Skip to main content

Determinant To Find Area Of A Triangle - Solved Examples-Class 12 Math Determinants Notes Study Material free pdf download

We know what a determinant is, let us know how to use Determinant to find Area of a Triangle

Suppose we are given three points in the Cartesian plane as

(x1,y1),(x2,y2) and (x3,y3)

. The area of the triangle obtained by joining these points is given by,

α=12[x1(y2–y3)+x2(y3–y1)+x3(y1–y2)]

Where  

α

denotes the area of the triangle and

(x1,y1),(x2,y2) and (x3,y3)

,  represent the vertices of the triangle.

Determinant To Find Area Of A Triangle

The formula for finding area could be represented in the form of determinants as given below.

α=12|x1y11x2y21x3y31|

As we know the value of a determinant can either be negative or a positive value but since we are talking about area and it can never be taken as a negative value, therefore we take the absolute value of the determinant so obtained.

If the area of the triangle is already given then we make use of both the positive and negative values of the determinant.

Also, if three points are collinear we would be left with a straight line instead of a triangle and as the area enclosed by a straight line is zero hence the value of the determinant will also be zero.

Keeping the above-mentioned points in mind,  let us try to expand the determinant which denoted the area by using determinant expansion techniques using minors and cofactors.

Therefore, 

α=12[x1(y2–y3)+x2(y3–y1)+x3(y1–y2)]

Hence we see that how determinants are applied to make calculations easy. Now let us try our hands at this application of determinants to find out the area of triangles.

Example To find Area of Triangle using Determinant

Example: Find out the area of the triangle whose vertices are given by A(0,0) , B (3,1) and C (2,4).

Solution: Using determinants we can easily find out the area of the triangle obtained by joining these points using the formula

α=12|x1y11x2y21x3y31|

.

Substituting the respective values in the determinant we have

α=12|001311241|

Expanding the above determinant by using expansion techniques of determinant we get,

α=12[0(1−4)–0(3−2)+1(12−2)]

⇒α=5units

Labels:

Comments

Post a Comment

Popular posts from this blog

Symmetric & Skew Symmetric Matrix-Properties, Solved Examples, Class 12 Matrices Chapter Notes Study Material Download pdf

A symmetric matrix and skew-symmetric matrix both are square matrices. But the difference between them is, the symmetric matrix is equal to its transpose whereas skew-symmetric matrix is a matrix whose transpose is equal to its negative. If A is a symmetric matrix, then A = A T   and if A is a skew-symmetric matrix then A T  = – A. Table of Contents Symmetric Matrix Properties of Symmetric Matrix Skew Symmetric Matrix Properties of Skew Symmetric Matrix Determinant of Skew Symmetric Matrix Eigenvalue of Skew Symmetric Matrix Frequently Asked Questions-FAQs What is a symmetric matrix? How do you know if a matrix is symmetric? Give an Example of a Matrix Which is Symmetric but not Invertible. Is Symmetric Matrix Diagonalizable? What is skew-symmetric matrix? What is the difference between symmetric and skew-symmetric matrix? Symmetric Matrix To understand if a matrix is a symmetric matrix, it is very important to know about transpose of a matrix and how to find it. If we in...
Post a Comment
Read more

Onto Functions(Surjective Functions)-Definition, Graph, Properties, Solved Examples, FAQs

  Onto Function is one of the many types of functions defined based on the relationship between its domain and codomain. For any function to be onto, it needs to relate two sets with a very specific mapping between elements, meaning that each element of the codomain has at least one element in the domain as its pre-image. In simple words, for any function, if all the elements of the codomain are mapped to some element of the domain, then the function is said to be an onto function.  In this article, we will discuss the concept of onto or surjective function in detail including its definition, example, and many more. We will also discuss key differences between one one, onto and into functions as well. Table of Contents What is an Onto Function? Onto Function Definition Representation for Onto Function Examples of Onto Function Properties of Onto Function Composition of Onto Function Onto Function Graph Number of Onto Functions One to One and Onto Functions Onto and Into Functi...
Post a Comment
Read more

Transpose of a Matrix-Addition & Multiplication Property of Transpose, Solved Examples, Class 12 Matrices Chapter Notes Study Material Download pdf

Transpose of a matrix is one of the most commonly used methods in matrix transformation. For a given matrix, the transpose of a matrix is obtained by interchanging rows into columns or columns to rows. In this article, we are going to learn the definition of the transpose of a matrix, steps to find the transpose of a matrix, properties and examples with a complete explanation. Before learning how to find the transpose of a matrix, first let us learn, what a matrix is? Table of Contents What is a Matrix? Transpose of a Matrix Definition How to Find the Transpose of a Matrix? Properties of Transpose of a Matrix (i) Transpose of the Transpose Matrix (ii) Addition Property of Transpose (iii) Multiplication by Constant (iv) Multiplication Property of Transpose Transpose of a Matrix Examples Practice Problems Frequently Asked Questions What is the transpose of a matrix? How to calculate the transpose of a Matrix? What is the Addition Property of Transpose? What is the Multiplication Property...
Post a Comment
Read more