Heron's Formula

Heron's Formula

Definition

Heron's Formula is a mathematical formula used to calculate the area of a triangle when the lengths of all three sides are known.

Formula

The formula is:

A = √(s(s-a)(s-b)(s-c))

where:

• A is the area of the triangle
• a, b, and c are the lengths of the three sides of the triangle
• s is the semi-perimeter, which is half the perimeter of the triangle: s = (a + b + c) / 2

How it Works

1. Calculate the semi-perimeter s by adding the lengths of the three sides and dividing by 2.
2. Plug the values of a, b, c, and s into the formula.
3. Calculate the area A by taking the square root of the product of the three terms.

Example

Given a triangle with sides of length 3, 4, and 5, calculate the area using Heron's Formula.

• s = (3 + 4 + 5) / 2 = 6
• A = √(6(6-3)(6-4)(6-5)) = √(6 × 3 × 2 × 1) = √36 = 6

The area of the triangle is 6 square units.

Applications

Heron's Formula is used in various fields, including:

• Geometry and trigonometry
• Physics and engineering
• Computer science and graphics
• Architecture and construction

Advantages

Heron's Formula is useful because it:

• Allows for the calculation of the area of a triangle without knowing the height or angles
• Is applicable to all types of triangles, including scalene, isosceles, and equilateral triangles
• Is a simple and efficient formula to use in calculations

Heron's Formula

Definition

• Heron's Formula is a mathematical formula that calculates the area of a triangle when the lengths of all three sides are known.

Formula

• The formula is: A = √(s(s-a)(s-b)(s-c))
• A is the area of the triangle
• a, b, and c are the lengths of the three sides of the triangle
• s is the semi-perimeter, which is half the perimeter of the triangle: s = (a + b + c) / 2

How it Works

• Calculate the semi-perimeter s by adding the lengths of the three sides and dividing by 2
• Plug the values of a, b, c, and s into the formula
• Calculate the area A by taking the square root of the product of the three terms

Example

• Given a triangle with sides of length 3, 4, and 5, calculate the area using Heron's Formula
• s = (3 + 4 + 5) / 2 = 6
• A = √(6(6-3)(6-4)(6-5)) = √(6 × 3 × 2 × 1) = √36 = 6
• The area of the triangle is 6 square units

Applications

• Geometry and trigonometry
• Physics and engineering
• Computer science and graphics
• Architecture and construction

Advantages

• Allows for the calculation of the area of a triangle without knowing the height or angles
• Is applicable to all types of triangles, including scalene, isosceles, and equilateral triangles
• Is a simple and efficient formula to use in calculations

Linear Equations in 2 Variables

Definition

• Linear equation in 2 variables takes the form ax + by = c
• a, b, and c are constants, while x and y are variables
• a and b cannot both be zero

Graphical Representation

• Graph of a linear equation in 2 variables is a straight line
• Line can slope upward or downward depending on the signs of a and b
• Line can be horizontal if b = 0 or vertical if a = 0

Solution of Linear Equations

• Solution is an ordered pair (x, y) that satisfies the equation
• Linear equation in 2 variables has infinitely many solutions
• Solutions can be found using:
  • Graphical method: finding the point of intersection of two or more lines
  • Substitution method: substituting the value of one variable into the equation
  • Elimination method: adding or subtracting the equations to eliminate one variable

Types of Linear Equations

• Consistent equations: have a unique solution
• Inconsistent equations: have no solution
• Dependent equations: have infinitely many solutions

Applications

• Linear equations in 2 variables are used to model real-world problems, such as:
  • Cost and revenue analysis
  • Supply and demand analysis
  • Motion along a straight line
  • Work and time problems