HMMT 二月 2004 · 几何 · 第 1 题

英文解析

1. In trapezoid ABCD , AD is parallel to BC . A = D = 45 , while B = C = 135 .
   If AB = 6 and the area of ABCD is 30, find BC .
   √
   Solution: 2 2
   Draw altitudes from B and C to AD and label the points of intersection X and Y ,
   √
   ° ° °
   respectively. Then ABX and CDY are 45 − 45 − 90 triangles with BX = CY = 3 2.
   So, the area of ABX and the area of CDY are each 9, meaning that the area of rectangle
   √ √ √
   BCY X is 12. Since BX = 3 2, BC = 12 / (3 2) = 2 2.
   B 2 2 3 C
   3 2 36
   9 9
   A D
   3 2 3 X Y
   Area BCYX = 30 − 18 = 12