HMMT 十一月 2016 · 冲刺赛 · 第 15 题

英文解析

15. [ 9 ] Let ABCD be an isosceles trapezoid with parallel bases AB = 1 and CD = 2 and height 1. Findthe area of the region containing all points inside ABCD whose projections onto the four sides of thetrapezoid lie on the segments formed by AB , BC , CD and DA .
    Proposed by: Allen Liu
    Answer:5
    Let E , F , be the projections of A , B on CD . A point whose projections lie on the sides must be 8
    contained in the square ABF E . Furthermore, the point must lie under the perpendicular to AD at A
    1 1
    and the perpendicular to BC at B , which have slopes and − . The area of the desired pentagon is
    2 2
    1 15
    1 − − = .
    4 8 8