Problem:
In the polygon shown, each side is perpendicular to its adjacent sides, and all of the sides are congruent. The perimeter of the polygon is . The area of the region bounded by the polygon is
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Answer Choices:
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Solution:
It is possible to sketch the polygon on a lattice of squares. Let the length of a side of the polygon be . The perimeter of the polygon is , so . The region bounded by the polygon consists of squares from the lattice, so its area is .
Instead of counting the squares as above, note that to form the figure from the lattice, squares are cut from each corner. Hence the area of the region is .
As shown, remove each of the four squares furthest from the center of the polygon, and use them to fill in the remaining four concave sections, . The area of the region bounded by the original polygon is the same as the area of the resulting segment by segment square. Since the length of each segment is , the required are is this .
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