Determinants

Area of a Triangle Using Determinants

Imagine a triangle with vertices at (x1,y1), (x2,y2), and (x3,y3). If the triangle was a right-angled triangle, it would be pretty easy to compute the area of a triangle by finding one-half the product of the base and the height (area of triangle formula). However, when the triangle is not a right-angled triangle there are multiple different ways to do so. It turns out that the area of triangle formula can also be found using determinants. Let us see in detail how do we go about it.

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Area of Triangle Formula Using Determinants

In earlier classes, we have studied that the area of a triangle whose vertices are  (x1, y1), (x2, y2) and (x3, y3), is given by the expression $$\frac{1}{2} [x1(y2–y3) + x2 (y3–y1) + x3 (y1–y2)]$$. Now this expression can be written in the form of a determinant as

Area of Triangle Formula

Points to be Noted

  • Since the area is a positive quantity, we always take the absolute value of the determinant in (1).
  • If the area is given, uses both positive and negative values of the determinant for calculation.
  • The area of the triangle formed by three collinear points is zero.

Learn more about Determinant of a Matrix here in detail.

Example: Find the area of the triangle whose vertices are (3, 8), (– 4, 2) and (5, 1).
Solution: The area of the triangle is given by
Area of Triangle Formula
= $$\frac{1}{2} [ 3(2 –1) – 8(– 4 – 5) + 1(– 4 –10)] $$
= $$\frac{1}{2} ( 3 + 72 – 14 ) = 61/2 $$

Derivation of the Area of Triangle Formula

Area of Triangle Formula

  • We know that the Area of Rectangle can be written as follows

Area of Triangle Formula

  • Area of Triangle A: Technically, each of those distances should be the absolute value of the difference. But the problem is much easier to work without the absolute values.

Area of Triangle Formula

  • Area of Triangle B: Realize, however, as if the points don’t lie in the same positions (point 2 is both the rightmost and the uppermost), that the area found using these formulas will be negative.

Area of Triangle Formula

  • Area of Triangle C: For that reason, caution should be exerted to always make the final answer non-negative. The area of a triangle, after all, can’t be negative.

Area of Triangle Formula

Learn more about Properties of Determinants here.

Let’s add the areas of the three outside triangles together.

Area of Triangle Formula

Simplifying further,

Area of Triangle Formula

Now, to subtract the areas of the three triangles from the area of the rectangle.

Area of Triangle Formula

Simplifying further,

Area of Triangle Formula

Let’s regroup those terms

Area of Triangle Formula ………….. (1)

Now, consider the determinant formed by placing the x-coordinates in the first column, the y-coordinates in the second column, and the constant 1 in the last column.

Area of Triangle Formula

Let’s evaluate the determinant by expanding along the 3rd column.

Area of Triangle Formula…………… (2)

On comparing both (1) and (2) we notice that the area of the triangle differs only in sign. The reason for this is because of the order the points were chosen in. If the points were chosen to be points 1, 2, and 3 in a different order, then the determinant would change only in sign.

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Solved Examples on Area of Triangle Formula

Question: If the lines p1x + q1y = 1, p2x + q2y = 1 and p3x + q3y = 1 be concurrent, then the points (p1,q1), (p2,q2) and (p3,q3) ,

  1. form scalene triangle
  2. form equilateral triangle
  3. are collinear
  4. form a right-angled triangle

Solution: p1q11, p2q21 p3q31. Given lines are concurrent $$\begin{vmatrix} p_{1} & q_{1} & 1\\ p_{2} & q_{2} & 1  \\ p_{3} & q_{3} & 1\end{vmatrix} = 0$$

The left-hand side of the above equation is also equal to twice the area of a triangle with coordinates (p1,q1)(p2,q2)(p3,q3). From the area of triangle formula, and since the area is equal to zero, (p1,q1)(p2,q2)(p3,q3are collinear.

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Is determinant available just for square matrix?

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