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Area of A Triangle using Determinants | Applied Mathematics for Class 12 - Commerce PDF Download

Determinant To Find Area of A Triangle

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,
Area of A Triangle using Determinants | Applied Mathematics for Class 12 - Commerce
Where α denotes the area of the triangle and (x1, y1), (x2, y2) and (x3, y3), represent the vertices of the triangle.


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

Area of A Triangle using Determinants | Applied Mathematics for Class 12 - Commerce

The determinant represents the signed area of a triangle formed by three points. To ensure a positive area, we take the absolute value of the determinant. If the area is known, both positive and negative determinants are considered. If the points are collinear, forming a straight line, the determinant is zero, indicating zero area. To calculate the area using determinant expansion techniques, minors and cofactors are employed.

Therefore,

Area of A Triangle using Determinants | Applied Mathematics for Class 12 - Commerce

Hence, we see 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.

Example1: Find out the area of the triangle whose vertices are given by A(0,0) , B (3,1) and C (2,4).
Sol: Using determinants we can easily find out the area of the triangle obtained by joining these points using the formula
Area of A Triangle using Determinants | Applied Mathematics for Class 12 - Commerce

Substituting the respective values in the determinant we have
Area of A Triangle using Determinants | Applied Mathematics for Class 12 - Commerce
Expanding the above determinant by using expansion techniques of determinant we get,
α = 1/2 [0(1−4)–0(3−2)+1(12−2)]
⇒ α = 5units

Example2: Find the area of the triangle whose vertices are A ( 1, 1 ), B ( 4, 2 ), and C ( 3, 5)
Sol: Using the formula that we have previously learnt, we can  find out the area of the triangle by joining the point given in the formula

Area of A Triangle using Determinants | Applied Mathematics for Class 12 - Commerce

Area of A Triangle using Determinants | Applied Mathematics for Class 12 - Commerce

When you substitute the given values in the above formula, we get:
k = ½ (1 ( 2 - 5 ) - 4 ( 4 - 3 ) + 3 ( 20 - 3 ))
k = ½ (1 ( -3 ) -4 ( 1 ) + 3 ( 17 ))
k = ½ (- 3 - 4 + 51)
k = ½ (44)
k = 22 units.
Since the area of the triangle cannot be negative, the value of k = 3 units.

The document Area of A Triangle using Determinants | Applied Mathematics for Class 12 - Commerce is a part of the Commerce Course Applied Mathematics for Class 12.
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FAQs on Area of A Triangle using Determinants - Applied Mathematics for Class 12 - Commerce

1. How can I use the determinant to find the area of a triangle?
Ans. To use the determinant to find the area of a triangle, you can construct a matrix with the coordinates of the triangle's vertices. Then, taking the determinant of the matrix and dividing it by 2 will give you the area of the triangle.
2. Can I find the area of any triangle using the determinant method?
Ans. Yes, the determinant method can be used to find the area of any triangle in a two-dimensional plane. However, it is important to ensure that the points of the triangle are correctly ordered when constructing the matrix to obtain the correct area value.
3. What is the significance of dividing the determinant by 2 when finding the area of a triangle?
Ans. Dividing the determinant by 2 is necessary to obtain the actual area of the triangle. The determinant gives the signed area, which can be negative depending on the orientation of the triangle. Dividing by 2 removes the negative sign and provides the absolute value of the area.
4. Are there any other methods to find the area of a triangle besides using the determinant?
Ans. Yes, there are other methods to find the area of a triangle. One common method is using the formula A = (1/2)bh, where A is the area, b is the base length, and h is the height of the triangle. Additionally, for triangles with known side lengths, you can use Heron's formula.
5. Can the determinant method be extended to find the area of triangles in three-dimensional space?
Ans. No, the determinant method described in the article only applies to triangles in a two-dimensional plane. In three-dimensional space, the concept of area is replaced by volume. Different formulas and methods, such as cross products, are used to find the volume of three-dimensional shapes.
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