Mathematics Foundations/13.2 Triangles and Polygons

13.2 Triangles and Polygons 13.2.1 Introduction to Triangles A triangle is a polygon with three sides and three vertices. It is one of the most fundamental shapes in Euclidean geometry. Basic Properties A triangle has 3 sides, 3 vertices, and 3 angles. The sum of the interior angles of any triangle is or radians. The area of a triangle can be calculated using the formula , where is the base and is the height. Triangle Inequality The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side: where , , and are the lengths of the sides of the triangle. 13.2.2 Classification of Triangles Triangles can be classified based on their sides and angles. By Sides Equilateral Triangle: All three sides have equal length. In an equilateral triangle, all angles are also equal (60°). Isosceles Triangle: Two sides have equal length. The angles opposite to the equal sides are also equal. Scalene Triangle: All three sides have different lengths. By Angles Acute Triangle: All three angles are less than . Right Triangle: One angle is exactly . Obtuse Triangle: One angle is greater than . 13.2.3 Important Triangle Theorems Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides: where is the length of the hypotenuse, and and are the lengths of the other two sides. Law of Sines The law of sines relates the sides of a triangle to the sines of the opposite angles: where is the radius of the circumscribed circle of the triangle. Law of Cosines The law of cosines generalizes the Pythagorean theorem: 13.2.4 Triangle Centers A triangle has several important points: Centroid: The point of intersection of the three medians. It divides each median in the ratio 2:1. Circumcenter: The center of the circumscribed circle, which is the point equidistant from the three vertices. Incenter: The center of the inscribed circle, which is the point equidistant from the three sides. Orthocenter: The point of intersection of the three altitudes. These four points lie on a single line called the Euler line, except in equilateral triangles where they coincide. 13.2.5 Polygons A polygon is a plane figure bounded by a finite number of line segments forming a closed path. Regular Polygons A regular polygon has all sides of equal length and all interior angles of equal measure. For a regular polygon with sides: Each interior angle measures Each exterior angle measures The sum of interior angles is Area of Regular Polygons The area of a regular polygon with sides, each of length , can be calculated as: Alternatively, if is the radius of the inscribed circle (apothem): Convex and Concave Polygons Convex Polygon: All interior angles are less than . Any line segment connecting two points within the polygon lies entirely within the polygon. Concave Polygon: At least one interior angle is greater than . Some line segments connecting points within the polygon may pass outside the polygon. 13.2.6 Polygon Theorems Sum of Interior Angles For any simple polygon with sides, the sum of interior angles is: or radians Pick's Theorem For a simple polygon with integer coordinates for all vertices, the area can be calculated as: where is the number of integer points in the interior of the polygon, and is the number of integer points on the boundary. 13.2.7 Triangulation Any simple polygon can be triangulated (divided into triangles) by adding diagonals. A polygon with sides can be divided into triangles. This property is useful in computational geometry and computer graphics for representing complex shapes.