HMMT 二月 2001 · GEN1 赛 · 第 6 题

英文解析

6. A right triangle has a hypotenuse of length 2, and one of its legs has length 1. Thealtitude to its hypotenuse is drawn. What is the area of the rectangle whose diagonal is thisaltitude?
   Solution: Call the triangle ABC , with AC = 2 and BC = 1. By the Pythagorean
   √
   theorem, AB = 3. Call the point at which the altitude intersects the hypotenuse D . Let
   E 6 = B be the vertex of the rectangle on AB and F 6 = B be the vertex of the rectangle on
   √
   BC . Triangle BDC is similar to triangle ABC , so BD = . Triangle DBF is similar to 3
   √ √ √2
   3 3 3 3 3 3
   triangle ABC , so DF = and BF = . The area of the rectangle is thus = .
   4 4 4 4 16
   2 4 8