HMMT 十一月 2011 · 冲刺赛 · 第 19 题
HMMT November 2011 — Guts Round — Problem 19
题目详情
英文原题
[10] LetABCD be a rectangle with AB= 3 and BC= 7. Let Wbe a point on segment ABsuch that AW = 1. Let X, Y, Z be points on segments BC, CD, DA , respectively, so that quadrilateral WXY Z is a rectangle, and BX < XC . Determine the length of segment BX.
解析
英文解析
- [ 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 Zis a rectangle, and BX < XC . Determine the length of segment BX .
√
7 − 41
Answer: We note that
∠ Y XC = 90 − ∠ W XB = ∠ XW B = 90 − ∠ AW Z = ∠ AZW2
∼
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 − BXfrom which we get
√
7 − 41
BX − 7 BX + 2 = 0 ⇒ BX =2
(since we have BX < CX ).2
A Z D
B X CYW
Guts Round