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. 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 rectangle is defined by 
where 
is the semi-perimeter. If the perimeter of the rectangle is , then the semi-perimeter will be 
. The area of the rectangle 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 rectangle is 
(How do you show that a side is maximized when the quadrilateral is a rectangle?) - megaboy6679
(Also, why is the quadrilateral cyclic? Brahmagupta's Formula only applies to cyclic quadrilaterals.) ~ Technodoggo
Video Solution by Math-X (First understand the problem!!!)
https://youtu.be/cMgngeSmFCY?si=YBa-pxkomHrJErSz&t=597
See Also
| 2023 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
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