HMMT 二月 2024 · 几何 · 第 4 题

HMMT February 2024 — Geometry — Problem 4

Let ABCD be a square, and let ℓ be a line passing through the midpoint of segment AB that intersects segment BC . Given that the distances from A and C to ℓ are 4 and 7 , respectively, compute the area of ABCD . ◦

解析

Let ABCD be a square, and let ℓ be a line passing through the midpoint of segment AB that intersects segment BC . Given that the distances from A and C to ℓ are 4 and 7 , respectively, compute the area of ABCD . Proposed by: Ethan Liu Answer: 185 Solution: P ′ ℓ A B ℓ D C ′ ′ ′ Consider the line ℓ through B parallel to ℓ , and drop perpendiculars from A to ℓ and C to ℓ . Note that because ℓ passes through the midpoint of segment AB , the distance from B to ℓ is 4. Thus, the ′ ′ distances from A to ℓ and from C to ℓ are 4 + 4 = 8 and 4 + 7 = 11 , respectively. Let P be the foot ′ ◦ ′ from A to ℓ . Rotating the square 90 from B to A sends the altitude from C to ℓ to the segment ′ ′ along ℓ between B and the foot from A to ℓ ; hence BP = 11 . So the side length of the square is √ √ 2 2 2 2 2 2 AP + BP = 8 + 11 , which means the area of the square is 8 + 11 = 185 . ◦