HMMT 二月 2022 · 几何 · 第 2 题

HMMT February 2022 — Geometry — Problem 2

Rectangle R has sides of lengths 3 and 4. Rectangles R , R , and R are formed such that: 0 1 2 3 • all four rectangles share a common vertex P , • for each n = 1 , 2 , 3, one side of R is a diagonal of R , n n − 1 • for each n = 1 , 2 , 3, the opposite side of R passes through a vertex of R such that the center n n − 1 of R is located counterclockwise of the center of R with respect to P . n n − 1 R 3 R 2 R 1 R 0 Compute the total area covered by the union of the four rectangles. 20

解析

Rectangle R has sides of lengths 3 and 4. Rectangles R , R , and R are formed such that: 0 1 2 3 • all four rectangles share a common vertex P , • for each n = 1 , 2 , 3, one side of R is a diagonal of R , n n − 1 • for each n = 1 , 2 , 3, the opposite side of R passes through a vertex of R such that the center n n − 1 of R is located counterclockwise of the center of R with respect to P . n n − 1 R 3 R 2 R 1 R 0 Compute the total area covered by the union of the four rectangles. Proposed by: Grace Tian Answer: 30 Solution: Let ABCD be R such that AB = 3 and BC = 4. Then, let AC be a side length of R and 0 1 let the other two vertices be E and F such that B lies on segment EF . Notice that the area of △ ABC is both half of the area of R and half of the area of R . This means forming R adds half of the area 0 1 1 of R to the union of rectangles. Similarly, forming R adds half of the area of R to the union of all 0 2 1 rectangles, and the same for R . This means the total area of the union of rectangles is given by 3 1 1 1 1 1 1 5 5 [ R ] + [ R ] + [ R ] + [ R ] = [ R ] + [ R ] + [ R ] + [ R ] = [ R ] = (3 · 4) = 30 . 0 1 2 3 0 0 0 0 0 2 2 2 2 2 2 2 2 Note that in the above equation, [ X ] denotes the area of shape X . 20