PUMaC 2012 · 几何（B 组） · 第 3 题

PUMaC 2012 — Geometry (Division B) — Problem 3

[ 4 ] Let A be a regular 12-sided polygon. A new 12-gon B is constructed by connecting the midpoints of the sides of A . The ratio of the area of B to the area of A can be written in √ simplest form as ( a + b ) /c , where a, b, c are integers. Find a + b + c .

解析

[ 4 ] Let A be a regular 12-sided polygon. A new 12-gon B is constructed by connecting the midpoints of the sides of A . The ratio of the area of B to the area of A can be written in √ simplest form as ( a + b ) /c , where a, b, c are integers. Find a + b + c . Solution: Note that the apothem of A is the circumradius of B , and A and B are similar. Therefore the ratio of the area of B to the area of A is the square of the ratio of the circumradius of B to the circumradius of A . Draw a right triangle with an apothem, half a side, and a 1 10 ◦ ◦ circumradius of A . A 12-gon has an interior angle of ( )180 = 150 , so the angle between 12 ◦ the side and the circumradius is 75 . Therefore the ratio of the apothem (circumradius of B ) ◦ to the circumradius (of A ) is sin 75 . Using the angle addition formula, ◦ ◦ ◦ ◦ ◦ sin 75 = sin 45 cos 30 + cos 45 sin 30 √ √ √ 2 3 2 1 = + 2 2 2 2 √ √ 2(1 + 3) ◦ sin 75 = 4 √ To find the ratio of the areas, we square this ratio, which becomes (2+ 3) / 4. So a + b + c = 9 .