HMMT 二月 2001 · GEN1 赛 · 第 6 题
HMMT February 2001 — GEN1 Round — Problem 6
题目详情
- A right triangle has a hypotenuse of length 2, and one of its legs has length 1. The altitude to its hypotenuse is drawn. What is the area of the rectangle whose diagonal is this altitude? 2 4 8
解析
- A right triangle has a hypotenuse of length 2, and one of its legs has length 1. The altitude to its hypotenuse is drawn. What is the area of the rectangle whose diagonal is this altitude? 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 √ 3 BC . Triangle BDC is similar to triangle ABC , so BD = . Triangle DBF is similar to 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