HMMT 十一月 2011 · 冲刺赛 · 第 19 题

HMMT November 2011 — Guts Round — Problem 19

[ 10 ] Let ABCD be a rectangle with AB = 3 and BC = 7. Let W be a point on segment AB such that AW = 1. Let X, Y, Z be points on segments BC, CD, DA , respectively, so that quadrilateral W XY Z is a rectangle, and BX < XC . Determine the length of segment BX .

解析

[ 10 ] Let ABCD be a rectangle with AB = 3 and BC = 7. Let W be a point on segment AB such that AW = 1. Let X, Y, Z be points on segments BC, CD, DA , respectively, so that quadrilateral W XY Z is a rectangle, and BX < XC . Determine the length of segment BX . √ 7 − 41 Answer: We note that 2 ∠ Y XC = 90 − ∠ W XB = ∠ XW B = 90 − ∠ AW Z = ∠ AZW ∼ gives us that XY C ZW A and XY Z ∼ W XB . Consequently, we get that Y C = AW = 1. From = XY Z ∼ W XB , we get that BX CY BX 1 = ⇒ = BW CX 2 7 − BX from which we get √ 7 − 41 2 BX − 7 BX + 2 = 0 ⇒ BX = 2 (since we have BX < CX ). A Z D W Y B X C Guts Round