HMMT 二月 2026 · 几何 · 第 1 题

HMMT February 2026 — Geometry — Problem 1

Let ABCD and W XY Z be squares such that W lies on segment AD , X lies on segment AB , and points Y and Z lie strictly inside ABCD . Given that AW = 4 , AX = 5 , and AB = 12 , compute the area of triangle △ BCY . ◦

解析

Let ABCD and W XY Z be squares such that W lies on segment AD , X lies on segment AB , and points Y and Z lie strictly inside ABCD . Given that AW = 4 , AX = 5 , and AB = 12 , compute the area of triangle △ BCY . Proposed by: Pitchayut Saengrungkongka Answer: 18 Solution: W A D Z X Y P B C Let P be the foot from Y to AB . Notice that ◦ • ∠ AW X = 90 − ∠ AXY = ∠ P XY ◦ • ∠ W AX = ∠ XP Y = 90 • W X = XY . Hence, triangles W AX and XP Y are congruent, so XP = AW = 4 and BP = AB − AX − XP = 12 − 5 − 4 = 3 . Thus, the altitude from Y to BC has length 3 , implying that the area of BCY is 1 · 12 · 3 = 18 . 2 ◦