HMMT 二月 2024 · 几何 · 第 9 题

HMMT February 2024 — Geometry — Problem 9

Let ABC be a triangle. Let X be the point on side AB such that ∠ BXC = 60 . Let P be the point on segment CX such that BP ⊥ AC . Given that AB = 6 , AC = 7 , and BP = 4 , compute CP .

解析

Let ABC be a triangle. Let X be the point on side AB such that ∠ BXC = 60 . Let P be the point on segment CX such that BP ⊥ AC . Given that AB = 6 , AC = 7 , and BP = 4 , compute CP . Proposed by: Pitchayut Saengrungkongka √ Answer: 38 − 3 ◦ ◦ Solution: Construct parallelogram BP CQ . We have CQ = 4 , ∠ ACQ = 90 , and ∠ ABQ = 120 . √ √ 2 2 2 2 Thus, AQ = AC + CQ = 65 , so if x = CP = BQ , then by Law of Cosine, x + 6 x + 6 = 65 . √ Solving this gives the answer x = 38 − 3 . A X ◦ 120 ◦ 60 6 7 P x 4 ◦ B 120 C x 4 Q