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HMMT 二月 2001 · 几何 · 第 6 题

HMMT February 2001 — Geometry — Problem 6

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. A point on a circle inscribed in a square is 1 and 2 units from the two closest sides of the square. Find the area of the square.
解析
  1. A point on a circle inscribed in a square is 1 and 2 units from the two closest sides of the square. Find the area of the square. Solution: Call the point in question A , the center of the circle O , and its radius r . Consider a right triangle BOA with hypotenuse OA : OA has length r , and BO and BA 2 2 2 have lengths r − 1 and r − 2. By the Pythagorean theorem, ( r − 1) + ( r − 2) = r ⇒ 2 2 r − 6 r + 5 = 0 ⇒ r = 5 since r > 4. The area of the square is (2 r ) = 100 .