Pick's Theorem The points in the coordinate plane having integer coordinates are calle...

Pick's Theorem The points in the coordinate plane having integer coordinates are called lattice points. If a polygon has lattice points as vertices, it is called a lattice polygon. A theorem discovered in 1900 by an English mathematician. George Pick, provides a formula for the area (K) of a lattice polygon in terms of the number of lattice points on the boundary (B) and the number inside (I). It is reasonable to assume that the area of a lattice polygon is linear in B and I; hence, there exist constants x, y, and z such that K = xB + yI + z for all polygons.

(a) Complete the table below, and find various entries for K, B, and I, to be substituted into the expression K = xB + yI + z. (Using the table, one equation is 2 = x • 6 + y • 0 + z or 6x + z = 2.)

(b) Solve the system of three equations you found in (a) for x, y, and z. You now have a plausible formula for the area of a lattice polygon, K = xB + yI + z (after substituting the values you found for x, y, and z). Try it out on the remaining polygon. Does the formula work? (The actual proof of Pick's Theorem may be found in Coxeter, Introduction to Geometry, p. 34.1 )