Area of Triangle using Determinant

• In coordinate geometry, the area of a triangle can be calculated if the coordinates of its vertices are known.

• Determinants provide a compact and systematic method to calculate this area using a 3 × 3 matrix formed from the coordinates of the vertices.

The same area can be expressed in determinant form using a 3 × 3 determinant:

Area of △ABC = \[ \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\]

• Area is always non-negative:

  The value of the determinant may be positive or negative, but the area is taken as its absolute value.

• Sign and $\pm$ in exam problems:

  When the area is given (for example, Area = 12 sq. units), the determinant value can be +24 or -24 because the formula uses \[\frac{1}{2}|\det|\].

• Condition for collinearity:

  If three points are collinear, the area of the triangle formed by them is zero, so

This is frequently used to check whether three given points lie on the same straight line.

• Area of triangle using determinant:

• Expanded coordinate form:

  \[\text{Area} = \frac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|\].

• Area is always taken as positive; use absolute value.

• For collinear points, determinant = 0, so area = 0.