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While calculating problems based on coordinate geometry, you will come across the problems such as calculating the distance between two points, to find the equation, midpoint, and slope of the line segment, or even more complex problems that require the use of the coordinate geometry formulas. But have you ever thought, how to construct a polygon on a cartesian plane?

A polygon is an area enclosed by multiple straight lines with a minimum of three straight lines known as a triangle, to an infinite number of straight lines. Calculating the area of the triangle using the coordinates of its vertices is a much-discussed topic in geometry.

Here, we will discuss how to use coordinate formulas to calculate the area of the triangle using coordinates of its vertices.

Coordinate Geometry, also called analytical geometry is a Mathematics subject in which algebraic methods and symbols are used to solve problems in Geometry. The importance of coordinate geometry is that it establishes the correlation between algebraic equations and geometric curves. This correlation helps to redevelop problems in Geometry as similar problems in Algebra, and vice versa. These methods can also be used to solve problems in other areas. For example, computers develop animations for display in games and films by using algebraic equations.

As we know, coordinate geometry is the study of geometry using the coordinate points. We can also determine the area of the triangle in coordinate geometry if the coordinates of the vertices of a triangle are given.

Example:

Consider the following points:

A = (x₁, y₁)

B = ( x₂, y₂)

C = (x₃, y₃)

If you plot these points in a plane, you will find that they are non - collinear, which means that they can be considered as the vertices of a triangle as shown below:

(Image will be uploaded soon)

In Coordinate geometry, we can determine the area of a triangle using coordinates of its vertices.

We can follow the following points if we want to find the area of the triangle using coordinates of its vertices.

Plot the given coordinates in a plane.

Locate the coordinates of the vertices of the triangle in a counter-clockwise direction, else the formula will give negative values.

Use the area of triangle formula given below.

(Image will be uploaded soon

Add the product of the diagonals (x₁, y₁), ( x₂, y₂), (x₃, y₃) as shown in the dark blue arrow.

Also, add the product of the diagonals (x₂, y₁), ( x₃, y₂), (x₁, y₃) as shown in the dark blue arrow.

Now, subtract the latter products of the diagonals with the former product to get the area of the triangle.

Hence, the area of a triangle using coordinates of its vertices can be calculated as:

½ {(x₁y₂ + x₂y₃ + x₃y₁) - (x₂y₁ + x₃y₂ + x₁y₃)}

If the coordinates of vertices of a triangle (x₁, y₁), ( x₂, y₂), (x₃, y₃) are given, then the area of triangle formula in coordinate geometry is given as:

½ {(x₁y₂ + x₂y₃ + x₃y₁) - (x₂y₁ + x₃y₂ + x₁y₃)}

Let us understand the concept with an example.

1. Find the area of the triangle whose vertices are (-2,1), (3,2), and (1,5).

Solution:

First, we will plot the given points on the graph as shown below:

(Image will be uploaded soon)

As we have to take the points in anticlockwise directions, we will take the points in the order C ( -3,5) , A (-2,1), and B (3,2).

x₁ = -3 x₂ = -2 x₃ = 3

y₁ = 5 y₂ = 1 y₃ = 2

The following pictorial representation helps us to calculate the area of the triangle easily.

(Image will be uploaded soon)

Area of △CAB = ½ {(x₁y₂ + x₂y₃ + x₃y₁) - (x₂y₁ + x₃y₂ + x₁y₃)}

Substituting the values, we get

= ½ {(-3 × 1) + (-2 × 2) + (3 × 5) - (-2 × 5 ) + (3 × 1) + (-3 × 2)}

= ½ {( -3) + (-4) + (15)} - (-10) + (3) + (-6)}

= ½ ( -3 - 4 + 15) - (-10 + 3 + -6)

= ½ (8 + 13)

= ½ (21)

= 10.5 square units

Hence, the area of △CAB = 10.5 square units.

2. Find the area of the triangle whose vertices are (3,1), (0,4), and (-3,1).

Solution:

First, we will plot the given points on a graph as shown below:

(Image will be uploaded soon)

As we have to take the points in anticlockwise directions, we will take the points in the order B (0,4), A (-3,1), and B (3,1).

x₁ = 0 x₂ = -3 x₃ = 3

y₁ = 4 y₂ = 1 y₃ = 1

The following pictorial representation helps us to calculate the area of the triangle easily.

(Image will be uploaded soon)

Area of △BCA = ½ {(x₁y₂ + x₂y₃ + x₃y₁) - (x₂y₁ + x₃y₂ + x₁y₃)}

Substituting the values, we get

= ½ {(0 × 1) + (-3 × 1) + ( 3 × 4) - ( -3 × 4) + (3 × 1) + ( 0 × 1)}

= ½ {(0) + (-3) + (12) - (-12) + (3) + (0)}

= ½ (9) - (-9)

= ½ (9 + 9)

= ½ (18)

= 18/2

= 9 square units.

Hence, the area of △ABC = 9 square units.

For finding the area of triangles using coordinates of their vertices, we need to be thorough with the coordinate geometry formulas and their related concepts. The formulas and method of finding the area of the triangle using coordinates given in this article will help you to solve the problems related to this topic easily. Hence, it is beneficial to refer to this while solving questions based on coordinate geometry.

FAQ (Frequently Asked Questions)

Q1. What are Coordinates?

Ans. Coordinates are the pair of values that help us to describe the exact position of points on a coordinate plane.

Q2. What is a Cartesian Plane?

Ans. A cartesian plane is referred to as the two-dimensional plane formed by the intersection of the vertical line known as the y-axis, and the horizontal line known as the x-axis. These are perpendicular lines that intersect each other at zero. The axes cut the coordinates into 4 equal parts, and each part is known as quadrants.

Q3. Who Introduced the Term Coordinate Geometry?

Ans. The coordinate geometry was introduced by French Mathematician Rene Descartes to show how algebra can be used to solve geometric problems.