Area of a Triangle using Coordinates

The area of a triangle can be calculated using the coordinates of its vertices in a coordinate plane. This method is particularly useful when the triangle’s base and height aren’t easily determined, or when the triangle is defined by its vertices rather than side lengths and angles. This formula directly utilizes the x and y coordinates of the vertices.

Formulae

The formula for calculating the area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is:

Area = $ \frac{1}{2} |x_1(y_2 – y_3) + x_2(y_3 – y_1) + x_3(y_1 – y_2)| $

The absolute value is used because area is always a positive quantity.

Examples

Example 1: Find the area of the triangle with vertices A(1, 2), B(4, 6), and C(7, 1).

Using the formula:

Area = $\frac{1}{2} |1(6 – 1) + 4(1 – 2) + 7(2 – 6)|$

Area = $\frac{1}{2} |1(5) + 4(-1) + 7(-4)|$

Area = $\frac{1}{2} |5 – 4 – 28|$

Area = $\frac{1}{2} |-27|$

Area = $\frac{1}{2} * 27 = 13.5$ square units.

Example 2: Determine the area of the triangle with vertices P(-2, 3), Q(1, -1), and R(4, 2).

Using the formula:

Area = $\frac{1}{2} |-2(-1 – 2) + 1(2 – 3) + 4(3 – (-1))|$

Area = $\frac{1}{2} |-2(-3) + 1(-1) + 4(4)|$

Area = $\frac{1}{2} |6 – 1 + 16|$

Area = $\frac{1}{2} |21|$

Area = $\frac{1}{2} * 21 = 10.5$ square units.

Theorem with Proof

Theorem: The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by $ \frac{1}{2} |x_1(y_2 – y_3) + x_2(y_3 – y_1) + x_3(y_1 – y_2)| $.

Proof: We can derive this formula using the concept of signed areas. Imagine extending lines from each vertex to the x-axis. These lines, together with the x-axis and segments of the triangle’s sides, form trapezoids. The area of the triangle can be found by subtracting the areas of the trapezoids formed outside the triangle from the area of the larger trapezoid which encompasses the triangle.

Consider the trapezoid formed by the points $(x_1, y_1)$, $(x_2, y_2)$, $(x_2, 0)$, and $(x_1, 0)$. The area of this trapezoid is $\frac{1}{2}(y_1 + y_2)(x_2 – x_1)$.

Similarly, we can form two other trapezoids with areas $\frac{1}{2}(y_2 + y_3)(x_3 – x_2)$ and $\frac{1}{2}(y_1 + y_3)(x_1 – x_3)$. The sum of the areas of these trapezoids can be rewritten as the algebraic sum of the areas to find the area of the triangle.

Simplifying and rearranging the expressions, and applying the absolute value to ensure a positive area, we obtain the given formula: $ \frac{1}{2} |x_1(y_2 – y_3) + x_2(y_3 – y_1) + x_3(y_1 – y_2)| $.

Common mistakes by students

* Incorrect order of coordinates: Students often mix up the order of the x and y coordinates or the order of the vertices, leading to an incorrect area.

* Forgetting the absolute value: Failing to take the absolute value of the result can lead to a negative area, which is impossible.

* Arithmetic errors: Careless mistakes in arithmetic operations, especially with negative numbers, are common.

Real Life Application

This formula is widely used in:

Computer Graphics: Calculating the area of triangles is fundamental in rendering 3D models and other graphical representations.
Surveying and Mapping: Determining the area of land parcels or irregularly shaped regions, using the coordinates of key points.
Engineering: Calculating cross-sectional areas in structural analysis or other engineering applications.

Fun Fact

The formula for the area of a triangle using coordinates can be easily extended to calculate the area of any polygon by dividing it into triangles and summing their areas.