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# rectangular region

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Consider the rectangular region with the following points as vertices: (5,4), (-5,4), (-5,-8), (5,-8) How many points with integer coordinates will be strictly in the interior of this rectangular region?

Jan 3, 2021

#1
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These poitns with integer coordinates  are known as lattice points

Let A  = (5,4)   B = (-5,4)  C  = (-5, -8)   D  = (5, -8)

AB =  10      BC = 12

The  number  = (AB  -1)  ( BC  -1)   =   (10 - 1)  ( 12 -1)  =   9 * 11   =   99   Jan 3, 2021
edited by CPhill  Jan 4, 2021
#2
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Lets lay out these points in their appropriate quadrants.

(-5, 4)                         (5, 4)

(-5, -4)                        (5, -4)

If we look at the horizontal direction, the x coordinates in the boundaries are -4, -3, -2, -1, 0, 1, 2, 3, 4

If we look at the vertical direction, the y coordinate in the boundaries are -3, -2, -1, 0, 1, 2, 3

If you draw them out, you will see.

The horizontal has 9 x-coordinates.  The vertical has 7 y-coordinates. These points form an array inside the rectangle. If we multiply 9 by 7, we get 63.

There are 63 integer coordinate points inside this rectangle.

Jan 4, 2021