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HMMT 二月 2024 · 冲刺赛 · 第 25 题

HMMT February 2024 — Guts Round — Problem 25

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [14] Point P is inside a square ABCD such that ∠ AP B = 135 , P C = 12 , and P D = 15 . Compute the area of this square.
解析
  1. [14] Point P is inside a square ABCD such that ∠ AP B = 135 , P C = 12 , and P D = 15 . Compute the area of this square. Proposed by: Pitchayut Saengrungkongka √ Answer: 123 + 6 119 Solution: C D x 12 √ Q ◦ 2 y 45 P y ◦ 135 y x A B ◦ Let x = AP and y = BP . Rotate △ BAP by 90 around B to get △ BCQ . Then, △ BP Q is right- ◦ ◦ isosceles, and from ∠ BQC = 135 , we get ∠ P QC = 90 . Therefore, by Pythagorean’s theorem, 2 2 2 2 2 2 P C = x + 2 y . Similarly, P D = y + 2 x . √ 2 2 2 2 P C − P D 2 Thus, y = = 21 , and similarly x = 102 = ⇒ xy = 3 238 . 3 Thus, by the Law of Cosines, the area of the square is 2 2 2 ◦ AB = AP + BP − 2 cos(135 )( AP )( BP ) √ 2 2 = x + y + 2 xy √ = 123 + 6 119 .