Area of Triangle - A Determinant Approach?

📌 CONCEPT: The area of a triangle can be calculated using the determinant approach, which involves using the formula involving the coordinates of the vertices of the triangle.

📐 RULE / FORMULA: The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by the formula: Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))| / 2.

💡 WORKED EXAMPLE: Given the vertices (1, 2), (3, 4), and (x, y) of a triangle with an area of 13 square units, we can use the determinant approach to show that the point (x, y) lies on the circle x^2 + y^2 - 6x - 8y = 0. The determinant formula for area is applied with the given vertices and the area is set equal to 13. This leads to an equation that simplifies to x^2 + y^2 - 6x - 8y = 0, showing that the point (x, y) lies on the circle.

⚠️ COMMON MISTAKE: Students may forget to apply the absolute value in the determinant formula for area, leading to incorrect calculations.