Consider the rectangle with vertices at (5,4), (-5, -4) and (5, -4). How many points with 2 integer coordinates will be strictly inside this rectangular region?

Question

Do we need graph paper for this, please help so confused.

Answer 1

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.

Answer 2

Yes, you must use graph paper and draw the
rectangle. You must draw the picture in order
to solve the problem. It is not easy to visualize
the rectangle, the coordinates and count the
points inside it all at the same time without
a picture to help you. You must draw it !!!1


Once you draw it, the vertices are (5,4), (5,-4),
(-5,-4) and (-5,4) in the clockwise directions starting
from the upper left corner.

Since we are interested in points that are strictly inside
the rectangle -4 < y < 4 and -5 < x < 5;
That is y = -3,-2,-1,0,1,2,3
and x = -4,-3,-2,-1,0,1,2,3,4


The points with integer coordinates that are strictly
inside the rectangle are then:



(-4,3)   (-3,3),   (-2,3),   (-1,3),   (0,3),   (1,3),   (2,3),   (3,3),  (4,3)
(-4,2)   (-3,2),   (-2,2),   (-1,2),   (0,2),   (1,2),   (2,2),   (3,2),  (4,2)
(-4,1)   (-3,1),   (-2,1),    (-1,1),  (0,1),   (1,1),    (2,1),  (3,1),  (4,1)
(-4,0)   (-3,0),   (-2,0),    (-1,0)   (0,0),   (1,0),    (2,0),   (3,0),  (4,0)
(-4,-1)  (-3,-1),  (-2,-1),  (-1,-1), (0,-1),  (1,-1),  (2,-1),  (3,-1), (4,-1)
(-4,-2) (-3,-2),   (-2,-2), (-1,-2),  (0,-2),  (1,-2),  (2,-2),  (3,-2), (4,-2)
(-4,-3) (-3,-3),   (-2,-3), (-1,-3),  (0,-3),  (1,-3),   (2,-3), (3,-3), (4,-3)