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.
What is Coordinate Geometry?
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.
Area of Triangle in Coordinate Geometry
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:
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A = (x₁, y₁)
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B = ( x₂, y₂)
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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:
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In Coordinate geometry, we can determine the area of a triangle using coordinates of its vertices.
Method to Find Area of Triangle Using Coordinates
We can follow the following points if we want to find the area of the triangle using coordinates of its vertices.
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Plot the given coordinates in a plane.
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Locate the coordinates of the vertices of the triangle in a counter-clockwise direction, else the formula will give negative values.
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Use the area of triangle formula given below.
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Add the product of the diagonals (x₁, y₁), ( x₂, y₂), (x₃, y₃) as shown in the dark blue arrow.
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Also, add the product of the diagonals (x₂, y₁), ( x₃, y₂), (x₁, y₃) as shown in the dark blue arrow.
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Now, subtract the latter products of the diagonals with the former product to get the area of the triangle.
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Hence, the area of a triangle using coordinates of its vertices can be calculated as:
½ {(x₁y₂ + x₂y₃ + x₃y₁) – (x₂y₁ + x₃y₂ + x₁y₃)}
Area of Triangle Formula in Coordinate Geometry
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.
Examples
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:
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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.
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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:
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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.
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.
Conclusion
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.