Area of a Triangle in Coordinate Geometry

Introduction

Area of a triangle in coordinate geometry is used to find the area of a triangle using coordinates. A triangle is a three-sided polygon with three edges and three vertices in geometry. The area of a triangle is the amount of space covered by the triangle in a two-dimensional plane.

A triangle is a closed two-dimensional form with three sides, three angles, and three vertices in geometry. Triangles are regarded as the polygon with the fewest sides. Triangles are one of those shapes that have a wide range of properties and applications in our world. A triangle is a closed two-dimensional figure. They are a form of the polygon in which the total of the three angles is usually equal to 180°.

In this tutorial, we will discuss triangles in coordinate geometry.

Triangles

Triangles are polygons in geometry that have three sides and three vertices. It is two-dimensional and has three straight sides. A triangle's three angles added together equals 180°. A single plane contains a triangle. The triangle is classified into six forms based on its sides and angles.

Properties of Triangles

Triangle properties help you understand the relationship between the various sides and angles of a triangle.

• Property of Angle Sum

  According to the angle sum property, the sum of a triangle's three internal angles is always 180°.

• A triangle is defined by its three sides, three angles, and three vertices.

• The sum of the lengths of the triangle's two sides is bigger than the length of its third side.

• A triangle's area equals half the product of its base and height.

• The perimeter of a triangle is the sum of the triangle's three sides.

• Property of Congruence

The Congruence Property states that two triangles are congruent if all of their related sides and angles are equal.

Area of a Triangle using coordinates

Coordinate geometry is the study of geometry through the use of coordinate points. In coordinate geometry, the area of a triangle can be determined if the triangle's three vertices are supplied in the coordinate plane.

Consider the following three points: A(1,3),B(4,1),and C.(6,4). When these three points are plotted in the plane, they are non-collinear, which means they can be the vertices of a triangle, as seen below −

We can now calculate the area of this triangle using coordinate geometry. Let us discover more about it in the section that follows.

Method to Calculate the Area of a Triangle in Coordinate Geometry

In coordinate geometry, we utilize the coordinates of the three vertices to compute the area of a triangle. Consider the triangle ABC, which has vertices A (x₁,y₁) B (x₂,y₂), and C (x₃,y₃).

Perpendiculars AE, CF, and BD have been traced from the triangle's vertices to the horizontal axis in this illustration. There are three trapeziums formed: BAED, ACFE, and BCFD.

The area of a triangle can be expressed in terms of the areas of these three trapeziums.

Area(ΔABC) = Area(Trap.BAED) + Area(Trap.ACFE) - Area(Trap.BCFD)

Consider any single trapezium, such as BAED. It has two bases, BD and AE, and a height of DE. BD and AE are the y coordinates of B and A, respectively, whereas DE is the difference between A and B's x coordinates. The bases and heights of the other two trapeziums are also simply computed. As a result, we have

Area(ΔABC) = Area(Trap.BAED) + Area(Trap.ACFE) - Area(Trap.BCFD)

$$\mathrm{Area(\Delta ABC) =\frac{1}{2}×[(y_2+y_1)×(x_1-x_2)]+\frac{1}{2}×[(y_1+y_3)×(x_3-x_1)]+\frac{1}{2}×[(y_2+y_3)×(x_3-x_2)]}$$

$$\mathrm{Area(Area(\Delta ABC) =\frac{1}{2}×|x_1 (y_2-y_3)+x_2 (y_3-y_1)+x_3 (y_1-y_2)|}$$

This formula can also be written in the form of determinant form as

$$\mathrm{Area = \begin{bmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{bmatrix}}$$

A triangle's area cannot be negative. If we get a negative result, we should evaluate the numerical value of the area without the negative sign.

Collinearity of 3 points using Area of Triangles

• Collinear points is those that are on the same straight line or in the same plane. In Euclidean geometry, two or more points on a line that is close to or far from each other are said to be collinear.

• Three points are collinear if the area of the triangle produced by the three points is equal to zero.

• In the area of the triangle formula, substitute the coordinates of the provided three points. If the area of the triangle yields 0, the provided points are said to be collinear.

Solved Examples

1) Find the area of the triangle whose coordinates are A (0,0) B 0,1), and C (1,0).

Answer: It is given that the coordinates are A (0,0) B 0,1), and C (1,0).

We know that the formula in the form of determinant form as

$$\mathrm{Area = \begin{bmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{bmatrix}}$$

Now put the values,

$$\mathrm{Area = \begin{bmatrix}0 & 0 & 1 \\0 & 1 & 1 \\1 & 0 & 1\end{bmatrix}}$$

Now expand along the first row $\mathrm{Area(\Delta ABC) =\frac{1}{2}× |0+0+1(0-1)|}$

$$\mathrm{\Rightarrow \frac{1}{2}}$$

2)Find the area of the given triangle.

Answer: It is given that the coordinates are A (-3,-1) B (2,-1), and C (2,3).

We know that the formula in the form of determinant form as

$$\mathrm{Area = \begin{bmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{bmatrix}}$$

Now put the values,

$$\mathrm{Area = \begin{bmatrix}-3 & -1 & 1 \\2 & -1 & 1 \\2 & 3 & 1\end{bmatrix}}$$

Now expand along the first row Area(ΔABC)

$$\mathrm{\Rightarrow \frac{1}{2} |[-3(-1-3)+1(2-2)+1(6+2)]|}$$

$$\mathrm{\Rightarrow 2}$$

3) Find the area of the given triangle.

Answer: It is given that the coordinates are A (1,2) B (5,2), and C (1,5).

We know that the formula in the form of determinant form as

$$\mathrm{Area = \begin{bmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{bmatrix}}$$

Now put the values,

$$\mathrm{Area = \begin{bmatrix}1 & 2 & 1 \\5 & 2 & 1 \\1 & 5 & 1\end{bmatrix}}$$

Now expand along the first row Area(ΔABC)

$$\mathrm{\Rightarrow \frac{1}{2} |[1(2-5)-2(5-1)+1(25-2)]|}$$

$$\mathrm{\Rightarrow 6}$$

Conclusion

A triangle is a three-sided polygon with three edges and three vertices in geometry. The area of a triangle is the space it occupies in a two-dimensional plane. Three points are collinear if the area of the triangle produced by the three points is equal to zero.

FAQs

1. What do you mean by coordinate geometry?

Coordinate geometry is a branch of mathematics that helps in presenting geometric figures on a two-dimensional plane and learning the features of these figures.

2. What is a triangle?

Triangles are polygons in geometry that have three sides and three vertices. It is two-dimensional and has three straight sides.

3. What is the area of the triangle?

The area of a triangle is the space it occupies in a two-dimensional plane.

4. What is the formula for finding the area of a triangle?

The formula for finding the area of triangle

$$\mathrm{Area = \begin{bmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{bmatrix}}$$

5. How do identify the any three given points are collinear or not?

Three points are collinear if the area of the triangle produced by the three points is equal to zero.