Maths Olympiad Prep

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Geometry Difficulty 3.1 AMC 10/12 Find the answer

A quadrilateral has all integer sides lengths, a perimeter of 2626, and one side of length 44. What is the greatest possible length of one side of this quadrilateral?

Pick one

Solution

Let's use the triangle inequality. We know that for a triangle, the sum of the 2 shorter sides must always be longer than the longest side. This is because if the longest side were to be as long as the sum of the other sides, or longer, we would only have a line.
Similarly, for a convex quadrilateral, the sum of the shortest 3 sides must always be longer than the longest side. Thus, the answer is 262−1=13−1=(D) 12\frac{26}{2}-1=13-1=\boxed {\textbf{(D) 12}}
~zhenghua

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.