Area of a Triangle

Area of a Triangle in Coordinate Geometry:

Using Coordinates:

  • Given Points: Three vertices of a triangle with coordinates (x1,y1), (x2,y2), and (x3,y3).
  • Formula: The area of the triangle formed by these points is given by the formula: Area=12x1(y2y3)+x2(y3y1)+x3(y1y2)

Steps to Find Area:

  1. Identify Coordinates:

    • Points A, B, and C with coordinates A(x1,y1), B(x2,y2), and C(x3,y3).
  2. Apply Area Formula: Area=12x1(y2y3)+x2(y3y1)+x3(y1y2)

  3. Perform Calculations: Substitute the values of coordinates into the formula and solve the expression.

Example:

Given the coordinates of vertices A(1, 3), B(4, 6), and C(7, 1), find the area of the triangle ABC.

Steps to Solve:

  1. Identify Coordinates:

    • A(1,3), B(4,6), and C(7,1).
  2. Apply Area Formula: Area=121(61)+4(13)+7(36)

  3. Perform Calculations: Area=12589 Area=1212 Area=122 Area=6 square units

Interpretation:

The area of the triangle formed by the points A(1, 3), B(4, 6), and C(7, 1) in the coordinate plane is 6 square units. This area formula based on coordinates allows for the calculation of a triangle's area without using the lengths of its sides, relying solely on the coordinates of its vertices.