HMMT 二月 2013 · 几何 · 第 2 题

HMMT February 2013 — Geometry — Problem 2

Let ABCD be an isosceles trapezoid such that AD = BC , AB = 3, and CD = 8. Let E be a point in the plane such that BC = EC and AE ⊥ EC . Compute AE .

解析

Let ABCD be an isosceles trapezoid such that AD = BC , AB = 3, and CD = 8. Let E be a point in the 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.