HMMT 二月 2006 · 几何 · 第 8 题
HMMT February 2006 — Geometry — Problem 8
题目详情
英文原题
- Triangle ABC has a right angle at B . Point D lies on side BC such that 3 ∠ BAD =
∠ BAC . Given AC = 2 and CD = 1, compute BD .
解析
英文解析
- Triangle ABC has a right angle at B . Point D lies on side BC such that 3 ∠ BAD =
∠ BAC . Given AC = 2 and CD = 1, compute BD .
Answer:3
′ ′8
Solution: Let BD = x . We reflect D over AB to D . Then DD = 2 x , but AD
√ √
′ ′
2 2 2 2 2
bisects CAD , so 4 x = AD = AD . Also, AD = x + AB = x + AC − BC =
√
√
2 22
x + 4 − ( x + 1) = 3 − 2 x . We have the quadratic 16 x = 3 − 2 x which givesx = 3 / 8.