Triangles in a grid
In the grid of dots above, each dot has equal vertical & horizontal spacing from the others. A small 45-45-90 triangle is drawn. Counting this triangle, how many triangles congruent to this one?
Note: Congruent includes rotated triangles, but not dilated (scaled) ones.
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1 solution
In each square, there are 4 such triangles. There are 1 + 2 + … + 7 = 2 8 such squares, so this gives 2 8 × 4 = 1 1 2 triangles.
Then, there is one such triangle along the diagonal edge, and there are 8 of these.
Thus, there are 1 1 2 + 8 = 1 2 0 triangles in all.