HMMT 十一月 2011 · 冲刺赛 · 第 14 题

HMMT November 2011 — Guts Round — Problem 14

[ 8 ] Let AM OL be a quadrilateral with AM = 10, M O = 11, and OL = 12. Given that the perpen- dicular bisectors of sides AM and OL intersect at the midpoint of segment AO , find the length of side LA .

解析

[ 8 ] Let AM OL be a quadrilateral with AM = 10, M O = 11, and OL = 12. Given that the perpen- dicular bisectors of sides AM and OL intersect at the midpoint of segment AO , find the length of side LA . √ Answer: 77 Let D be the midpoint of AM and E be the midpoint of AO . Then, we note that ADE ∼ AM O , so M is a right angle. Similarly, L is a right angle. Consequently, we get that √ √ √ 2 2 2 2 2 2 2 2 AO = OM + AM ⇒ AL = AO − OL = 11 + 10 − 12 = 77 . Guts Round M D E A O L