How many points with integer coordinates are in the triangle with vertices #(0, 0)#, #(0, 21)# and #(21, 0)#, including the points on its boundary?

1 Answer
George C.
Jan 6, 2018

#253#

Explanation:

The number of integral points on the diagonal line segment joining #(0, n)# and #(n, 0)# is #n+1#, being #(0, n)#, #(1, n-1)#, #(2, n-2)#,..., #(n, 0)#.

So there are #22# integral points on the line segment #x+y = 21# between #(0, 21)# and #(21, 0)#, then #21# points on the next diagonal joining #(0, 20)# and #(20, 0)#, and so on down to one point at #(0, 0)#.

So the total number of integral points in the triangle is:

#sum_(n=1)^22 n = 1/2 (color(blue)(22))(color(blue)(22)+1) = 253#

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