HMMT 二月 2015 · 几何 · 第 7 题

HMMT February 2015 — Geometry — Problem 7

Let ABCDE be a square pyramid of height with square base ABCD of side length AB = 12 (so 2 E is the vertex of the pyramid, and the foot of the altitude from E to ABCD is the center of square ◦ ◦ ABCD ). The faces ADE and CDE meet at an acute angle of measure α (so that 0 < α < 90 ). Find tan α .

解析

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