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HMMT 二月 2002 · 冲刺赛 · 第 34 题

HMMT February 2002 — Guts Round — Problem 34

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [7] Points P and Q are 3 units apart. A circle centered at P with a radius of 3 unitsintersects a circle centered at Q with a radius of 3 units at points A and B . Find the areaof quadrilateral APBQ.
解析

英文解析

  1. Points P and Q are 3 units apart. A circle centered at P with a radius of 3 unitsintersects a circle centered at Q with a radius of 3 units at points A and B . Find the areaof quadrilateral APBQ.
    Solution: The area is twice the area of triangle AP Q , which is isosceles with side lengths
    √
    √ √ √
    2 2
    3 , 3 , 3. By Pythagoras, the altitude to the base has length 3 − ( 3 / 2) = 33 / 2, so
    √ √
    99 3 11
    the triangle has area . Double this to get .
    4 2