# Area of a Triangle

### Area of a Triangle in Coordinate Geometry:

#### Using Coordinates:

• Given Points: Three vertices of a triangle with coordinates $\left({x}_{1},{y}_{1}\right)$, $\left({x}_{2},{y}_{2}\right)$, and $\left({x}_{3},{y}_{3}\right)$.
• Formula: The area of the triangle formed by these points is given by the formula: $\text{Area}=\frac{1}{2}\mathrm{\mid }{x}_{1}\left({y}_{2}-{y}_{3}\right)+{x}_{2}\left({y}_{3}-{y}_{1}\right)+{x}_{3}\left({y}_{1}-{y}_{2}\right)\mathrm{\mid }$

#### Steps to Find Area:

1. Identify Coordinates:

• Points A, B, and C with coordinates $A\left({x}_{1},{y}_{1}\right)$, $B\left({x}_{2},{y}_{2}\right)$, and $C\left({x}_{3},{y}_{3}\right)$.
2. Apply Area Formula: $\text{Area}=\frac{1}{2}\mathrm{\mid }{x}_{1}\left({y}_{2}-{y}_{3}\right)+{x}_{2}\left({y}_{3}-{y}_{1}\right)+{x}_{3}\left({y}_{1}-{y}_{2}\right)\mathrm{\mid }$

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:

• $A\left(1,3\right)$, $B\left(4,6\right)$, and $C\left(7,1\right)$.
2. Apply Area Formula: $\text{Area}=\frac{1}{2}\mathrm{\mid }1\left(6-1\right)+4\left(1-3\right)+7\left(3-6\right)\mathrm{\mid }$

3. Perform Calculations: $\text{Area}=\frac{1}{2}\mathrm{\mid }5-8-9\mathrm{\mid }$ $\text{Area}=\frac{1}{2}\mathrm{\mid }-12\mathrm{\mid }$ $\text{Area}=\frac{12}{2}$

### 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.