The lattice polygon with vertices (in this order): (0, 0), ( 1,2), (3,1), (2, –1), (0, 0) has area 5.
True ; False
True. You can also see this directly: the polygon is a square of side length 5.
2.
The area of the lattice polygon with vertices (in this order): (0, 0), (3, 1), (2, –1), (1, 2), (0, 0) is given by Pick’s Theorem.
True ; False
False. The standard form of Pick’s Theorem only applies to simple lattice polygons.
3.
Any simple lattice polygon can be split into lattice triangles having B = 3 and I = 0.
True ; False
True. We have discussed splitting polygons into triangles with B = 3. If there are interior points, further segments can be added to obtain this result.
4.
Pick’s Theorem continues to hold when the polygon is just a unit segment.
True ; False
True, but it appears to be a fluke. Try a longer segment.
5.
There is a lattice polygon which has area equal to half the number of boundary points.
True ; False
True. Using Pick’s Theorem we require I = 1. Now take the square with vertices (0, 0),
(2, 0), (2, 2), and (0, 2). This has A = 4, B = 8.
6.
There is a lattice polygon which has area equal to half the number of interior points.
True ; False
False. We require A = c. Now Pick’s Theorem tells us that B = 2. A polygon requires at least three vertices. (A line segment has no true ‘interior’ points.)
5.