HMMT 十一月 2013 · GEN 赛 · 第 9 题

英文解析

9. [ 7 ] Let ABC be a triangle and D a point on BC such that AB = 2, AC = 3, ∠ BAD = 30 , and
   °
   ∠ CAD = 45 . Find AD .
   √ √
   6 3
   √
   Answer: OR Note that [ BAD ] + [ CAD ] = [ ABC ]. If α = ∠ BAD , α = ∠ CAD , then
   1 2
   sin( α + α )22
   sin α sin α
   1 2
   1 2
   we deduce = + upon division by AB · AC · AD . Now
   AD AC AB
   ° °
   sin(30 + 45 )
   AD = .
   ° °
   sin 30 sin 45
   √ √
   3 2+
   √ √
   ° °
   1 3 6 sin 30 sin 45
   ° ° ° ° ° ° ° °
   √ √ √
   But sin(30 + 45 ) = sin 30 cos 45 + sin 45 cos 30 = sin 30 + sin 45 = ( + ), so
   2 2
   2 3 2
   √
   our answer is .6 2