Prove that if a is a natural number and the square root of a is
rational, then it is a square number (an integer n^2 for some
Proof of Pick's Theorem
Stage: 5 Challenge Level:
Follow the five steps in the question. These steps lead you to the
(1) Here you are glueing 2 polygons along a common edge and the
area of the 'sum' of the polygons is the sum of the areas of the
individual polygons. Decide which lattice points on the common
edges become interior lattice points to the new polygon and which
are on the edge of the new polygon and make sure none of these is
(2) Proving Pick's Theorem for a rectangle simply involves counting
(3) Now deduce Pick's Theorem for right-angled triangles.
(4) Now you know Pick's Theorem holds for rectangles and
right-angled triangles deduce that it holds for the general
(5) Finally deduce Pick's Theorem for the general plane polygon
using the earlier parts of the proof.
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