HMMT 二月 2024 · 冲刺赛 · 第 11 题

HMMT February 2024 — Guts Round — Problem 11

[7] Let ABCD be a rectangle such that AB = 20 and AD = 24 . Point P lies inside ABCD such that triangles P AC and P BD have areas 20 and 24 , respectively. Compute all possible areas of triangle P AB .

解析

[7] Let ABCD be a rectangle such that AB = 20 and AD = 24 . Point P lies inside ABCD such that triangles P AC and P BD have areas 20 and 24 , respectively. Compute all possible areas of triangle P AB . Proposed by: Pitchayut Saengrungkongka Answer: 98 , 118 , 122 , 142 Solution: A D P P P O P B C There are four possible locations of P as shown in the diagram. Let O be the center. Then, [ P AO ] = 10 and [ P BO ] = 12 . Thus, [ P AB ] = [ AOB ] ± [ P AO ] ± [ P BO ] = 120 ± 10 ± 12 , giving the four values 98 , 118 , 122 , and 142 .