HMMT 二月 2023 · 几何 · 第 2 题

HMMT February 2023 — Geometry — Problem 2

Points X , Y , and Z lie on a circle with center O such that XY = 12. Points A and B lie on segment XY such that OA = AZ = ZB = BO = 5. Compute AB .

解析

Points X , Y , and Z lie on a circle with center O such that XY = 12. Points A and B lie on segment XY such that OA = AZ = ZB = BO = 5. Compute AB . Proposed by: Rishabh Das √ Answer: 2 13 Solution: Let the midpoint of XY be M . Because OAZB is a rhombus, OZ ⊥ AB , so M is the 1 midpoint of AB as well. Since OM = OX , △ OM X is a 30 − 60 − 90 triangle, and since XM = 6, 2 √ √ √ OM = 2 3. Since OA = 5, the Pythagorean theorem gives AM = 13, so AB = 2 13.