HMMT 二月 2006 · 几何 · 第 4 题

英文解析

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