we hit $12?$ The sides $4, 12, 9, 1$ work, since $12 \lt 4 + 9 + 1.$ So the greatest length is $12.$ Therefore, the answer is D.