Difference between revisions of "2023 AMC 10A Problems/Problem 4"

Revision as of 16:35, 11 November 2023

Problem

A quadrilateral has all integer sides lengths, a perimeter of $26$ , and one side of length $4$ . What is the greatest possible length of one side of this quadrilateral?

$\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }12\qquad\textbf{(E) }13$

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 $\frac{26}{2}-1=13-1=\boxed {\textbf{(D) 12}}$

~zhenghua

Solution 2

Say the chosen side is $a$ and the other sides are $b,c,d$ .

By the Generalised Polygon Inequality, $a<b+c+d$ . We also have $a+b+c+d=26\Rightarrow b+c+d=26-a$ .

Combining these two, we get $a<26-a\Rightarrow a<13$ .

The smallest length that satisfies this is $a=\boxed {\textbf{(D) 12}}$

~not_slay

Solution 3 (Fast)

By Brahmagupta's Formula, the area of the quadrilateral is defined by $\sqrt{(s-a)(s-b)(s-c)(s-d)}$ where $s$ is the semi-perimeter. If the perimeter of the quadrilateral is $26$ , then the semi-perimeter will be $13$ . The area of the quadrilateral must be positive so the difference between the semi-perimeter and a side length must be greater than $0$ as otherwise, the area will be $0$ or negative. Therefore, the longest a side can be in this quadrilateral is $\boxed {\textbf{(D) 12}}$

~South

(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

Video Solution

https://youtu.be/HCUAbodk_NA

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

@@ Line 23: / Line 23: @@
 == Solution 3 (Fast) ==
-By Brahmagupta's Formula, the area of the rectangle is defined by <math>\sqrt{(s-a)(s-b)(s-c)(s-d)}</math> where <math>s</math> is the semi-perimeter. If the perimeter of the rectangle is <math>26</math>, then the semi-perimeter will be <math>13</math>. The area of the rectangle must be positive so the difference between the semi-perimeter and a side length must be greater than <math>0</math> as otherwise, the area will be <math>0</math> or negative. Therefore, the longest a side can be in this rectangle is <math>\boxed {\textbf{(D) 12}}</math>
+By Brahmagupta's Formula, the area of the quadrilateral is defined by <math>\sqrt{(s-a)(s-b)(s-c)(s-d)}</math> where <math>s</math> is the semi-perimeter. If the perimeter of the quadrilateral is <math>26</math>, then the semi-perimeter will be <math>13</math>. The area of the quadrilateral must be positive so the difference between the semi-perimeter and a side length must be greater than <math>0</math> as otherwise, the area will be <math>0</math> or negative. Therefore, the longest a side can be in this quadrilateral is <math>\boxed {\textbf{(D) 12}}</math>
 ~[https://artofproblemsolving.com/wiki/index.php/User:South South]
-(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

2023 AMC 10A (Problems • Answer Key • Resources)
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