HMMT 二月 2015 · 几何 · 第 7 题

英文解析

7. Let ABCDE be a square pyramid of height with square base ABCD of side length AB = 12 (so
   E is the vertex of the pyramid, and the foot of the altitude from E to ABCD is the center of square 2
   ° °
   ABCD ). The faces ADE and CDE meet at an acute angle of measure α (so that 0 < α < 90 ).
   Find tan α .
   Answer: Let X be the projection of A onto DE . Let b = AB = 12.17
   The key fact in this computation is that if Y is the projection of A onto face CDE , then the projection 144
   of Y onto line DE coincides with the projection of A onto line DE (i.e. X as defined above). Web
   √
   compute AY = by looking at the angle formed by the faces and the square base (via 1 / 2- b/ 2-
   b +12
   √
   √
   b b +1 / 22
   √2
   b + 1 / 2 right triangle). Now we compute AX = 2[ AED ] /ED = .
   2 b +1 / 22
   √
   √
   2 b +12
   2 2 2 2 2 17
   But α = ∠ AXY , so from ( b + 1) − ( 2 b + 1) = ( b ) , it easily follows that tan α = = .2
   b 1442