HMMT 二月 2014 · 几何 · 第 3 题
HMMT February 2014 — Geometry — Problem 3
题目详情
英文原题
- ABC is a triangle such that BC = 10, CA = 12. Let M be the midpoint of side AC . Given that BMis parallel to the external bisector of \ A , find area of triangle ABC . (Lines AB and AC form twoangles, one of which is \ BAC . The external bisector of \ A is the line that bisects the other angle.)
解析
英文解析
- ABC is a triangle such that BC = 10, CA = 12. Let M be the midpoint of side AC . Given that BMis parallel to the external bisector of ∠ A , find area of triangle ABC . (Lines AB and AC form twoangles, one of which is ∠ BAC . The external bisector of ∠ A is the line that bisects the other angle.)
√
Answer: 8 14 Since BM is parallel to the external bisector of ∠ A = ∠ BAM , it is perpendicularto the angle bisector of ∠ BAM . Thus BA = BM = BC = 6. By Heron’s formula, the area of 1
√ √2
∆ ABC is therefore (14)(8)(4)(2) = 8 14.
° ° °