HMMT 十一月 2009 · 冲刺赛 · 第 20 题

HMMT November 2009 — Guts Round — Problem 20

[ 11 ] Consider an equilateral triangle and a square both inscribed in a unit circle such that one side of the square is parallel to one side of the triangle. Compute the area of the convex heptagon formed by the vertices of both the triangle and the square. 2

解析

[ 11 ] Consider an equilateral triangle and a square both inscribed in a unit circle such that one side of the square is parallel to one side of the triangle. Compute the area of the convex heptagon formed by the vertices of both the triangle and the square. √ 3+ 3 Answer: 2 Consider the diagram above. We see that the shape is a square plus 3 triangles. The top and bottom √ √ √ 1 triangles have base 2 and height ( 3 − 2), and the triangle on the side has the same base and 2 √ 2 height 1 − Adding their areas, we get the answer. 2 2