Skip to main content

2023 AMC 10A Problem 4

Problem 4 of 25EasierGeometryProblem-Solving Techniques

A quadrilateral has all integer side lengths, a perimeter of 26,26, and one side of length 4.4. What is the greatest possible length of one side of this quadrilateral?

Answer choices

Show solution

Solution

In any quadrilateral each side is shorter than the sum of the other three. Call the longest side s.s. The rest sum to 26−s,26 - s, so s<26−s,s \lt 26 - s, which gives s<13s \lt 13 and hence s≤12.s \le 12. Can we hit 12?12? The sides 4,12,9,14, 12, 9, 1 work, since 12<4+9+1.12 \lt 4 + 9 + 1. So the greatest length is 12.12. Therefore, the answer is D.
AoPS wiki

Tagged: triangle inequality · bounding to limit cases

More practice