Difference between revisions of "2023 AMC 10A Problems/Problem 4"
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| + | ==Problem== | ||
| + | A quadrilateral has all integer sides lengths, a perimeter of <math>26</math>, and one side of length <math>4</math>. What is the greatest possible length of one side of this quadrilateral? | ||
| + | <cmath>\textbf{(A)}~9\qquad\textbf{(B)}~10\qquad\textbf{(C)}~11\qquad\textbf{(D)}~12\qquad\textbf{(E)}~13</cmath> | ||
| + | |||
| + | ==Solution 1== | ||
| + | Lets 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 <math>\frac{26}{2}-1=13-1=\text{\boxed{(D)12}}</math> | ||
Revision as of 15:09, 9 November 2023
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?
![\[\textbf{(A)}~9\qquad\textbf{(B)}~10\qquad\textbf{(C)}~11\qquad\textbf{(D)}~12\qquad\textbf{(E)}~13\]](http://latex.artofproblemsolving.com/c/a/a/caa0c870b300a94184a7cea172ecb5dba7a38f2c.png)
Solution 1
Lets 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 
