Triangle

A polygon with three edges and three vertices

Definition

A triangle is a polygon with three edges and three vertices. It is one of the most basic shapes in geometry and the simplest polygon. A triangle with vertices A, B, and C is denoted as △ABC.

In Euclidean geometry, any three non-collinear points determine a unique triangle and a unique plane. This means that every triangle is contained in some plane, and the study of triangles is fundamental to both plane geometry and trigonometry.

Types of Triangles

By Side Lengths

  • Equilateral Triangle: All three sides are equal in length, and all three angles measure 60°.
  • Isosceles Triangle: Two sides are equal in length. The angles opposite the equal sides are also equal.
  • Scalene Triangle: All sides have different lengths, and all angles have different measures.

By Angles

  • Acute Triangle: All three angles are less than 90°.
  • Right Triangle: One angle is exactly 90°. The side opposite the right angle is called the hypotenuse.
  • Obtuse Triangle: One angle is greater than 90°.

Fundamental Properties

  • The sum of interior angles is always 180° (π radians).
  • The sum of any two sides is greater than the third side (Triangle Inequality).
  • The difference of any two sides is less than the third side.
  • The exterior angle equals the sum of the two opposite interior angles.
  • The area of a triangle equals half the area of any parallelogram with the same base and height.

Formulas

Area Formulas

Area = ½ × base × height
Area = ½ × a × b × sin(C)
Area = √[s(s-a)(s-b)(s-c)] (Heron's Formula)

where s = (a + b + c) / 2 is the semi-perimeter.

Perimeter

Perimeter = a + b + c

Pythagorean Theorem (Right Triangles)

a² + b² = c²

where c is the hypotenuse, and a, b are the legs.

Law of Sines

a/sin(A) = b/sin(B) = c/sin(C) = 2R

where R is the circumradius.

Law of Cosines

c² = a² + b² - 2ab × cos(C)

Special Points and Lines

  • Centroid: Intersection of medians; center of mass.
  • Circumcenter: Center of the circumscribed circle; equidistant from all vertices.
  • Incenter: Center of the inscribed circle; intersection of angle bisectors.
  • Orthocenter: Intersection of altitudes.
  • Euler Line: In any non-equilateral triangle, the centroid, orthocenter, and circumcenter are collinear.

Examples

Example 1: Find the area of a triangle with base 8 cm and height 5 cm.

Solution:
Using the formula: Area = ½ × base × height
Area = ½ × 8 × 5 = 20 cm²

Example 2: In a right triangle, the two legs measure 3 cm and 4 cm. Find the hypotenuse.

Solution:
Using the Pythagorean theorem: a² + b² = c²
3² + 4² = c²
9 + 16 = c²
25 = c²
c = 5 cm

Example 3: Find the area of a triangle with sides 5 cm, 6 cm, and 7 cm.

Solution:
Using Heron's formula:
s = (5 + 6 + 7) / 2 = 9
Area = √[9(9-5)(9-6)(9-7)]
Area = √[9 × 4 × 3 × 2]
Area = √216 = 6√6 ≈ 14.7 cm²

Applications

Triangles have numerous real-world applications:

  • Architecture and Construction: Triangular trusses provide structural stability in bridges and buildings.
  • Navigation: Triangulation is used to determine positions and distances.
  • Computer Graphics: 3D models are often composed of triangular meshes.
  • Surveying: Land measurement relies heavily on triangular calculations.
  • Physics: Vector addition uses triangular methods.
  • Engineering: Stress analysis in materials uses triangular elements.

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