PUMaC 2014 · 几何（B 组） · 第 2 题

PUMaC 2014 — Geometry (Division B) — Problem 2

[ 3 ] Consider the pyramid OABC . Let the equilateral triangle ABC with side length 6 be the base. Also 9 = OA = OB = OC . Let M be the midpoint of AB . Find the square of the distance from M to OC .

解析

[ 3 ] Consider the pyramid OABC . Let the equilateral triangle ABC with side length 6 be the base. Also 9 = OA = OB = OC . Let M be the midpoint of AB . Find the square of the distance from M to OC . Solution: Let D be the center of the equilateral triangle. Take a slice of the pyramid that goes through p the apex and M C . Then we get a triangle with base 3 3 and OC = 9. Dropping the p p 2 2 2 perpendicular from O to D , we have that OD = 9 (2 3) ) OD = 69. Now if we let the perpendicular from M to OC intersect OC at E , we have that M E · OC = M C · OD = p p p p 2 3 3 69 = 9 23 ) M E = 23 ) M E = 23 .