PSL4#11

the line that marks the top of the triangle for x<0 is given by 

y = 5+x/2

we can list the points such that y>0, y< 5+x/2

let's list the y boundary for each of the values -5 < x < 0

(-9, 0.5), (-8, 1), (-7, 1.5), (-6, 2), (-5, 2.5), (-4, 3), (-3, 3.5), (-2, 4), (-1, 4.5)

so we see the following points are in the interior for x<0

(-7, 1), (-6, 1), (-5, 1), (-5, 2), (-4,1), (-4, 2), (-3, 1), (-3, 2), (-3, 3), (-2, 1), (-2, 2), (-2, 3), (-1, 1), (-1, 2), (-1, 3), (-1, 4)

16 points.  There are 16 simllar points for x>0.  That brings us to 32 points

Then on the y axis there is (0,1), (0,2), (0,3), (0,4) which brings the total number to 36.

Here's another way to solve this......

The area  of the triangle  =   

Integer coordinates on the boundary / 2  +  integer coordinates in the interior   - 1

The area  of the  triangle  is   20 * 5 / 2 =    50   units ^2

The line  joining   (-10,0)  and (0, 5)   has the equation     y  = x/2  + 5

So we will have integer  coodinates on this line when x=  -10, x = -8, x = -6,  x = -4,  x  = -2 and  x  = 0

Using symmety....we will also have integer coordinates when  x = 2, x = 4 ,  x = 6  x  = 8  and x  = 10

And we have  19 additional  integer coordinates on the  base

So.....the number of  integer coordinates on the  boundary  = 11 + 19  = 30

So

50 =  30 / 2  +   integer  coordinates in the interior  - 1

50  = 30/2  +  integer coordinates in the interior   - 1

50  =  15  - 1   + integer coordinates in the  interior

50 =  14   +  integer coodinates in the interior

36   =    integer coodinates in the interior