HMMT 十一月 2025 · 冲刺赛 · 第 12 题

HMMT November 2025 — Guts Round — Problem 12

[8] Let ABCD be a right trapezoid such that ∠ ABC = ∠ BCD = 90 and the circle with diameter AD is tangent to side BC . Given that AB = 7 and BC = 8, compute CD . © 2025 HMMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2025, November 08, 2025 — GUTS ROUND Organization Team Team ID#

解析

[8] Let ABCD be a right trapezoid such that ∠ ABC = ∠ BCD = 90 and the circle with diameter AD is tangent to side BC . Given that AB = 7 and BC = 8, compute CD . Proposed by: Pitchayut Saengrungkongka 16 Answer: 7 Solution: © 2025 HMMT A B P O D C Note that the center O of the circle with diameter AD is the midpoint of AD . Let P denote the tangency point of BC . Because OP is perpendicular to BC , we must have that P is the midpoint of BC , implying ◦ ◦ BP = P C = 4. Next, notice that because ∠ ABP = ∠ P CD = 90 and ∠ BP A + ∠ DP C = 90 , we have △ ABP ∼ △ P CD , so AB P C BP · P C 16 = = ⇒ CD = = . BP CD AB 7