HMMT 十一月 2008 · 冲刺赛 · 第 27 题
HMMT November 2008 — Guts Round — Problem 27
题目详情
- [ 13 ] ABCDE is a regular pentagon inscribed in a circle of radius 1. What is the area of the set ofpoints inside the circle that are farther from A than they are from any other vertex?
1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUNDst
英文原题
[ 13 ] ABCDE is a regular pentagon inscribed in a circle of radius 1. What is the area of the set of
points inside the circle that are farther from A than they are from any other vertex?
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1 st HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUND
解析
英文解析
- [ 13 ] ABCDE is a regular pentagon inscribed in a circle of radius 1. What is the area of the set ofpoints inside the circle that are farther from A than they are from any other vertex?
Answer: Draw the perpendicular bisectors of all the sides and diagonals of the pentagon withπone endpoint at A . These lines all intersect in the center of the circle, because they are the set of 5
points equidistant from two points on the circle. Now, a given point is farther from A than from point
X if it is on the X side of the perpendicular bisector of segment AX . So, we want to find the area ofthe set of all points which are separated from A by all of these perpendicular bisectors, which turnsout to be a single 72 sector of the circle, which has area .oπ
1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 SATURDAY 2008 — GUTS ROUNDst