AMC8 2005

AMC8 2005 · Q15

AMC8 2005 · Q15. It mainly tests Triangles (properties), Ratios in geometry.

How many different isosceles triangles have integer side lengths and perimeter 23?

有多少不同等腰三角形具有整数边长且周长为 23？

(A) 2 2

(B) 4 4

(D) 9 9

(E) 11 11

Answer

Correct choice: (C)

正确答案：（C）

Solution

(C) Because the perimeter of such a triangle is 23, and the sum of the two equal side lengths is even, the length of the base is odd. Also, the length of the base is less than the sum of the other two side lengths, so it is less than half of 23. Thus the six possible triangles have side lengths 1, 11, 11; 3, 10, 10; 5, 9, 9; 7, 8, 8; 9, 7, 7 and 11, 6, 6.

（C）因为这种三角形的周长为 23，且两条相等边的长度之和为偶数，所以底边长度为奇数。另外，底边长度小于另外两边长度之和，因此它小于 23 的一半。于是，六种可能的三角形边长分别为：1，11，11；3，10，10；5，9，9；7，8，8；9，7，7；以及 11，6，6。

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