HMMT 二月 2006 · 几何 · 第 4 题

HMMT February 2006 — Geometry — Problem 4

Let ABC be a triangle such that AB = 2 , CA = 3, and BC = 4. A semicircle with its diameter on BC is tangent to AB and AC . Compute the area of the semicircle. √

解析

Let ABC be a triangle such that AB = 2 , CA = 3, and BC = 4. A semicircle with its diameter on BC is tangent to AB and AC . Compute the area of the semicircle. 27 π Answer: 40 Solution: Let O , D , and E be the midpoint of the diameter and the points of 1 tangency with AB and AC respectively. Then [ ABC ] = [ AOB ] + [ AOC ] = ( AB + 2 AC ) r , where r is the radius of the semicircle. Now by Heron’s formula, [ ABC ] = √ √ √ 9 1 3 5 3 15 3 15 1 2 27 π · · · = . We solve for r = and compute πr = . 2 2 2 2 4 10 2 40 √