HMMT 二月 2008 · 冲刺赛 · 第 2 题
HMMT February 2008 — Guts Round — Problem 2
题目详情
- [ 5 ] Given right triangle ABC , with AB = 4 , BC = 3 , and CA = 5. Circle ω passes through A and is tangent to BC at C . What is the radius of ω ?
解析
- [ 5 ] Given right triangle ABC , with AB = 4 , BC = 3 , and CA = 5. Circle ω passes through A and is tangent to BC at C . What is the radius of ω ? 25 Answer: Let O be the center of ω , and let M be the midpoint of AC . Since OA = OC , 8 OM ⊥ AC . Also, ∠ OCM = ∠ BAC , and so triangles ABC and CM O are similar. Then, CO/CM = 25 AC/AB , from which we obtain that the radius of ω is CO = . 8