HMMT 十一月 2024 · 冲刺赛 · 第 14 题

HMMT November 2024 — Guts Round — Problem 14

[9] Let ABCD be a trapezoid with AB ∥ CD . Point X is placed on segment BC such that ∠ BAX = ∠ XDC . Given that AB = 5 , BX = 3 , CX = 4 , and CD = 12 , compute AX .

解析

[9] Let ABCD be a trapezoid with AB ∥ CD . Point X is placed on segment BC such that ∠ BAX = ∠ XDC . Given that AB = 5, BX = 3, CX = 4, and CD = 12, compute AX . Proposed by: Pitchayut Saengrungkongka √ √ Answer: 3 6 = 54. Solution: B A P X D C Let P = DX ∩ AB . Then, from the angle condition, we get that ∠ XAP = ∠ XAB = ∠ XDC = ∠ XP A, XB so △ XAP is isosceles. Moreover, △ XCD and △ XBP are similar, so BP = CD · = 9. Thus, if M XC √ √ 2 2 is the midpoint of AP , then Pythagorean theorem on △ XM P gives XM = 3 − 2 = 5. Finally, √ √ √ 2 Pythagorean theorem on △ XM A gives AX = 7 + 5 = 54 = 3 6.