1995 AHSME Problems/Problem 22
Problem
A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths and , although this is not necessarily their order around the pentagon. The area of the pentagon is
Solution
Since the pentagon is cut from a rectangle, the cut-off triangle must be right. Since all of the lengths given are integers, it follows that this triangle is a Pythagorean Triple. We know that and either must be the dimensions of the rectangle, since they are the largest lengths. With some trial-and-error, if we assign the shortest side, , to be the hypotenuse of the triangle, we see the triple. Indeed this works, by placing the side opposite from the side and the side opposite from the side, leaving the cutaway side to be, as before, .
To find the area of the pentagon, we subtract the area of the triangle from that of the big rectangle: .
See also
1995 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.