PUMaC 2014 · 团队赛 · 第 4 题

PUMaC 2014 — Team Round — Problem 4

[ 5 ] ABC is a right triangle with AC = 3, BC = 4, AB = 5. Squares are erected externally on the sides of the triangle. Evaluate the area of the hexagon PQRSTU.

解析

[ 5 ] ABC is a right triangle with AC = 3, BC = 4, AB = 5. Squares are erected externally on the sides of the triangle. Evaluate the area of the hexagon PQRSTU. Solution: The hexagon is made up of 3 triangles and 3 squares and ABC . The area of each of these 1 triangles is the same as the area of ABC because the area of each of these triangles is ab sin θ . 2 The angle is supplementary to one of the angles in ABC and sin(180 − θ ) = sin θ while the 2 2 2 side lengths are the same. Hence the total area is: 3 + 4 + 5 + 4[ ABC ] = 74 .