HMMT 二月 2014 · 几何 · 第 2 题
HMMT February 2014 — Geometry — Problem 2
题目详情
英文原题
- Point P and line ` are such that the distance from P to ` is 12. Given that T is a point on
such that P T = 13, find the radius of the circle passing through P and tangent toat T .
解析
英文解析
- Point P and line ℓ are such that the distance from P to ℓ is 12. Given that T is a point on ℓ such that
P T = 13, find the radius of the circle passing through P and tangent to ℓ at T .
Answer: 169/24 Let O be the center of the given circle, Q be the foot of the altitude from P toℓ , and M be the midpoint of P T . Then since OM ⊥ P T and ∠ OT P = ∠ T P Q , ∆ OM P ∼ ∆ T QP .
13 / 2
P M 169
Thus the OP = T P ·
= 13 · =
P Q 12 24