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 2 shorter sides must always be longer than the longest side. Similarly for a convex quadrilateral, 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 possibly be in this rectangle is
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 | |
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All AMC 10 Problems and Solutions |
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