HMMT 二月 1998 · 几何 · 第 7 题
HMMT February 1998 — Geometry — Problem 7
题目详情
英文原题
Question Seven . [5 points]
Pyramid EARLY has rectangular base EARL and apex Y , and all of itsedges are of integer length. The four edges from the apex havelengths 1, 4, 7, 8 (in no particular order), and EY is perpendicular to
YR .
Find the area of rectangle EARL .
解析
英文解析
- Let E = ( a, b, 0), A = ( − c, b, 0), R = ( − c, − d, 0), L = ( a, − d, 0), Y = (0 , 0 , h ), and observe
2 2 2 2 2 2 2 2 2
that EY + RY = a + b + c + d + 2 h = AY + LY , which can only be satisfied by
EY = 1, AY = 4, RY = 8, LY = 7 (or the symmetric configurations). Since EA is anintegral side of a triangle whose other sides are 1 and 4, we must have EA = 4; similarly,
EL = 7. Therefore, the area of rectangle EARL is 28 . (Such a pyramid may be constructed
√
1 1 31 97 3071
by taking a = , b = , c = , d = , h = .)
8 14 8 14 56
°1
6 √