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# Help with counting

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Determine the number of ways of choosing three points from the grid below, so that they form an isosceles right triangle.

May 3, 2023

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Let's split this problem into cases:

Case 1 (1 x 1 isosceles triangle) - There are 36 1 x 1 squares, and each square can have a total of 4 triangles which makes for $$36 \times 4 = 144$$ triangles.

Case 2 (2 x 2 isosceles triangle) - There are 25 2 x 2 squares, and each square can have a total of 4 triangles, which makes for $$25 \times 4 = 100$$ triangles.

Case 3 (3 x 3 isosceles triangle) - There are 16 2 x 2 squares, and each square can have a total of 4 triangles, which makes for $$16\times 4 = 64$$ triangles.

Case 4 (4 x 4 isosceles triangle) - There are 9 4 x 4 squares, and each square can have a total of 4 triangles, which makes for $$9 \times 4 = 36$$ triangles.

Case 5 (5 x 5 isosceles triangle) - There are 4 5 x 5 squares, and each square can have a total of 4 triangles, which makes for $$4 \times 4 = 16$$ triangles.

Case 6 (6 x 6 isosceles triangle) - There is 1 6 x 6 square, and each square can have a total of 4 triangles, which makes for $$1 \times 4 = 4$$ triangles.

So, there are $$144 + 100 + 64 + 36 + 16 + 9 + 1 = \color{brown}\boxed{370}$$ isosceles right triangles.

May 3, 2023