AMC10 2023 A

AMC10 2023 A · Q4

AMC10 2023 A · Q4. It mainly tests Polygons, Geometry misc.

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?

一个四边形的所有边长均为整数，周长为 $26$，有一边长为 $4$。该四边形的最长一边可能的最大长度是多少？

(A) 9 9

(B) 10 10

(D) 12 12

(E) 13 13

Answer

Correct choice: (D)

正确答案：（D）

Solution

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 $\frac{26}{2}-1=13-1=\boxed {\textbf{(D) 12}}$ Sidenote: If there weren't a restriction on integer side lengths, the answer would be the decimal just less than 13, so the sum of the other 3 sides could be just more than 13. That would make the longest side 12.99999..., stopping at who knows how many 9's.

使用三角不等式。对于三角形，较短两边之和必须大于最长边。因为如果最长边等于或大于其他两边之和，就只能成一条直线。类似地，对于凸四边形，最短三边之和必须大于最长边。因此，答案为 $\frac{26}{2}-1=13-1=\boxed {\textbf{(D) 12}}$ 附注：如果没有整数边长限制，答案将是略小于 $13$ 的小数，这样其他三边之和略大于 $13$。最长边将是 $12.99999\dots$，小数点后有无数个 $9$。

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