1980 AMC 12 第 26 题
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26.
四个半径为 的球两两相切，其中三个放在地面上，第四个放在其余三个球上。用一个每条棱长均为 的正四面体外切于这四个球，则 等于
Four balls of radius are mutually tangent, three resting on the floor and the fourth resting on the others. A tetrahedron, each of whose edges has length is circumscribed around the balls. Then equals
答案：E
小提示：
四个球心组成棱长为 的正四面体
The four ball centers form a regular tetrahedron of edge
大提示：
外部正四面体的每个面都与相应的球心四面体面平行，并向外相距一个半径
The outer faces are parallel to the corresponding center-tetrahedron faces and one radius farther away
解答：
四个球心组成一个棱长为 的正四面体。它的内切球半径为 外切正四面体的每个面都与球心四面体的相应面平行，并且比该面到共同中心的距离多一个单位。由相似关系，因此 。
因此，正确答案是 E。
The ball centers form a regular tetrahedron of edge Its inradius is Each face of the circumscribed tetrahedron is parallel to the corresponding face of this center tetrahedron and lies one unit farther from the common center. By similarity, Hence
Therefore, the correct answer is E.
其他年份的第 26 题
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- 1951 AMC 12
- 1952 AMC 12
- 1953 AMC 12
- 1954 AMC 12
- 1955 AMC 12
- 1956 AMC 12
- 1957 AMC 12
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- 1967 AMC 12
- 1968 AMC 12
- 1969 AMC 12
- 1970 AMC 12
- 1971 AMC 12
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- 1978 AMC 12
- 1979 AMC 12
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- 1999 AMC 12