HMMT 二月 2012 · 冲刺赛 · 第 6 题

HMMT February 2012 — Guts Round — Problem 6

[ 3 ] Let rectangle ABCD have lengths AB = 20 and BC = 12. Extend ray BC to Z such that CZ = 18. Let E be the point in the interior of ABCD such that the perpendicular distance from E to AB is 6 and the perpendicular distance from E to AD is 6. Let line EZ intersect AB at X and CD at Y . Find the area of quadrilateral AXY D .

解析

[ 3 ] Let rectangle ABCD have lengths AB = 20 and BC = 12. Extend ray BC to Z such that CZ = 18. Let E be the point in the interior of ABCD such that the perpendicular distance from E to AB is 6 and the perpendicular distance from E to AD is 6. Let line EZ intersect AB at X and CD at Y . Find the area of quadrilateral AXY D . Answer: 72 Draw the line parallel to AD through E , intersecting AB at F and CD at G . It is clear that XF E and Y GE are congruent, so the area of AXY D is equal to that of AF GD . But AF GD is simply a 12 by 6 rectangle, so the answer must be 72. (Note: It is also possible to directly compute the values of AX and DY , then use the formula for the area of a trapezoid.)