Numerical answer
How many points with both coordinates integers lie strictly inside the triangle whose vertices are (0, 0), (10, 0) and (0, 10)?
Show answer and solution
Answer: 36
Interior points need x ≥ 1, y ≥ 1 and x + y ≤ 9 (x + y = 10 lies on the hypotenuse).
For x = 1, y can be 1..8 (8 values); x = 2 gives 7; ... ; x = 8 gives 1.
Total = 8 + 7 + ... + 1 = 36.
Check with Pick's theorem: Area = 50, boundary lattice points B = 30 (10 on each side), so interior I = A − B/2 + 1 = 50 − 15 + 1 = 36 ✓.
For x = 1, y can be 1..8 (8 values); x = 2 gives 7; ... ; x = 8 gives 1.
Total = 8 + 7 + ... + 1 = 36.
Check with Pick's theorem: Area = 50, boundary lattice points B = 30 (10 on each side), so interior I = A − B/2 + 1 = 50 − 15 + 1 = 36 ✓.
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