Geometry: Understanding Triangles

Study Notes

A triangle is a shape with three sides and three angles.
Triangles are crucial in geometry because they are used in many geometry proofs and are the basis for trigonometry and the Pythagorean theorem.

A triangle has three parts: sides, angles, and vertices.
Sides are line segments that form the triangle and are labeled with lowercase letters (e.g., a, b, and c).
Angles are formed by two sides and are labeled with uppercase letters (e.g., A, B, and C).
Vertices are the points where two sides meet and are labeled with uppercase letters.
The base of a triangle is one side, typically the horizontal side.
The height (or altitude) is the perpendicular distance from the base to the top vertex (or apex).

Triangles can be categorized based on sides or angles.
Types of triangles based on sides:
- Scalene triangle: three sides of different lengths.
- Isosceles triangle: two sides of equal length.
- Equilateral triangle: three sides of equal length.
Types of triangles based on angles:
- Obtuse triangle: one angle greater than 90°.
- Right triangle: one angle of 90°.
- Acute triangle: three angles less than 90°.

Formulas for finding the area of a triangle:
- A = (b * h) / 2.
- A = (a * b * sin(C)) / 2.
- A = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter.
The Pythagorean theorem: a² + b² = c², where c is the hypotenuse.
The law of cosines: c² = a² + b² - 2ab * cos(C).
The law of sines: a / sin(A) = b / sin(B) = c / sin(C).

Trigonometry is the study of relationships between angles and side ratios of triangles.
Basic trigonometric ratios: SOH CAH TOA.
Trigonometry formulas apply to all triangles, including right triangles.
Angles are often measured in radians, with 2π radians in a circle.

Example 1: Finding the hypotenuse of a right triangle using the Pythagorean theorem.
Example 2: Finding an angle of a right triangle using SOH CAH TOA.
Example 3: Finding an angle of a non-right triangle using the law of cosines.