HMMT 十一月 2014 · GEN 赛 · 第 2 题
HMMT November 2014 — GEN Round — Problem 2
题目详情
- Let ABC be a triangle with ∠ B = 90 . Given that there exists a point D on AC such that AD = DC AB and BD = BC , compute the value of the ratio . BC 8 12
解析
- Let ABC be a triangle with ∠ B = 90 . Given that there exists a point D on AC such that AD = DC AB and BD = BC , compute the value of the ratio . BC √ Answer: 3 D is the circumcenter of ABC because it is the midpoint of the hypotenuse. Therefore, DB = DA = DC because they are all radii of the circumcircle, so DBC is an equilateral √ AB ◦ ◦ ◦ ◦ triangle, and ∠ C = 60 . This means that ABC is a 30 − 60 − 90 triangle, with = 3 . BC 8 12