PUMaC 2015 · 几何（B 组） · 第 1 题

PUMaC 2015 — Geometry (Division B) — Problem 1

[ 3 ] Find the distance CF in the diagram below where ABDE is a square and angles and lengths are as given: E D 8 F 15 15 C 8 A B ” The length CF is of the form a b for integers a, b such that no integer square greater than 1 divides b . What is a + b ?

解析

[ 3 ] Find the distance CF in the diagram below where ABDE is a square and angles and lengths are as given: E D 8 F 15 15 C 8 A B ” The length CF is of the form a b for integers a, b such that no integer square greater than 1 divides b . What is a + b ? Answer: 25 Solution: Note by symmetry that F E and CD can be extended until they intersect at point G such that GE has length 15 and GD has length 8. Furthermore, ∠ G is a right angle. Thus ” F GC is an isosceles, right triangle, and CF has length 23 2.