HMMT 二月 2013 · 几何 · 第 2 题
HMMT February 2013 — Geometry — Problem 2
题目详情
英文原题
Let ABCD be an isosceles trapezoid such that AD=BC,AB= 3, and CD = 8. Let Ebe a point in
the plane such that BC=EC andAE⊥EC. Compute AE.
解析
英文解析
- Let ABCD be an isosceles trapezoid such that AD = BC , AB = 3, and CD = 8. Let E be a point inthe plane such that BC = EC and AE ⊥ EC . Compute AE .
√
Answer: 2 6 Let r = BC = EC = AD . △ ACE has right angle at E , so by the Pythagorean
Theorem,
2 2 2 2 2
AE = AC − CE = AC − r
Let the height of △ ACD at A intersect DC at F . Once again, by the Pythagorean Theorem,
( ) ( ) ( )
2 2 2
8 − 3 11 5
2 2 2 2 2 2
AC = F C + AF = + 3 + AD − DF = + r −
2 2 2
Plugging into the first equation,
( ) ( )
2 2
11 5
2 2 2
AE = + r − − r ,
2 2
√
so AE = 2 6.