AMC 8 1996 Problem 8
AMC8 1996 年第 8 题
坐标几何★★☆☆☆
Points and are 10 units apart. Points and are 4 units apart. Points and are 3 units apart. If and are as close as possible, then the number of units between them is
点 和 相距 10 个单位。点 和 相距 4 个单位。点 和 相距 3 个单位。如果 和 尽可能接近，则它们之间的距离是多少个单位？
- A.0
- B.3
- C.9
- D.11
- E.17
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
If AB = 10 and BC=4, then by the triangle inequality. In the triangle inequality, the equality is only reached when the "triangle" is really a degenerate triangle, and are collinear.
Simplifying, this means the smallest value can be is 6.
Applying the triangle inequality on with AC = 6 and CD = 3, we know that when is minimized. If were larger, then could be larger, but we want the smallest possible, and not the largest. Thus, must be at least 3, but cannot be smaller than 3. Therefore, is the answer.
This answer comes when are all on a line, with A=0, B=10, C = 6, and D=3.
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