Difference between revisions of "1998 AJHSME Problems/Problem 6"
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Bottom-right: Square with area <math>1</math> | Bottom-right: Square with area <math>1</math> | ||
− | Adding all of these together, we get <math> | + | Adding all of these together, we get <math>6</math> which is the same as <math>\boxed{B}</math> |
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===Solution 2=== | ===Solution 2=== |
Latest revision as of 16:52, 26 October 2020
Problem
Dots are spaced one unit apart, horizontally and vertically. The number of square units enclosed by the polygon is
Solutions
Solution 1
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 which is the same as
Solution 2
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
Solution 3
Notice that the extra triangle on the top with area can be placed (like a jigsaw puzzle) at the bottom of the grid where there is a triangular hole, also with area . This creates a rectangle, with a area of . The answer is ~sakshamsethi
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 |
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