HMMT 二月 2005 · 冲刺赛 · 第 19 题
HMMT February 2005 — Guts Round — Problem 19
题目详情
英文原题
- [8] Regular tetrahedron ABCD is projected onto a plane sending A , B , C , and D
′ ′ ′ ′ ′ ′ ′ ′
to A , B , C , and D respectively. Suppose A B C D is a convex quadrilateral with
′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′
A B = B C ’ and C D = D A , and suppose that the area of A B C D = 4. Giventhese conditions, the set of possible lengths of AB consists of all real numbers in theinterval [ a, b ). Compute b .
解析
英文解析
- Regular tetrahedron ABCD is projected onto a plane sending A , B , C , and D to
′ ′ ′ ′ ′ ′ ′ ′
A , B , C , and D respectively. Suppose A B C D is a convex quadrilateral with
′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′
A B = A D and C B = C D , and suppose that the area of A B C D = 4. Giventhese conditions, the set of possible lengths of AB consists of all real numbers in theinterval [ a, b ). Compute b .
√
Solution: 2 64
′ ′ ′ ′
The value of b occurs when the quadrilateral A B C D degenerates to an isoscelestriangle. This occurs when the altitude from A to BCD is parallel to the plane. Lets = AB . Then the altitude from A intersects the center E of face BCD . Since
√
√
s s s 62
′ ′
√2
EB = , it follows that A C = AE = s − = . Then since BD is parallel to
3 3
√3
√
1 s 62
′ ′ ′ ′ ′ ′ 2
the plane, B D = s . Then the area of A B C D is 4 = · , implying s = 4 6, or
2 3
√
s = 2 6.4