HMMT 二月 2004 · 几何 · 第 1 题

HMMT February 2004 — Geometry — Problem 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 . B ? C 135 135 6 45 45 A D Area = 30

解析

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 6 3 2 3 9 9 A D 3 2 3 X Y Area BCYX = 30 − 18 = 12