1998 AJHSME Problems/Problem 6
Problem
Dots are spaced one unit apart, horizontally and vertically. The number of square units enclosed by the polygon is
Solutions
Solution 1
By inspection, you can notice that the triangle on the top row matches the hole in the bottom row.
This creates a box, which has area
Solution 2
We could count the area contributed by each square on the grid:
Top-left:
Top: Triangle with area
Top-right:
Left: Square with area
Center: Square with area
Right: Square with area
Bottom-left: Square with area
Bottom: Triangle with area
Bottom-right: Square with area
Adding all of these together, we get or
Solution 3
By Pick's Theorem, we get the formula, where is the number of lattice points in the interior and being the number of lattice points on the boundary. In this problem, we can see that and . Substituting gives us Thus, the answer is
See Also
1998 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.