HMMT 十一月 2013 · GEN 赛 · 第 9 题

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 2 2 sin( α + α ) 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 √ 6 our answer is . 2