Question

All vertices of a convex pentagon are lattice points, and its sides have integral length. Show that its perimeter is even.​

Answer 1

Answer:

All vertices of a convex pentagon are lattice points, and its sides have integral length. Show that its perimeter is even." - Problem Solving Strategies, Arthur Engel, pg. 27.

I have proven the theorem through two methods. I thought about a third method, but it appears faulty, since it shows that the theorem is false. Please indicate me where I have made the error.

Consider two vertices of the pentagon. Since the points are in a lattice, and the sides are integers, Δx,Δy,d∈N . It is also obvious that Δx2+Δy2=d2.

That means that those three integers form a Pythagorean triplet. ∴d is odd, which applies to all sides.

Since the perimeter is the sum of all sides,

p=∑i=15di∀di,2∤di⎫⎭⎬⎪⎪⇒p is odd.

I can't seem to find the mistake. Help would be appreciated.

Answer 2

Answer:

All vertices of a convex pentagon are lattice points, and its sides have integral length. Show that its perimeter is even." - Problem Solving Strategies, Arthur Engel, pg. 27.

I have proven the theorem through two methods. I thought about a third method, but it appears faulty, since it shows that the theorem is false. Please indicate me where I have made the error.

Consider two vertices of the pentagon. Since the points are in a lattice, and the sides are integers, Δx,Δy,d∈N . It is also obvious that Δx2+Δy2=d2.

That means that those three integers form a Pythagorean triplet. ∴d is odd, which applies to all sides.

Since the perimeter is the sum of all sides,

p=∑i=15di∀di,2∤di⎫⎭⎬⎪⎪⇒p is odd.

I can't seem to find the mistake. Help would be appreciated.

Step-by-step explanation:

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