How do I find the area of a triangle when I only know the coordinates of its vertices?

Use the coordinate area formula: Area = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. Each vertex's x is multiplied by the difference of the other two vertices' y-coordinates, then you add them, halve, and take the positive value. For vertices (1, 1), (4, 5), (7, 1): Area = ½ |1(5 − 1) + 4(1 − 1) + 7(1 − 5)| = ½ |4 + 0 − 28| = ½ × 24 = 12 square units. Always use the modulus so the area is never negative. A bonus: if the area comes out as 0, the three points are collinear, meaning they lie on one straight line. Substitute carefully and follow the cyclic pattern of the subscripts.