HMMT 二月 2002 · 冲刺赛 · 第 34 题

HMMT February 2002 — Guts Round — Problem 34

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

解析

Points P and Q are 3 units apart. A circle centered at P with a radius of 3 units intersects a circle centered at Q with a radius of 3 units at points A and B . Find the area of 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