how to do triangle math problems

Planar Geometry: Triangles

Fundamental Concepts

Triangles are fundamental geometric shapes defined by three sides and three angles. Their properties are governed by axioms and theorems of Euclidean geometry. Understanding these properties is crucial for solving problems involving triangles.

Classifying Triangles

By Sides

  • Equilateral: All three sides are equal in length.
  • Isosceles: Two sides are equal in length.
  • Scalene: All three sides are of different lengths.

By Angles

  • Acute: All three angles are less than 90 degrees.
  • Right: One angle is exactly 90 degrees.
  • Obtuse: One angle is greater than 90 degrees.

Key Theorems and Relationships

  • Pythagorean Theorem (for right-angled triangles): The square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (a² + b² = c²).
  • Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
  • Angle Sum Theorem: The sum of the interior angles of any triangle is always 180 degrees.
  • Similar Triangles: Triangles are similar if their corresponding angles are congruent and their corresponding sides are proportional.
  • Congruent Triangles: Triangles are congruent if their corresponding sides and angles are equal.

Trigonometric Functions

Trigonometric functions (sine, cosine, and tangent) relate the angles of a right-angled triangle to the ratios of its sides. These functions are essential for solving problems involving unknown angles or side lengths.

Solving Triangle Problems

Techniques for solving problems often involve applying the theorems and trigonometric functions mentioned above, along with algebraic manipulation to find unknown values. The specific approach depends on the information provided and the unknowns to be determined.

Area Calculations

The area of a triangle can be calculated using various formulas, depending on the available information. Common formulas include: Area = (1/2) base height; Heron's formula (using side lengths).