A triangle has vertices at points A(2, 3), B(6, 7), and C(8, 2). Calculate the area of this triangle using the determinant method

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In this problem, we need to calculate the area of a triangle with vertices at points A(2, 3), B(6, 7), and C(8, 2). We'll use the determinant method, which is a powerful technique for finding the area of a triangle when we know the coordinates of its vertices. Now, let's substitute the coordinates of our triangle vertices into the determinant formula. We have A at (2, 3), B at (6, 7), and C at (8, 2). Plugging these values into our formula, we get a 3 by 3 determinant with the x-coordinates in the first column, y-coordinates in the second column, and all ones in the third column. Now, let's calculate the value of the determinant. We'll expand along the first row using the cofactor expansion method. For each element in the first row, we multiply by its cofactor, which is the determinant of the submatrix obtained by removing the row and column of that element. After expanding and calculating each term, we get negative 28 as the value of our determinant. Now that we have the determinant value of negative 28, we can calculate the area of the triangle. The area formula requires the absolute value of the determinant multiplied by one-half. Taking the absolute value of negative 28 gives us 28, and multiplying by one-half gives us 14 square units. Therefore, the area of the triangle with vertices at A(2, 3), B(6, 7), and C(8, 2) is 14 square units. To summarize what we've learned: The determinant method provides an elegant way to calculate the area of a triangle when we know the coordinates of its vertices. We set up a 3 by 3 determinant with the x-coordinates in the first column, y-coordinates in the second column, and ones in the third column. The area equals one-half times the absolute value of this determinant. For our triangle with vertices at (2,3), (6,7), and (8,2), we calculated an area of 14 square units. This method can be extended to find the area of any polygon by dividing it into triangles.