HMMT 二月 2002 · 几何 · 第 5 题
HMMT February 2002 — Geometry — Problem 5
题目详情
英文原题
- Consider a square of side length 1. Draw four lines that each connect a midpoint of aside with a corner not on that side, such that each midpoint and each corner is touched byonly one line. Find the area of the region completely bounded by these lines.
解析
英文解析
- Consider a square of side length 1. Draw four lines that each connect a midpoint of aside with a corner not on that side, such that each midpoint and each corner is touched byonly one line. Find the area of the region completely bounded by these lines.
Solution: In unit square ABCD , denote by E, F, G, H the respective midpoints of sides 1
AB, BC, CD, DA . Let I be the intersection of AF and DE , let J be the intersection of BGand AF , let K be the intersection of CH and BG , and let L be the intersection of DE and
CH . We want to find the area of square IJKL . The area of ABF is , which is equal to 1
√4
1 5 1 1
√
AF · BJ = BJ , so BJ = . Using similar triangles, GK = JF = BJ . Thus the
2 4 2
√5
5 1 1 1 1 1
√ √ √
length of a side of IJKL is JK = − − = , and the area of IJKL is .
2 2 5
5 5 5