Notes On Area of a Triangle - CBSE Class 12 Maths
The area of a triangle formed by the vertices - (x1, y1), (x2, y2) and (x3, y3) is given by ½[x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)] sq.units It can be expressed in the form of a determinant. $\left|\begin{array}{ccc}{\text{x}}_{\text{1}}& {\text{y}}_{\text{1}}& \text{1}\\ {\text{x}}_{\text{2}}& {\text{y}}_{\text{2}}& \text{1}\\ {\text{x}}_{\text{3}}& {\text{y}}_{\text{3}}& \text{1}\end{array}\right|$ Note: We always take 1 in the last column of the determinant. Expanding along the first column, we get  = x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) The area of a triangle formed by the vertices - (x1, y1), (x2, y2) and (x3, y3) - is given by ½ $\left|\begin{array}{ccc}{\text{x}}_{\text{1}}& {\text{y}}_{\text{1}}& \text{1}\\ {\text{x}}_{\text{2}}& {\text{y}}_{\text{2}}& \text{1}\\ {\text{x}}_{\text{3}}& {\text{y}}_{\text{3}}& \text{1}\end{array}\right|$. Note: In calculating the area of a triangle by using this formula, we need to take the absolute value of the above determinant to avoid negative values, if any. If area is given, use both positive and negative values of the determinant for the calculation. The area of a triangle formed by three collinear points is equal to zero.

#### Summary

The area of a triangle formed by the vertices - (x1, y1), (x2, y2) and (x3, y3) is given by ½[x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)] sq.units It can be expressed in the form of a determinant. $\left|\begin{array}{ccc}{\text{x}}_{\text{1}}& {\text{y}}_{\text{1}}& \text{1}\\ {\text{x}}_{\text{2}}& {\text{y}}_{\text{2}}& \text{1}\\ {\text{x}}_{\text{3}}& {\text{y}}_{\text{3}}& \text{1}\end{array}\right|$ Note: We always take 1 in the last column of the determinant. Expanding along the first column, we get  = x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) The area of a triangle formed by the vertices - (x1, y1), (x2, y2) and (x3, y3) - is given by ½ $\left|\begin{array}{ccc}{\text{x}}_{\text{1}}& {\text{y}}_{\text{1}}& \text{1}\\ {\text{x}}_{\text{2}}& {\text{y}}_{\text{2}}& \text{1}\\ {\text{x}}_{\text{3}}& {\text{y}}_{\text{3}}& \text{1}\end{array}\right|$. Note: In calculating the area of a triangle by using this formula, we need to take the absolute value of the above determinant to avoid negative values, if any. If area is given, use both positive and negative values of the determinant for the calculation. The area of a triangle formed by three collinear points is equal to zero.

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