A triangle has vertices at points A(2, 3), B(7, 8), and C(9, 3). 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(7,8), and C(9,3). We'll use the determinant method, which is a powerful technique for finding the area of a polygon given its vertices. The formula for the area of a triangle using determinants is shown here. We'll substitute the coordinates of our three points into this formula to find the area. Now, let's substitute the coordinates of our triangle vertices into the determinant formula. For point A, we have x1 equals 2 and y1 equals 3. For point B, x2 equals 7 and y2 equals 8. And for point C, x3 equals 9 and y3 equals 3. Next, we calculate the determinant by expanding along the first row. This gives us negative 35. Finally, we take the absolute value of the determinant and multiply by one-half to get the area. The area of our triangle is 17.5 square units. Let's explore the geometric interpretation of the determinant. The determinant actually represents the area of a parallelogram formed by the vectors from the origin to our points. For a triangle, we take half of this area. When we calculate the determinant, we got negative 35, which means the area of the parallelogram is 35 square units, but with a negative sign indicating the orientation of the vertices. The area of our triangle is therefore half of 35, which is 17.5 square units. This geometric interpretation helps us understand why the determinant method works for calculating areas. Besides the determinant method, there are several other ways to calculate the area of a triangle. The first alternative is the base-height method, where we multiply half the base length by the height. In our triangle, the base AC is 7 units, and the height from B to AC is 5 units, giving us an area of 17.5 square units. Another approach is Heron's formula, which uses the semi-perimeter and the lengths of all three sides. After calculating the side lengths and applying the formula, we again get 17.5 square units. A third method is the Shoelace formula, which is another way to calculate the area of any polygon using the coordinates of its vertices. All these methods give us the same result of 17.5 square units, confirming our answer. To summarize what we've learned: We calculated the area of a triangle with vertices at points A(2,3), B(7,8), and C(9,3) using the determinant method, finding it to be 17.5 square units. The determinant method is particularly useful when we know the coordinates of the vertices. Geometrically, the determinant represents the area of a parallelogram formed by vectors, so we take half of this value to get the triangle's area. We verified our result using multiple approaches: the base-height method, Heron's formula, and the shoelace formula, all confirming the same answer. Additionally, we learned that the sign of the determinant indicates the orientation of the vertices, which can be useful in more advanced geometric applications.