Prove that the formula is valid for rectangles with sides parallel to the coordinate axes

In summary, the conversation discusses a formula for calculating the area of a polygon with lattice point vertices, where Z represents the number of lattice points inside the polygon and B represents the number of points on the boundary. Part (a) suggests using the exhaustion property to find two regions that are subsets and supersets of the rectangle to prove the formula's validity. However, it is actually more straightforward to imagine a rectangle with lattice point vertices and sides parallel to the axes, and count the number of Z and B points to verify the formula. Part (b) prompts using induction to prove the formula for general polygons.
  • #1
courtrigrad
1,236
2
Let P be a polygon whose vertices are lattice points. The area of P is [tex] Z + \frac{1}{2}B - 1 [/tex]. Z is the number of lattice points inside the polygon, and B is the number on the boundary.

(a) Prove that the forumula is valid for rectangles with sides parallel to the coordinate axes.
(b) Use induction on the number of edges to construct a proof for general polygons.

(a) Would you have to use the exhaustion property? We have to find two regions such that the area of a rectangle with sides parallel to the coordinate axes is a subet and superset of. There can be only one c such that c = [tex] Z + \frac{1}{2}B - 1 [/tex]. The question is, how do we determine the two step regions?
(b) I don't know that this question is asking me to prove.

Thanks
 
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  • #2
For b, what do you mean you don't know what it's asking you to prove. It's entirely clear: prove that for any polygon whose vertices are lattice points that its area is Z + B/2 - 1 where Z is the number of lattice points inside the polygon and B is the number on the boundary. Moreover, it says to use induction to prove it.

For a, it's really quite straight forward. There's no need for any "exhaustion property"s or anything to do with subsets or supersets. Just imagine a rectangle whose vertices are lattice points and whose sides are parallel to the axes. You should know how to compute the area of an arbitrary rectangle of this form, it's just length x width. But the length and with of such a rectangle uniquely determine the number of Z points and B points, and you can easily just count these numbers, and verify that the formula holds.

For example, consider the square whose vertices are (0,0), (1,0), (0,1), (1,1). It has 0 interior points, and 4 boundary points. The formula predicts its area to be 0 + 4/2 - 1 = 1, and indeed you know the are of such a square is just 1 x 1 = 1, so the formula does work. Prove this for an arbitrary rectangle.
 

1. How do you prove the formula for rectangles with sides parallel to the coordinate axes?

To prove the formula for rectangles with sides parallel to the coordinate axes, we use the properties of rectangles and the coordinate plane. We can also use algebraic equations to show that the formula holds true for all rectangles with sides parallel to the coordinate axes.

2. What is the formula for finding the area of a rectangle with sides parallel to the coordinate axes?

The formula for finding the area of a rectangle with sides parallel to the coordinate axes is A = length x width. This means that the area is equal to the length of one side multiplied by the length of the other side.

3. Can the formula be applied to all rectangles?

Yes, the formula for finding the area of a rectangle with sides parallel to the coordinate axes can be applied to all rectangles, as long as the sides are parallel to the coordinate axes. This includes squares, which are a special type of rectangle with all sides equal in length.

4. How does the formula relate to the concept of Cartesian coordinates?

The formula for rectangles with sides parallel to the coordinate axes is closely related to the concept of Cartesian coordinates. The x and y coordinates of a rectangle's vertices can be used to determine the length and width of the rectangle, which are then used in the formula to find the area.

5. Can the formula be extended to find the area of other shapes?

Yes, the formula for rectangles with sides parallel to the coordinate axes can be extended to find the area of other shapes, such as parallelograms and triangles, by breaking them down into smaller rectangles. However, this may require additional calculations and adjustments to the formula.

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