HMMT 二月 2012 · 冲刺赛 · 第 6 题
HMMT February 2012 — Guts Round — Problem 6
题目详情
英文原题
[3] Let rectangle ABCD have lengths AB= 20 and BC= 12. Extend ray BCtoZsuch that CZ= 18. LetEbe the point in the interior of ABCD such that the perpendicular distance from EtoABis 6 and the perpendicular distance from EtoADis 6. Let line EZintersect ABatXandCDatY. Find the area of quadrilateral AXY D .
解析
英文解析
- [ 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 . Itis 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 directlycompute the values of AX and DY , then use the formula for the area of a trapezoid.)