HMMT 二月 2013 · 几何 · 第 1 题

HMMT February 2013 — Geometry — Problem 1

Jarris the triangle is playing in the ( x, y ) plane. Let his maximum y coordinate be k . Given that he has side lengths 6, 8, and 10 and that no part of him is below the x -axis, find the minimum possible value of k .

解析

Jarris the triangle is playing in the ( x, y ) plane. Let his maximum y coordinate be k . Given that he has side lengths 6, 8, and 10 and that no part of him is below the x -axis, find the minimum possible value of k . 24 Answer: By playing around, we find that Jarris should have his hypotenuse flat on the x - 5 axis. The desired minimum value of k is then the length of the altitude to the hypotenuse. Thus, by 1 1 24 computing the area of the triangle in two ways, · 10 · k = · 6 · 8 and so k = . 2 2 5