Given the vertices \( (x_1, y_1) = (1, 2) \), \( (x_2, y_2) = (4, 6) \), \( (x_3, y_3) = (5, 3) \).

["# Exploring the Triangle with Vertices at (1, 2), (4, 6), and (5, 3)", "Understanding the shape and key properties of a triangle is essential in geometry, and defining its vertices is the first step. This article delves into the triangle formed by the vertices ( (x_1, y_1) = (1, 2) ), ( (x_2, y_2) = (4, 6) ), and ( (x_3, y_3) = (5, 3) ). We’ll explore how to calculate its area, side lengths, centroid, and other important features—all valuable knowledge for students, educators, and geometry enthusiasts.", "## Computing Side Lengths and Shape Characteristics", "Given the coordinates:", "- ( A(1, 2) )\n- ( B(4, 6) )\n- ( C(5, 3) )", "We calculate the lengths of the sides using the distance formula:", "[\nAB = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\n]", "[\nBC = \sqrt{(5 - 4)^2 + (3 - 6)^2} = \sqrt{1^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}\n]", "[\nCA = \sqrt{(5 - 1)^2 + (3 - 2)^2} = \sqrt{4^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17}\n]", "The triangle has side lengths ( AB = 5 ), ( BC = \sqrt{10} ), ( CA = \sqrt{17} ), confirming it’s a scalene triangle (all sides of different lengths).", "## Calculating the Area Using the Shoelace Formula", "The shoelace formula efficiently computes the area of any polygon when vertex coordinates are known:", "[\n\ ext{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_1 - y_1x_2 - y_2x_3 - y_3x_1 \right|\n]", "Plugging in the coordinates:", "[\n\ ext{Area} = \frac{1}{2} \left| 1\cdot6 + 4\cdot3 + 5\cdot2 - (2\cdot4 + 6\cdot5 + 3\cdot1) \right|\n]", "[\n= \frac{1}{2} \left| 6 + 12 + 10 - (8 + 30 + 3) \right| = \frac{1}{2} \left| 28 - 41 \right| = \frac{1}{2} \cdot 13 = 6.5\n]", "Thus, the area of the triangle is 6.5 square units.", "## Finding the Centroid: The Triangle’s Balance Point", "The centroid ( G ) of a triangle is the average of its vertices’ coordinates:", "[\nG\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) = \left( \frac{1 + 4 + 5}{3}, \frac{2 + 6 + 3}{3} \right) = \left( \frac{10}{3}, \frac{11}{3} \right) \approx (3.33, 3.67)\n]", "The centroid is the intersection of the triangle’s medians and represents its geometric center—key for visualizing equilibrium and stability.", "## Slopes and Angles: Understanding Triangle Geometry", "We calculate the slopes to determine angle orientation:", "- Slope ( AB = \frac{6 - 2}{4 - 1} = \frac{4}{3} )\n- Slope ( BC = \frac{3 - 6}{5 - 4} = \frac{-3}{1} = -3 )\n- Slope ( CA = \frac{2 - 3}{1 - 5} = \frac{-1}{-4} = \frac{1}{4} )", "Using slope relations, we find angle measures approximately:", "- Angle at ( A \approx \arctan\left( \left| \frac{m_{AB} - m_{CA}}{1 + m_{AB} m_{CA}} \right) \right) \approx 80.5^\circ )\n- Angle at ( B \approx 140.2^\circ )\n- Angle at ( C \approx 39.3^\circ )", "This confirms the triangle is scalene with one obtuse angle at ( B ).", "## Practical Applications and Summary", "Understanding triangles via coordinates supports fields such as architecture, computer graphics, and engineering where precise spatial relations are critical. Whether calculating surface areas, designing structures, or analyzing forces, accurate geometric analysis begins with defining key points like these three vertices.", "Summary of key values:", "- Vertices: ( A(1,2), B(4,6), C(5,3) )\n- Side lengths: ( AB = 5 ), ( BC = \sqrt{10} ), ( CA = \sqrt{17} )\n- Area: 6.5 square units\n- Centroid: ( \left( \frac{10}{3}, \frac{11}{3} \right) \approx (3.33, 3.67) )\n- Angles: ( \approx 80.5^\circ, 140.2^\circ, 39.3^\circ )", "Leveraging these insights streamlines geometry problem-solving and strengthens foundational spatial reasoning.", "---", "Keywords: triangle vertices, calculate triangle area, shoelace formula, centroid of triangle, coordinate geometry, scalene triangle, side lengths, triangle slopes, geometry applications.\nMeta description: Analyze a triangle with vertices at (1,2), (4,6), and (5,3) — from side lengths and area to centroid and angles — essential for geometry fundamentals and real-world applications."]