2023 AMC 10A Problems/Problem 4
Contents
Problem
A quadrilateral has all integer sides lengths, a perimeter of , and one side of length
. What is the greatest possible length of one side of this quadrilateral?
Solution 1
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
~zhenghua
Solution 2
Say the chosen side is and the other sides are
.
By the Generalised Polygon Inequality, . We also have
.
Combining these two, we get .
The smallest length that satisfies this is
~not_slay
Solution 3 (Fast)
By Brahmagupta's Formula, the area of the quadrilateral is defined by where
is the semi-perimeter. If the perimeter of the quadrilateral is
, then the semi-perimeter will be
. The area of the quadrilateral must be positive so the difference between the semi-perimeter and a side length must be greater than
as otherwise, the area will be
or negative. Therefore, the longest a side can be in this quadrilateral is
(Also, why is the quadrilateral cyclic? Brahmagupta's Formula only applies to cyclic quadrilaterals.) ~ Technodoggo Quadrilateral is not necessarily cyclic - sepehr2010
Solution 4
This is an AMC 10 problem 4, so there is no need for any complex formulas. The largest singular side length from a quadrilateral comes from a trapezoid. So we can set the sides of the trapezoid equal to
. Next we can split the trapezoid into
triangles, where each base length of the triangle equals
. So the top side equals
, and the bottom side length equals
~ kabbybear
Video Solution by Math-X (First understand the problem!!!)
https://youtu.be/cMgngeSmFCY?si=YBa-pxkomHrJErSz&t=597
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=Iu0AJ2rof7k
See Also
2023 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.