triangles,triangles,triangles......
Geometry
Level
4
The no. of integral points (both the coordinates should be integers) that lie exactly in the interior of the triangles with vertices , is
The answer is 190.
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1 solution
Since in 21 distances there are 21+1=22 lines. But two of them are on the boundaries. So inside there are only 20 lines in a 21x21 square.
So there will be 20x20=400 intersections. But for a triangle, the diagonal of the square is an additional boundary line robbing 20 intersections, 400-20=380.
Half of this will be in one triangle. So the no. of integral points = 380/2=190.
OR.
In the triangle we will have 3 boundaries. So interior intersections will be 22-3=19.
The top most line will have one, next 2, and so on. The last 19th line will have 19. So total= 2 n ∗ ( n + 1 ) = 2 1 9 ∗ ( 1 9 + 1 ) = 1 9 0 .