Triangles: Properties, Theorems, and Formulas

Definition

• A triangle is a polygon with three sides and three vertices.
• It is a closed shape with three straight sides.

Types of Triangles

• Equilateral Triangle: A triangle with all sides of equal length.
• Isosceles Triangle: A triangle with two sides of equal length.
• Scalene Triangle: A triangle with all sides of different lengths.
• Right Triangle: A triangle with one right angle (90 degrees).
• Oblique Triangle: A triangle that is not a right triangle.
• Acute Triangle: A triangle with all angles acute (less than 90 degrees).
• Obtuse Triangle: A triangle with one obtuse angle (greater than 90 degrees).

Properties

• Perimeter: The sum of the lengths of all three sides.
• Area: The region inside the triangle, calculated using various formulas (e.g. base × height, Heron's formula).
• Angles: The sum of the interior angles is always 180 degrees.
• Altitude: The distance from a vertex to the opposite side.
• Median: The line segment from a vertex to the midpoint of the opposite side.
• Angle Bisector: The line segment that divides an angle into two equal parts.

Theorems and Formulas

• Pythagorean Theorem: a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse of a right triangle.
• Triangle Inequality Theorem: The sum of the lengths of any two sides is greater than the length of the third side.
• Heron's Formula: The area of a triangle can be calculated using the lengths of its sides.
• Law of Cosines: A formula relating the lengths of the sides and the cosine of one of the angles.

Special Triangles

• 30-60-90 Triangle: A right triangle with angles of 30, 60, and 90 degrees.
• 45-45-90 Triangle: A right triangle with angles of 45, 45, and 90 degrees.
• 3-4-5 Triangle: A right triangle with sides of length 3, 4, and 5 units.