Calculate the area of a triangle with vertices at \( (1, 2) \), \( (4, 6) \), and \( (5, 3) \) using the formula:
["# How to Calculate the Area of a Triangle Using Coordinates: A Step-by-Step Guide", "When given a triangle defined by its vertices’ coordinates, calculating its area becomes simple with the right formula. If you’re working with points ( A(1, 2) ), ( B(4, 6) ), and ( C(5, 3) ), this article walks you through computing the area using a well-known mathematical formula.", "## What Formula to Use?", "The most efficient method to find the area of a triangle when vertex coordinates are known is the Shoelace Formula (also known as determinant-based area calculation or Gauss’s area formula). For a triangle with vertices ( A(x_1, y_1) ), ( B(x_2, y_2) ), and ( C(x_3, y_3) ), the formula is:", "[\n\ ext{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|\n]", "Alternatively, using determinants, the area can be written as:", "[\n\ ext{Area} = \frac{1}{2} \left| \n\begin{vmatrix}\nx_1 & y_1 & 1 \\nx_2 & y_2 & 1 \\nx_3 & y_3 & 1 \\n\end{vmatrix}\n\right|\n]", "Both yield the same correct result.", "## Step-by-Step Calculation", "Given:\n- ( A(1, 2) \Rightarrow x_1 = 1, y_1 = 2 )\n- ( B(4, 6) \Rightarrow x_2 = 4, y_2 = 6 )\n- ( C(5, 3) \Rightarrow x_3 = 5, y_3 = 3 )", "Plug into the Shoelace Formula:", "[\n\ ext{Area} = \frac{1}{2} \left| 1(6 - 3) + 4(3 - 2) + 5(2 - 6) \right|\n]", "Simplify inside the absolute value:", "[\n= \frac{1}{2} \left| 1(3) + 4(1) + 5(-4) \right|\n= \frac{1}{2} \left| 3 + 4 - 20 \right|\n= \frac{1}{2} \left| -13 \right|\n= \frac{13}{2} = 6.5\n]", "## Conclusion", "The area of the triangle with vertices at ( (1, 2) ), ( (4, 6) ), and ( (5, 3) ) is ( 6.5 ) square units. Using the Shoelace Formula provides an accurate, efficient way to compute triangle areas based on coordinate geometry—ideal for students, engineers, and designers who rely on precise geometric calculations.", "---", "Keywords for SEO: Calculate area of a triangle, triangle area formula, Shoelace formula, triangle coordinates area, geometry formula, determine triangle area with coordinates, math calculation, coordinate geometry area, formula for triangle area from vertices."]
