PUMaC 2015 · 几何（B 组） · 第 2 题

PUMaC 2015 — Geometry (Division B) — Problem 2

[ 3 ] Let ABCD be a regular tetrahedron with side length 1. Let EF GH be another regular 1 tetrahedron such that the volume of EF GH is -th the volume of ABCD . The height of 8 EF GH (the minimum distance from any of the vertices to its opposing face) can be written ’ a as , where a and b are positive coprime integers. What is a + b ? b

解析

[ 3 ] Let ABCD be a regular tetrahedron with side length 1. Let EF GH be another regular 1 tetrahedron such that the volume of EF GH is -th the volume of ABCD . The height of 8 EF GH (the minimum distance from any of the vertices to its opposing face) can be written ’ a as , where a and b are coprime integers. What is a + b ? b Solution: 7 . WLOG let ABC be the base of tetrahedron ABCD , and let Z be the circumcenter of ABC . Thus the height of tetrahedron ABCD is the length of DZ . Note that AZ is 2 / 3 the length ” 3 1 2 of the altitude from A to BC . Thus AZ = ⋅ = . Thus by the Pythagorean Theorem, ” 3 2 3 ‘ ‘ 3 DZ = 1 − 1 / 3 = 2 / 3. Note that since the volume of EF GH is ( 1 / 2 ) as much as the volume of ABCD , by scaling we know that any lengths associated to EF GH is 1 / 2 as much as the ÷ ‘ 1 bijective length associated to ABCD . Namely, the height of EF GH is 1 / 2 ⋅ 2 / 3 = , and 6 our answer is 7 .