Area of Triangle in Determinant Form – Definition, Examples

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What Is the Area of a Triangle in Determinant Form?

In coordinate geometry, the area of a triangle in determinant form can be calculated when the coordinates of the vertices of the triangle are provided. Calculating the area of a triangle in determinant form is an important application of determinants.

If the vertices of a triangle PQR are P(x1, y1), Q(x2, y2), and R(x3, y3), the area of the triangle can be calculated as follows:

Note that the process of expanding to find the determinant of a matrix remains consistent regardless of the chosen row or column. The result remains the same, making the choice of row or column arbitrary.

Expanding the determinant along the first column, we get

Area $= \frac{1}{2} \times \left[x_{1} (y_{2} – y_{3}) + x_{2} (y_{3} – y_{1}) + x_{3} (y_{1} – y_{2})\right]$

A determinant of a square matrix A is a real value denoted by | A | or det (A). It is a scalar value held by every square matrix.

Formula of Area of Triangle in Determinant Form

3

Area of the triangle ABC is 0. It means that

A, B, C are collinear.
A, B, C are non-collinear.
A, B, C have negative coordinates.
A, B, C have positive coordinates
CorrectIncorrect
Correct answer is: A, B, C are collinear.
If the area of a triangle is 0, it means that the vertices of the triangle are collinear.

Area of a triangle $= \frac{1}{2} \times$ base $\times$ height
Heron’s formula for finding the area of a triangle $= A = \sqrt{s(s – a)(s – b)(s – c)}$, where s is the semiperimeter of the triangle; a, b, and c are the lengths of the sides.